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 Volterra Integro-Differential EquationsLeigh C. BeckerDepartment of Mathematics
Christian Brothers University
650 E. Parkway South
Memphis, TN 38104
lbecker@cbu.edu
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">0. Initializations</Font></Text-field>Before proceeding to the Introduction in Section 1, execute the following commands to invoke Maple packages that are needed for this Maple worksheet to function properly in its entirety. restart:
with(LinearAlgebra, LinearSolve, VectorNorm);
with(plots, pointplot, textplot, display);
with(VectorCalculus, Jacobian);
Digits := 10;
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">1. Introduction</Font></Text-field>This worksheet can be used as an experimental tool to investigate the general behavior of solutions of scalar Volterra integro-differential equations of the form(1.1) 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by numerically solving them---with the implicit trapezoid rule in conjunction with Newton's method for nonlinear systems---and then graphing their solutions. The author has used a similar worksheet and one by Becker and Wheeler [1] to do just this and to also illustrate theorems and examples in recent papers (cf. [2] and [3]). We assume 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 \342\206\222 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= are continuous functions and that the kernel function 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 is continuous for 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 and 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 To launch a solution of (1.1), so to speak, requires either an initial value 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, where 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, or an initial function 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRJSZsZTtGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJWZvcm1HUSZpbmZpeEYnLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSi8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1JI21pR0YkNiVRJyYjOTY0O0YnLyUnaXRhbGljR0Y4Ri8tRjM2MFEiPEYnRi9GNkY5RjtGPUY/RkFGQ0ZFRkhGS0ZNRlAtRlQ2JVEiVEYnL0ZYUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYzNjBRIi5GJ0YvRjZGOUY7Rj1GP0ZBRkNGRS9GSVEmMC4wZW1GJy9GTEZhb0ZNRlAtRjM2MFEifkYnRi9GNkY5RjtGPUY/RkFGQy9GRlEhRidGYG9GYm9GTUZQRmNvThroughout this worksheet, any function with the name LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== (psi) will denote a continuous initial function. A solution of equation (1.1) is a continuous function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn that satisfies it on some interval to the right of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== such that(1.2) 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 for 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For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjI=, we interpret (1.2) to be the initial value 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 where 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. As regards the existence of solutions of Volterra equations, one classic result states that if the kernel 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 satisfies the Lipschitz condition(1.3) 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== is independent of 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, then equation (1.1) has a unique solution for all finite LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==. Further details can be found in Linz [5, p. 52] and Burton [4, sec. 3.2].
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">2. Incorporating the initial function into the forcing function</Font></Text-field>In the Maple worksheet [1], a procedure called Volterra is presented. In this worksheet, it is found in Section 4. This procedure employs the implicit trapezoidal rule in conjunction with Newton's method for nonlinear systems to numerically solve Volterra integral equations of the second kind, namely, equations of the form(2.1) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ0RidGMi9GNVEnaXRhbGljRicvRjVRJ25vcm1hbEYnLUkjbW9HRiQ2MFEiPUYnRkQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkwvJSlzdHJldGNoeUdGTC8lKnN5bW1ldHJpY0dGTC8lKGxhcmdlb3BHRkwvJS5tb3ZhYmxlbGltaXRzR0ZMLyUnYWNjZW50R0ZMLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zobi8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1GLDYnUSJmRidGL0YyRjRGN0Y6LUZHNjBRIitGJ0ZERkpGTUZPRlFGU0ZVRldGWS9GZ25RLDAuMjIyMjIyMmVtRicvRmpuRmhvRltvRl5vLUkobXN1YnN1cEdGJDYnLUZHNjFRKyZJbnRlZ3JhbDtGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGRC9GS1EmdW5zZXRGJy9GTkZkcC9GUEYxL0ZSRmRwL0ZURjEvRlZGZHAvRlhGZHAvRlpRJ3ByZWZpeEYnL0ZnblEmMC4wZW1GJy9Gam5GXnFGW29GXm8tRiM2Iy1JI21uR0YkNiRRIjBGJ0ZERj0vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRmhxLUYsNidRImtGJ0YvRjJGNEY3LUY7NiQtRiM2KEY/LUZHNjBRIixGJ0ZERkovRk5GMUZPRlFGU0ZVRldGWUZdcS9Gam5RLDAuMzMzMzMzM2VtRidGW29GXm8tRiw2JVEic0YnRjJGQkZickYrLUY7NiQtRiM2I0ZockZERkQtRkc2MFEifkYnRkRGSkZNRk9GUUZTRlVGVy9GWlEhRidGXXFGX3FGW29GXm8tRkc2MVEwJkRpZmZlcmVudGlhbEQ7RidGYHBGREZjcEZlcC9GUEZkcEZncC9GVEZkcEZpcEZqcEZbcUZdcUZfcUZbb0Zeb0ZockZicg==where 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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2J1Eia0YnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRic= denote vector-valued functions with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== components. In this section, we will show how to convert a scalar integro-differential equation (1.1) to a vector integral equation (2.1) with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMkYnRjk=. In other words, we will reduce (1.1) of order one to an equivalent system of two integral equations of the form(2.2) 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 (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 that we can use the Volterra procedure to numerically solve (2.2)---and thereby also (1.1). To effect this reduction, we start with the translation (2.3) 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This will incorporate the initial function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==into the forcing function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== of (1.1) (cf. Burton [4, sec. 2.1]), which has the intended consequence of changing the "initial function problem" to an "initial value problem." To show this, let us differentiate (2.3) with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==. By the chain rule and (1.1), we 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 (2.3), this 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or (2.4) 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By defining the 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(2.5) 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 can rewrite (2.4) as (2.6) 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 to the initial condition(2.7) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+LUkjbW9HRiQ2MFEiPUYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkYvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZYLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUYsNiVRInpGJ0YvRjItRjY2JC1GIzYjLUYsNiVRJyYjOTY0O0YnL0YwRkZGPkY+RkAtRiw2JVEnJiM5Njg7RidGZW9GPkZeby1GQTYwUSIuRidGPkZERkdGSUZLRk1GT0ZRRlMvRldRJjAuMGVtRicvRlpGXXBGZW5GaG4= To sum up, we have changed the initial function problem consisting of equation (1.1) and condition (1.2) to the equivalent initial value problem consisting of equation (2.6) and the initial condition (2.7). The respective solutions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnLUYsNiVRIUYnRi9GMg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn are related as follows:(2.8) 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Example 1. Change the initial function problem
(2.9) 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
such that
(2.10) 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRJSZsZTtGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJWZvcm1HUSZpbmZpeEYnLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSi8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1JI21pR0YkNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGMi1GLDYkRk9GLw==to an equivalent initial value problem. Solution. In the notation of (1.1), 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 and 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 Execute the following commands to input these functions: a := t -> -3;
b := (t,s,u) -> -exp(-(t-s))*u;
g := t -> 4*sin(t-1);
Now input equation (2.9):zEqn := diff(z(t),t) = a(t)*z(t) + int(b(t,s,z(s)), s=0..t) + g(t);
By (2.10), 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 for 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tau := 1;
psi := t -> 1;
To incorporate the initial function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== into the forcing function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, define the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== by
(2.11) 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 ( cf. (2.3) ).
Then by (2.5),
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(2.12) 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Execute the following commands to input (2.12): A := t -> a(t+tau);
B := (t,s,u) -> b(t+tau,s+tau,u);
G := t -> g(t+tau) + int(b(t+tau,s,psi(s)), s=0..tau);
'A(t)' = A(t); 'B(t,s,u)' = B(t,s,u); 'G(t)'= G(t);
By (2.6), the translated equation is
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.
Execute the following command to input this equation:wEqn := diff(w(t),t) = A(t)*w(t) + int(B(t,s,w(s)), s=0..t) + G(t);
Since the initial function is 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRJSZsZTtGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJWZvcm1HUSZpbmZpeEYnLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSi8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1JI21pR0YkNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGMi1GLDYkRk9GLw==, it follows from (2.7) thatJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRis=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.Therefore, the initial function problem (2.9) and (2.10) is equivalent to the initial value problem(2.13) 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
such that(2.14) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+LUkjbW9HRiQ2MFEiPUYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkYvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZYLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUY7NiRRIzEuRidGPg==Generally speaking, analytical methods for solving integro-differential equations fail; consequently, in those situations one must resort to numerical methods. However, Example 1 was constructed so that it does have an analytical, closed form solution---which we will obtain next in Example 2. This was done for the purpose of comparing it to the numerical solution that we will compute later in Section 5 with the aid of the Volterra procedure.
Example 2. Find an analytical solution of Example 1.Solution. Analytical solutions of equation (2.9) will be found by changing its counterpart, the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-equation (2.13), to a nonhomogeneous second-order ordinary differential equation with constant coefficients and then solving it. This can easily be done with paper and pencil by finding the characteristic roots of the associated homogeneous differential equation and using the method of undetermined coefficients. However, we can readily accomplish the same thing with a few Maple commands. First multiply (2.13) by 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(t)*wEqn;
Differentiate with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== and simplify to obtain the following second-order ordinary differential equation:diff(%,t):
ODE := simplify(%);
This has the general solutiondsolve(ODE,w(t));
By (2.13) and (2.14), the initial conditions are 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 and 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 := w(0)=1, D(w)(0) = exp(-1) - 4;
Thus, the initial value problem isIVP := {ODE, IC};
Its solution isSOLN := dsolve(IVP, w(t)):
Soln := combine(SOLN);
Use the unapply command to change the above expression for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnLUYsNiVRIUYnRi9GMg== to a Maple function. Name it LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnLUYsNiVRIUYnRi9GMg== to make it easy to remember that it is really the analytical solution 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 := rhs(Soln):
W := unapply(RS,t);
Now that we have determined the analytical solution 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we can obtain the analytical solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2MFEifkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==of (2.9) satisfying condition (2.10) from (2.8):z := t -> piecewise(t >= 0 and t <= tau, psi(t), t > tau, W(t-tau)):
'z(t)' = z(t);
A graph of the analytical solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2MFEifkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==is
P[1] := plot(z(t), t=0..8.6, z=-1.5..1.5, labels=[t,z], labelfont=[times, bold, 12], color=blue, title="z(t) (analytic)", titlefont=[times, bold, 12], legend = "z(t) (analytic)", thickness=1):
display(P[1]);
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">3. Reduction to a system of integral equations</Font></Text-field>In this section, we show how to reduce a scalar integro-differential equation of the form (2.6) to an equivalent system of two integral equations of the form (2.2) so that we can compute numerical solutions with the Volterra procedure in Section 4. Begin by defining the functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYkLUYjNiMtRi82JVEidEYnRjJGNUY+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYkLUYjNiMtRi82JVEidEYnRjJGNUY+LUYvNiVRIUYnRjJGNQ== as follows:
(3.1) 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 and 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
Integrating the latter, we 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 \342\207\222 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or (3.2) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYkLUYjNiMtRi82JVEidEYnRjJGNUY+LUkjbW9HRiQ2MFEiPUYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlEvJSlzdHJldGNoeUdGUS8lKnN5bW1ldHJpY0dGUS8lKGxhcmdlb3BHRlEvJS5tb3ZhYmxlbGltaXRzR0ZRLyUnYWNjZW50R0ZRLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zdby8lKG1pbnNpemVHRj0vJShtYXhzaXplR1EpaW5maW5pdHlGJy1GLzYlUScmIzk2ODtGJy9GM0ZRRj4tRkQ2JC1GIzYjLUYvNiVRJyYjOTY0O0YnRmhvRj5GPi1GTDYwUSIrRidGPkZPRlJGVEZWRlhGWkZmbkZobi9GXG9RLDAuMjIyMjIyMmVtRicvRl9vRmRwRmBvRmJvLUkobXN1YnN1cEdGJDYnLUZMNjBRKyZJbnRlZ3JhbDtGJ0Y+Rk9GUi9GVUY0RlYvRllGNEZaRmZuL0ZpblEncHJlZml4RicvRlxvUSYwLjBlbUYnL0Zfb0ZhcUZgb0Ziby1GIzYjLUY7NiRRIjBGJ0Y+RkYvJTFzdXBlcnNjcmlwdHNoaWZ0R0ZCRkAtRiw2JUYuLUYjNiMtRjs2JFEiMkYnRj5GQC1GRDYkLUYjNiMtRi82JVEic0YnRjJGNUY+LUZMNjBRIn5GJ0Y+Rk9GUkZURlZGWEZaRmZuL0ZpblEhRidGYHFGYnFGYG9GYm8tRkw2MFEwJkRpZmZlcmVudGlhbEQ7RidGPi9GUFEmdW5zZXRGJy9GU0Zhcy9GVUZhcy9GV0Zhcy9GWUZhcy9GZW5GYXMvRmduRmFzRl5xRmBxRmJxRmBvRmJvRmVyas 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From (2.6), we 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 short, the scalar integro-differential equation (2.6) with the initial 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 is equivalent to the system of integral equations
(3.3) 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
with the initial conditions(3.4) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYkLUYjNiMtRjs2JFEiMEYnRj5GPi1JI21vR0YkNjBRIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZRLyUpc3RyZXRjaHlHRlEvJSpzeW1tZXRyaWNHRlEvJShsYXJnZW9wR0ZRLyUubW92YWJsZWxpbWl0c0dGUS8lJ2FjY2VudEdGUS8lJWZvcm1HUSZpbmZpeEYnLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGXW8vJShtaW5zaXplR0Y9LyUobWF4c2l6ZUdRKWluZmluaXR5RictRi82JVEnJiM5Njg7RicvRjNGUUY+LUZENiQtRiM2Iy1GLzYlUScmIzk2NDtGJ0Zob0Y+Rj4tRi82JVEhRidGMkY1 and 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Observe that (3.3) has the form of (2.1) where(3.5) 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and (3.7) 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.Example 3. Change the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-integro-differential equation, namely (2.13), to a system of integral equations.Solution. The functions 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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2MFEifkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==for (2.13) have already been entered with the Maple commands immediately following (2.12). By (2.12) and (3.5), the vector function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2J1EiZkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYkLUYjNiMtSSNtaUdGJDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJw== is
(3.8) 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Execute the following commands to input LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEiZkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSxtYXRodmFyaWFudEdRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUkobWZlbmNlZEdGJDYlLUYjNiMtRiw2JlEidEYnRjJGNC9GOFEnaXRhbGljRidGNC9GOFEnbm9ybWFsRic=:f := proc(t)
Vector([psi(tau), A(t)*psi(tau) + G(t)])
end proc:
f(t);
By (2.12) and (3.7), the kernel 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 is
(3.9) 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2MFEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkYvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5Ric=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.
Execute the following commands to input 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:k := proc(t, s, x::Vector)
Vector([x[2], B(t,s,x[1]) + A(t)*x[2]]);
end proc:
v := Vector(2, symbol=x): vlist := convert(v, 'list'):
k(t,s,v);
It follows from (2.12) and (3.3) that an equivalent system of integral equations is
(3.10) 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.
Execute the following commands to check that the system (3.10) has been correctly entered:for i from 1 to 2 do
x[i](t) = f(t)[i] + Int(k(t, s, v)[i], s = 0..t)
end do;
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">4. Volterra procedure</Font></Text-field>Next we will numerically solve the system of integral equations (3.10) by implementing the Volterra procedure that is given in the next paragraph. (Detailed information about the procedure may be found in Becker and Wheeler [1].) The arguments of the procedure are 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, Iter_Max, wherem := number of equations; also, the number of unknown scalar functions 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 := step size;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== := total number of steps of size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEkVG9sRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn := tolerance;Iter_Max := maximum number of Newton iterations.Note that the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== in this worksheet is always 2 since we only consider scalar Volterra integro-differential equations of the form (1.1); consequently, there are 2 integral equations in the associated system (cf. (3.3)).m := 2;
The Volterra procedure (immediately below) must be executed now before numerically solving (3.10) in Section 5 and the other integro-differential equations in subsequent sections.
The Volterra procedureVolterra := proc(m::posint, h::positive, N::posint, Tol::positive, Iter_Max::posint)
local i, j, p, Matrix_Partials;
global n, t, k, kpartial, X, sigma, J, phi, y, epsilon;
Let 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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiWEYnLyUlc2l6ZUdRIzExRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRi82JlEibkYnRjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= be vectors with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2J1EibUYnLyUlc2l6ZUdRIzExRicvJSdpdGFsaWNHUSV0cnVlRicvJStiYWNrZ3JvdW5kR1EoWzAsMCwwXUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== components:
y := Vector(m); sigma := Vector(m); phi := Vector(m);
for n from 0 to N do
X[n] := Vector(m) JSFH
end do;
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== is the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlEibUYnLyUlc2l6ZUdRIzExRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlEibUYnLyUlc2l6ZUdRIzExRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Jacobian matrix defined by equation (2.9) in reference [1]:
J := Matrix(m,m);
The matrix kpartial consists of the first-order partial derivatives of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRInRGJ0YvRjItSSNtb0dGJDYwUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJWZvcm1HUSZpbmZpeEYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRicvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictRiw2JVEic0YnRi9GMkY9LUYsNiVRInhGJ0YvRjJGQS1GLDYlUSFGJ0YvRjI=:
Matrix_Partials := Jacobian(k(t, s, v), vlist);
kpartial := unapply(Matrix_Partials, t, s, x);
t[0] := 0;
X[0] := f(t[0]); (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 and 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)
for n from 1 to N do
t[n] := t[n-1] + h;
sigma := f(t[n]) + 0.5*h*k(t[n], t[0], X[0]); (cf. (2.4) in [1])
for j from 1 to n-1 do
sigma := sigma + h*k(t[n], t[j], X[j])
end do;
X[n] := X[n-1];
p := 0; epsilon := 1; (To begin, choose a value for epsilon greater than Tol.)
while (epsilon >= Tol) and (p <= Iter_Max) do
p := p + 1;
for i from 1 to m do ( cf. (2.9) in [1] )
for j from 1 to m do
if i = j then
J[i,j] := 1 - 0.5*h*kpartial(t[n], t[n], X[n])[i,j]
else
J[i,j] := - 0.5*h*kpartial(t[n], t[n], X[n])[i,j]
end if;
end do;
end do;
phi := X[n] - 0.5*h*k(t[n], t[n], X[n]) - sigma; (cf. (2.7) in [1])
LinearSolve (J,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlEnJiM5ODE7RicvJSVzaXplR1EjMTFGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn) computes the solution y of the linear 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:
y := LinearSolve(J,phi);
VectorNormLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYkLUYjNiMtSSNtaUdGJDYmUSJ5RicvJSVzaXplR1EjMTFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjtRJ25vcm1hbEYn computes the infinity norm of y (cf. explanations in [1]):
epsilon := VectorNorm(y);
X[n] := X[n] - y; (cf. (2.12) in [1])
end do; (end of the "while" loop)
Test for convergence:
if epsilon >= Tol then
"The maximum number of Newton iterations has been exceeded.";
break
end if;
end do; (end of the "for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== from 1 to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== do" loop)
end proc:
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">5. Integro-differential equation 1</Font></Text-field>Execute the following commands to numerically solve and graph the solution of the initial function problem of Example 1 in Section 2, namely:(2.9) 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
such that
(2.10) 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRJSZsZTtGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJWZvcm1HUSZpbmZpeEYnLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSi8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1JI21pR0YkNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGMi1GLDYkUSMxLkYnRi8=Enter a step size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnh := 0.2;Enter the desired number of steps of size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnN := 38;Execution of the next command invoking Volterra with 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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEjMzhGJ0Y5LUY2NjBRIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRIUYnL0ZOUSYwLjBlbUYnL0ZRRlxvRlJGVQ==takes the author's computer (1400 MHz processor) slightly more than 4 seconds. Accuracy can be increased by decreasing the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== but that increases the time to complete the execution of the command. For example, changing to 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and 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roughly triples the time to 13 seconds).Volterra(m, h, N, 10^(-8), 5);Table of values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn:
d := 2; (Every value is displayed if d = 1; every other value if d = 2, etc.) M := floor(N/d); (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJ0Y1is the number of values greater than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==that will be displayed in each column.) Tabular comparison of the analytical values to the numerical approximations of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn:
printf(" analytic approximations \134n");
printf(" t z(t) z(t) \134n");
for j from 0 to ceil(tau/(d*h)) - 1 do
printf("%10.2f %14.4f %14.4f\134n", t[d*j], z(t[d*j]), psi(t[d*j]))
end do;
for j from 0 to M do
printf("%10.2f %14.4f %14.4f\134n", t[d*j]+tau, z(t[d*j]+tau), X[d*j][1])
end do;
Graphs:
Minimum and maximum values of the initial function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0Yy for 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psiMin := minimize(psi(t), [t=0..tau]);
psiMax := maximize(psi(t), [t=0..tau]);
Minimum and maximum values of the numerical approximations of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn:JSFHXMin := min(seq(X[n][1], n=0..N));
XMax := max(seq(X[n][1], n=0..N));
Minimum and maximum values that will be displayed on the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-axis: zMin := min(psiMin,XMin) - 0.5;
zMax := max(psiMax,XMax) + 0.5;
Graph of the initial function and the numerical approximation ofJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EiekYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYnUSJ0RidGL0YyRjRGN0YvL0Y1RjlGNw==:
P[2] := plot(psi(t), t=0..tau, view=[0..tau, zMin..zMax], labels=[t,z], labelfont=[times,bold,12], legend="psi(t) = 1", title="Numerical approximations of z(t) using the Volterra procedure", titlefont=[times,bold,12], color=black, thickness=1):
P[3] := pointplot([seq([tau + t[n], X[n][1]], n = 0..N)], color=black, labels=[t,z], labelfont=[times,bold,12], view=[0..tau + t[N], zMin..zMax], symbol=circle, symbolsize=6, legend = "z(t) (numeric)"):
display([P[2], P[3]]);
Comparison of the graphs of the analytical and numerical solutions:
display([P[1], P[3]], title = "z(t) (analytic vs. numeric)");
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">6. Integro-differential equation 2</Font></Text-field>The next integro-differential equation for which we will compute numerical solutions is the homogeneous equation
(6.1) 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.It was chosen for this worksheet because it can be proven, with the aid of a Liapunov functional (cf. Becker [3]), that all solutions of (6.1) tend to 0 irrespective of the initial values or functions. We illustrate this next by computing numerical solutions for the following two initial functions: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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkovJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictSSNtaUdGJDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRjItRiw2JFEiMkYnRi8=and 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for 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 with the formerLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2MVEiLEYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==let us compute the numerical solution of (6.1) such that(6.2) 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkovJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictSSNtaUdGJDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRjItRiw2JFEiMkYnRi8=.JSFHBefore we begin, we must reset the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==since its value from previous calculations isz(t);
To reset LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, we use unevaluation quotes (cf. Section 7.3 in the User Manual):z := 'z';
Let us verify that it has been reset:z(t);
Now reset LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnt := 't';
In the notation of (1.1), the functions 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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn are:a := t -> -1/(1+t);
b := (t,s,u) -> cos(s)/(1+t)^3*u;
g := t -> 0;
Let's verify that (6.1) has been correctly entered:zEqn := diff(z(t),t) = a(t)*z(t) + int(b(t,s,z(s)), s=0..t) + g(t);Now enter the initial function (6.2):tau := 2;
psi := t -> 3*sin(2*t)-2*t;
Execute the following commands to obtain the functions involved with the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-equation (cf.(2.6)):A := t -> a(t+tau);
B := (t,s,u) -> b(t+tau,s+tau,u);
'A(t)' = A(t); 'B(t,s,u)' = B(t,s,u); G := t -> g(t+tau) + int(b(t+tau,s,psi(s)), s=0..tau):
'G(t)' = G(t);The LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-equation iswEqn := diff(w(t),t) = A(t)*w(t) + int(B(t,s,w(s)), s=0..t) + G(t);Execute the following commands to input LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2KFEiZkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSxtYXRodmFyaWFudEdRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUkobWZlbmNlZEdGJDYlLUYjNiMtRiw2JlEidEYnRjJGNC9GOFEnaXRhbGljRidGNC9GOFEnbm9ybWFsRic=:f := proc(t)
Vector([psi(tau), A(t)*psi(tau) + G(t)])
end proc:
f(t);Execute the following commands to input 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:k := proc(t, s, x::Vector)
Vector([x[2], B(t,s,x[1]) + A(t)*x[2]]);
end proc:
v := Vector(2, symbol=x): vlist := convert(v, 'list'):
k(t,s,v);The corresponding system of integral equations (cf. (3.3)) isfor i from 1 to 2 do
x[i](t) = f(t)[i] + Int(k(t, s, v)[i], s = 0..t)
end do;Enter a step size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnh := 0.25;Enter the desired number of steps of size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnN := 32;Invoke the Volterra procedure:Volterra(m, h, N, 10^(-8), 5);Table of values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn:
d := 2; (Every value is displayed if d = 1; every other value if d = 2, etc.) M := floor(N/d); (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJ0Y1is the number of values greater than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==that will be displayed in each column.) Numerical approximations of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn:
printf(" t z(t) \134n");
for j from 0 to ceil(tau/(d*h)) - 1 do
printf("%10.2f %14.4f\134n", t[d*j], psi(t[d*j]))
end do;
for j from 0 to M do
printf("%10.2f %14.4f\134n", t[d*j]+tau, X[d*j][1])
end do;
Graphs:
Minimum and maximum values of the initial function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0Yy for 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psiMin := minimize(psi(t), [t=0..tau]): evalf(%);
psiMax := maximize(psi(t), [t=0..tau]): evalf(%);
Minimum and maximum values of the numerical approximations of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn:JSFHXMin := min(seq(X[n][1], n=0..N)): evalf(%);
XMax := max(seq(X[n][1], n=0..N)): evalf(%);
Minimum and maximum values that will be displayed on the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-axis: zMin := min(psiMin,XMin) - 0.5: evalf(%);
zMax := max(psiMax,XMax) + 0.5: evalf(%);
Graph of the initial function and the numerical approximation ofJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EiekYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYnUSJ0RidGL0YyRjRGN0YvL0Y1RjlGNw==:
Q[1] := plot(psi(t), t=0..tau, view=[0..tau, zMin..zMax], labels=[t,z], labelfont=[times,bold,12], legend="psi(t) = 3sin(2t) - 2t", title="A numerical solution of (6.1)", titlefont=[times,bold,12], color=blue, thickness=1):
Q[2] := pointplot([seq([tau + t[n], X[n][1]], n = 0..N)], color=blue, labels=[t,z], labelfont=[times,bold,12], style=point, view=[0..tau + t[N], zMin..zMax], symbol=circle,symbolsize=6):
display([Q[1], Q[2]]); Now let us find the numerical solution corresponding to the second initial function 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tau := 2;
psi := t -> exp(t)-1;
Reset LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnz := 'z'; t := 't';Invoke the Volterra procedure:Volterra(m, h, N, 10^(-8), 5);Table of values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn:d := 2; (Every value is displayed if d = 1; every other value if d = 2, etc.) M := floor(N/d); (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJ0Y1is the number of values greater than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==that will be displayed in each column.) Numerical approximations of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn:
printf(" t z(t) \134n");
for j from 0 to ceil(tau/(d*h)) - 1 do
printf("%10.2f %14.4f\134n", t[d*j], psi(t[d*j]))
end do;
for j from 0 to M do
printf("%10.2f %14.4f\134n", t[d*j]+tau, X[d*j][1])
end do;
Graphs:
Minimum and maximum values of the initial function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0Yy for 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psiMin := minimize(psi(t), [t=0..tau]): evalf(%);
psiMax := maximize(psi(t), [t=0..tau]): evalf(%);
Minimum and maximum values of the numerical approximations of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn:JSFHXMin := min(seq(X[n][1], n=0..N)): evalf(%);
XMax := max(seq(X[n][1], n=0..N)): evalf(%);
Minimum and maximum values that will be displayed on the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-axis: zMin := min(psiMin,XMin) - 0.5: evalf(%);
zMax := max(psiMax,XMax) + 0.5: evalf(%);
Graph of the initial function and the numerical approximation ofJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EiekYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYnUSJ0RidGL0YyRjRGN0YvL0Y1RjlGNw==:
R[1] := plot(psi(t), t=0..tau, view=[0..tau, zMin..zMax], labels=[t,z], labelfont=[times,bold,12], legend="psi(t) = exp(t) - 1", title="A second numerical solution of (6.1)", titlefont=[times,bold,12], color=red, thickness=1):
R[2] := pointplot([seq([tau + t[n], X[n][1]], n = 0..N)], color=red, labels=[t,z], labelfont=[times,bold,12], style=point, view=[0..tau + t[N], zMin..zMax], symbol=circle, symbolsize=6):
display([R[1], R[2]]); Comparison of the graphs of the two numerical solutions:display([Q[1], Q[2], R[1], R[2]], title = "Two numerical solutions of (6.1)");
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">7. Integro-differential equation 3</Font></Text-field>In Sections 5 and 6, we considered integro-differential equations with kernels of the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUkjbW9HRiQ2MFEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSVmb3JtR1EhRicvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZGLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnRitGK0YrRitGK0YrLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUZQNiVRInRGJ0ZTRlYtRiw2MFEiLEYnRi9GMi9GNkZVRjdGOUY7Rj1GPy9GQlEmaW5maXhGJ0ZEL0ZIUSwwLjMzMzMzMzNlbUYnRklGTC1GUDYlUSJzRidGU0ZWRmpuLUZQNiVRInVGJ0ZTRlZGL0YrLUYsNjBRIj1GJ0YvRjJGNUY3RjlGO0Y9Rj9GXm8vRkVRLDAuMjc3Nzc3OGVtRicvRkhGXHBGSUZMRitGTy1GWTYkLUYjNiVGZ25Gam5GYm9GL0Zlb0Zqbg==which makes them linear equations. The kernels of the equations considered from now on will not have this special form. Our first example is taken from Linz [5, p. 178] where he considers the initial value problem
(7.1) 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where (7.2) 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It has the analytical solution 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, which the reader can easily verify. Solution. First reset LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==:z := 'z'; t := 't';
In the notation of (1.1), LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2MFEoJmVxdWl2O0YnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUZANjBRIn5GJ0Y9RkNGRkZIRkpGTEZORlAvRlNRIUYnL0ZWUSYwLjBlbUYnL0ZZRmBvRlpGZ24tSSNtbkdGJDYkRmZuRj0=, 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 Execute the following commands to input these functions:
a := t -> 1;
b := (t,s,u) -> -2*t*s*exp(-u^2);
g := t -> 1 - t*exp(-t^2);
Let's verify that (7.1) has been correctly entered:zEqn := diff(z(t),t) = a(t)*z(t) + int(b(t,s,z(s)), s=0..t) + g(t);Execute the following commands to enter the initial value (7.2):tau := 0;
psi := t -> 0;
Execute the following commands to obtain the functions involved with the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-equation:A := t -> a(t+tau);
B := (t,s,u) -> b(t+tau,s+tau,u);
'A(t)' = A(t); 'B(t,s,u)' = B(t,s,u);
G := t -> g(t+tau) + int(b(t+tau,s,psi(s)), s=0..tau):
'G(t)' = G(t);The LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-equation iswEqn := diff(w(t),t) = A(t)*w(t) + int(B(t,s,w(s)), s=0..t) + G(t);Execute the following commands to input 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:f := proc(t)
Vector([psi(tau), A(t)*psi(tau) + G(t)])
end proc:
f(t);Execute the following commands to input 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:k := proc(t, s, x::Vector)
Vector([x[2], B(t,s,x[1]) + A(t)*x[2]]);
end proc:
v := Vector(2, symbol=x): vlist := convert(v, 'list'):
k(t,s,v);The corresponding system of integral equations isfor i from 1 to 2 do
x[i](t) = f(t)[i] + Int(k(t, s, v)[i], s = 0..t)
end do;
Enter a step size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnh := 0.05;
Enter the desired number of steps of 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N := 40;
Invoke the Volterra procedure:Volterra(m, h, N, 10^(-8), 5);Table of values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn:d := 4; (Every value is displayed if d = 1; every other value if d = 2, etc.) M := floor(N/d); (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJ0Y1is the number of values greater than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==that will be displayed in each column.) Numerical approximations of the exact solution 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:
printf(" t z(t) \134n");
for j from 0 to M do
printf("%10.2f %14.8f\134n", t[d*j]+tau, X[d*j][1])
end do;Remark. The above numerical values are precisely what Linz [5] lists in a table on p. 179.Graphs:
Minimum and maximum values of the initial function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0Yy for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRJSZsZTtGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJWZvcm1HUSZpbmZpeEYnLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSi8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1JI21pR0YkNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGMi1GVDYlUScmIzk2NDtGJy9GWEY4Ri8tRjM2MFEiOkYnRi9GNkY5RjtGPUY/RkFGQ0ZFRkhGS0ZNRlA=psiMin := minimize(psi(t), [t=0..tau]): evalf(%);
psiMax := maximize(psi(t), [t=0..tau]): evalf(%);
Minimum and maximum values of the numerical approximations of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn:JSFHXMin := min(seq(X[n][1], n=0..N)): evalf(%);
XMax := max(seq(X[n][1], n=0..N)): evalf(%);
Minimum and maximum values that will be displayed on the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-axis: zMin := min(psiMin,XMin) - 0.5: evalf(%);
zMax := max(psiMax,XMax) + 0.5: evalf(%);
Graph of the numerical solution of (7.1) satisfying (7.2):
S := pointplot([seq([tau + t[n], X[n][1]], n = 0..N)], color=blue, labels=[t,z], labelfont=[times,bold,12], style=point, view=[0..tau + t[N], zMin..zMax], symbol=circle, symbolsize=6, title = "Numerical solution of (7.1) satisying (7.2)", titlefont=[times,bold,12]):
display(S);
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">8. Integro-differential equation 4</Font></Text-field>Solve the nonlinear integro-differential equation
(8.1) 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such that (8.2) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2MFEiPUYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUklbXN1cEdGJDYlRjotRiM2Iy1JI21uR0YkNiRRIjJGJ0Y9LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJw== for 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. First reset LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==:z := 'z'; t := 't';
In the notation of (1.1), 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, 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and 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. Execute the following commands to input these functions:
a := t -> -1 + cos(2*t);
b := (t,s,u) -> t*sin(u)/(1+s^2);
g := t -> 1/(1+t)^2;
Let's verify that (8.1) has been correctly entered:zEqn := diff(z(t),t) = a(t)*z(t) + int(b(t,s,z(s)), s=0..t) + g(t);Execute the following commands to enter the initial function (8.2):tau := 2;
psi := t -> t^2;
Execute the following commands to obtain the functions involved with the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-equation:A := t -> a(t+tau);
B := (t,s,u) -> b(t+tau,s+tau,u);
'A(t)' = A(t); 'B(t,s,u)' = B(t,s,u);
G := t -> g(t+tau) + int(b(t+tau,s,psi(s)), s=0..tau):
'G(t)' = G(t);The LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-equation iswEqn := diff(w(t),t) = A(t)*w(t) + int(B(t,s,w(s)), s=0..t) + G(t);Execute the following commands to input 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:f := proc(t)
Vector([psi(tau), A(t)*psi(tau) + G(t)])
end proc:
f(t);Execute the following commands to input 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:k := proc(t, s, x::Vector)
Vector([x[2], B(t,s,x[1]) + A(t)*x[2]]);
end proc:
v := Vector(2, symbol=x): vlist := convert(v, 'list'):
k(t,s,v);The corresponding system of integral equations isfor i from 1 to 2 do
x[i](t) = f(t)[i] + Int(k(t, s, v)[i], s = 0..t)
end do;
Enter a step size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnh := 0.1;
Enter the desired number of steps of size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJ0Y1
(85 steps with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEkMC4xRidGOQ== takes the author's computer (1400 MHz processor) about 30 seconds):N := 85;
Invoke the Volterra procedure:Volterra(m, h, N, 10^(-8), 5);Table of values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn:d := 4; (Every value is displayed if d = 1; every other value if d = 2, etc.) M := floor(N/d); (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJ0Y1is the number of values greater than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJw==that will be displayed in each column.) Numerical approximations of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn:
printf(" t z(t) \134n");
for j from 0 to ceil(tau/(d*h)) do
printf("%10.2f %14.4f\134n", t[d*j], psi(t[d*j]))
end do;
for j from 0 to M do
printf("%10.2f %14.4f\134n", t[d*j]+tau, X[d*j][1])
end do;Graphs:
Minimum and maximum values of the initial function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Njg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0Yy for 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psiMin := minimize(psi(t), [t=0..tau]): evalf(%);
psiMax := maximize(psi(t), [t=0..tau]): evalf(%);
Minimum and maximum values of the numerical approximations of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYn:JSFHXMin := min(seq(X[n][1], n=0..N)): evalf(%);
XMax := max(seq(X[n][1], n=0..N)): evalf(%);
Minimum and maximum values that will be displayed on the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-axis: zMin := min(psiMin,XMin) - 0.5: evalf(%);
zMax := max(psiMax,XMax) + 0.5: evalf(%);
Graph of the numerical solution of (8.1) satisfying (8.2):
T[1] := plot(psi(t), t=0..tau, view=[0..tau, zMin..zMax], labels=[t,z], labelfont=[times,bold,12], titlefont=[times,bold,12], color=blue, thickness=1):
T[2] := pointplot([seq([tau + t[n], X[n][1]], n = 0..N)], color=blue, labels=[t,z], labelfont=[times,bold,12], style=point, view=[0..tau + t[N], zMin..zMax], symbol=circle,symbolsize=6):
T[3] := textplot([5.3,7,"Numerical solution of (8.1)"]):
T[4] := textplot([5.3,6.5, " with initial function psi(t) = t^2"]):
display([T[1],T[2],T[3],T[4]]);
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">9. References</Font></Text-field>[1] Leigh C. Becker and Micah Wheeler, Numerical and Graphical Solutions of Volterra Equations of the Second Kind, Maple Application Center, 2005.[2] Leigh C. Becker, Function bounds for solutions of Volterra equations and exponential asymptotic stability, Nonlinear Analysis: Theory, Methods & Applications 67, No. 2 (July 2007), pp. 382-397. [3] Leigh C. Becker, Uniformly Continuous LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRic= Solutions of Volterra Equations and Global Asymptotic Stability (work in progress). [4] T. A. Burton, Volterra Integral and Differential Equations, 2nd ed., Mathematics in Science & Engineering, 202, Elsevier, 2005.[5] Peter Linz, Analytical and Numerical Methods for Volterra Equations, Studies in Applied Mathematics 7, SIAM, Philadelphia, 1985.Legal Notice: The copyright for this application is owned by the author. Neither Maplesoft nor the author is responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit 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