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Classroom Tips and Techniques: Diffusion with a Generalized Robin Condition Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow MaplesoftLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Recently, while perusing the backlog of unanswered questions on MaplePrimes, I came across an interesting boundary value problem in one-dimensional diffusion. In particular, the problem could be interpreted as describing heat transfer in a rod, with the left end insulated, and the right end subject to a generalized Robin condition. In the typical nonhomogeneous Robin condition, the "driving term" is known. In this problem, the nonhomogeneous term was the solution of an initial value problem for an ordinary differential equation that in turn, was driven by the temperature on the right end of the rod. Table 1 (where subscripts denote partial derivatives; and the prime, differentiation with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) states the problem in detail. The diffusion equation 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 The initial temperature distribution 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 The insulated left end 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 Robin condition at right end 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 Initial value problem for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUY7NiRRIjFGJ0Y+LUZBNi1RIi9GJ0Y+RkRGRy9GSkYxRktGTUZPRlEvRlRRLDAuMTY2NjY2N2VtRicvRldGam4tRjs2JFEiNUYnRj5GPg== Table 1 One-dimensional diffusion with a Robin condition whose nonhomogeneous term is determined by an initial value problem for an ordinary differential equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Maple's pdsolve command will solve one-dimensional evolution equations numerically (by finite difference techniques), and Maple's dsolve command will solve (numerically) initial value problems for ordinary differential equations, no provision exists in Maple for the kind of coupling that occurs in the BVP given in Table 1. But the problem does yield to a simple finite difference scheme, as we will show below. In addition, since the BVP in Table 1 is linear with constant coefficients, it yields to a Laplace transform solution. While LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInNGJ0YvRjJGQUZBRkE= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==, the Laplace transforms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGQUZBRkE= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==, respectively, are easily found, the expressions are sufficiently complex that inverting them analytically is apparently impossible. But at least two techniques for inverting the Laplace transform numerically succeed. The first is just a brute-force numerical evaluation of the Bromwich inversion integral; and the second, a more sophisticated numerical evaluation of this integral along a deformed contour of finite length. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The general one-dimensional Robin boundary condition is of the form 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is a known function, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPUY+ is a bounding edge of the interval over which the accompanying BVP has been set, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= are constants. In the BVP of Table 1, 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, but LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUYsNiVRImZGJ0YvRjJGNUY9 is not a "known" function. The normal derivative, denoted by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn, becomes 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 for the BVP in Table 1. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Nearly every numerical analysis text that discusses the numeric solution of partial differential equations will exposit the simple explicit method for solving the one-dimensional heat equation. For the BVP in Table 1, we elected this technique on the grounds of its utter simplicity, even though there is a stepsize restriction for stability. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= For the sake of completeness, we summarize this technique here. Discretize with stepsizes 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, and 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, where 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 is required for stability. (We chose 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, and with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzEwRidGOUY5, we had 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.) Taking the discrete grid as 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, and 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, and using simple forward differences to approximate the derivatives in the one-dimensional heat equation, we have 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRInBGJ0YyRjUtSSNtb0dGJDYtUSIsRicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYvNiVRInFGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGQQ== approximates 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. Defining 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, and isolating 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, we have 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Notice that it takes three values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm1GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn to determine one new value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRIm1GJ0YyRjUtSSNtb0dGJDYtUSIrRicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtSSNtbkdGJDYkUSIxRidGQUYyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZB. With this formula, the nodal values LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRIm5GJ0YyRjUtSSNtb0dGJDYtUSIsRicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYvNiVRInNGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPUY6LUY+Ni1RIj1GJ0ZBRkMvRkdGRUZIRkpGTEZORlAvRlNRLDAuMjc3Nzc3OGVtRicvRlZGXW8tSSNtbkdGJDYkUSIxRidGQUY9LUZgbzYkUSIyRidGQUY9LUY+Ni1RKSZoZWxsaXA7RidGQUZDRltvRkhGSkZMRk5GUC9GU1EsMC4xNjY2NjY3ZW1GJy9GVkZqb0Y9LUYvNiVRIk5GJ0YyRjUtRj42LVEoJm1pbnVzO0YnRkFGQ0Zbb0ZIRkpGTEZORlAvRlNRLDAuMjIyMjIyMmVtRicvRlZGY3BGX29GQQ==, can be determined at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdWJHRiQ2JUYrLUYjNiUtRiw2JVEic0YnRi9GMkYvRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y5, provided the nodal values are known at time 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. When LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=, we have 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= To obtain the value 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, which falls outside the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YyLUYsNiRRIjFGJ0YvRi8=, write the homogeneous Neumann condition 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 in terms of the forward difference 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= so that 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. At the right end of the interval, when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJORidGL0YyRjk=, we have 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= To find a value for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYpLUYvNiVRIk5GJ0YyRjUtSSNtb0dGJDYtUSIrRicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtSSNtbkdGJDYkUSIxRidGQS1GPjYtUSIsRidGQUZDL0ZHRjRGSEZKRkxGTkZQL0ZTUSYwLjBlbUYnL0ZWUSwwLjMzMzMzMzNlbUYnLUYvNiVRIm1GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGQQ==, we discretize the derivative appearing in the Robin condition, obtaining 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm1GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn approximates LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUklbXN1YkdGJDYlLUYsNiVRInRGJ0YvRjItRiM2JS1GLDYlUSJtRidGL0YyRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0YzUSdub3JtYWxGJ0ZILUYsNiNRIUYnRkg=. From this, we get 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 so that we finally have 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 Since 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, values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRIm5GJ0YyRjUtSSNtb0dGJDYtUSIsRicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYvNiVRIm1GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGQQ== are computed in conjunction with a corresponding finite-difference solution of the ODE. For this, we write 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 so that 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 Of course, 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. After a few experiments, it became apparent that the solution need not be extended beyond LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjRGJ0Y5Rjk=, so 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. Table 2 summarizes the Maple code that implements the finite-difference calculations. Initialize problem parameters. 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 Initialize 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZBLUkjbW9HRiQ2L1EkZm9yRicvJSVib2xkR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y0LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkgtRiw2LlEifkYnLyUnaXRhbGljR0YxL0YzUSdpdGFsaWNGJ0Y3RjpGPEY+RkBGQkZERkZGSS1JI21pR0YkNiVRIm5GJ0ZORlBGSy1GLDYvUSVmcm9tRidGL0YyRjVGN0Y6RjxGPkZARkJGREZGRklGSy1JI21uR0YkNiRRIjBGJy9GM1Enbm9ybWFsRidGSy1GLDYvUSN0b0YnRi9GMkY1RjdGOkY8Rj5GQEZCRkRGRkZJRkstRlM2JVEiTkYnRk5GUEZLLUYsNi9RI2RvRidGL0YyRjVGN0Y6RjxGPkZARkJGREZGRkktSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGSC8lJmRlcHRoR0Znby8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GY282JkZlb0Zob0Zqby9GXXBRJWF1dG9GJy1GUzYlUSJnRidGTkZQLUkobWZlbmNlZEdGJDYnLUYjNihGUi1GLDYuUSIsRidGTkZQRjcvRjtGMUY8Rj5GQEZCRkRGRi9GSlEsMC4zMzMzMzMzZW1GJ0ZZRk4vJStleGVjdXRhYmxlR0Y5RlBGTkZQLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRkstRiw2LVEqJmNvbG9uZXE7RidGZ25GN0Y6RjxGPkZARkJGRC9GR1EsMC4yNzc3Nzc4ZW1GJy9GSkZdckZLRlktRiw2LVEiO0YnRmduRjdGXnFGPEY+RkBGQkZERkZGXnJGYm8tRlM2I1EhRictRiw2L1EkZW5kRidGL0YyRjVGN0Y6RjxGPkZARkJGREZGRklGS0Zfby1GLDYtUSI6RidGZ25GN0Y6RjxGPkZARkJGREZcckZeckZORmFxRlA= Initialize 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 Advance LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= Advance LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at "interior nodes" Advance LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZjcy1JI21pR0YkNiVRImZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNictRiM2Ji1JI21uR0YkNiRRIjBGJy9GM1Enbm9ybWFsRidGLy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRjJGL0YyLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUkjbW9HRiQ2LlEifkYnRi9GMi8lJmZlbmNlR0ZCLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmduLUZKNi1RKiZjb2xvbmVxO0YnRj5GTUZPRlFGU0ZVRldGWS9GZm5RLDAuMjc3Nzc3OGVtRicvRmluRl5vRkktRjs2JFEiMUYnRj4tRko2LVEiL0YnRj5GTUZPL0ZSRjFGU0ZVRldGWS9GZm5RLDAuMTY2NjY2N2VtRicvRmluRmhvLUY7NiRRIjVGJ0Y+LUZKNi1RIjpGJ0Y+Rk1GT0ZRRlNGVUZXRllGXW9GX28tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGZ24vJSZkZXB0aEdGZXAvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRko2L1EkZm9yRicvJSVib2xkR0YxL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0ZjcUZNRk9GUUZTRlVGV0ZZRmVuRmhuRkktRiw2JVEibUYnRi9GMkZJLUZKNi9RJWZyb21GJ0ZgcUZicUZkcUZNRk9GUUZTRlVGV0ZZRmVuRmhuRklGOkZJLUZKNi9RI3RvRidGYHFGYnFGZHFGTUZPRlFGU0ZVRldGWUZlbkZobkZJLUYsNiVRIk1GJ0YvRjJGSS1GSjYvUSNkb0YnRmBxRmJxRmRxRk1GT0ZRRlNGVUZXRllGZW5GaG5GYHAtRmFwNiZGY3BGZnBGaHAvRltxUSVhdXRvRictRiw2JVEiZ0YnRi9GMi1GNjYnLUYjNipGOi1GSjYuUSIsRidGL0YyRk0vRlBGMUZRRlNGVUZXRllGZW4vRmluUSwwLjMzMzMzMzNlbUYnRmZxLUZKNi1RIitGJ0Y+Rk1GT0ZRRlNGVUZXRlkvRmZuUSwwLjIyMjIyMjJlbUYnL0ZpbkZqc0Zgb0YvRkBGMkYvRjJGQ0ZGRklGam5GSS1GLDYlUSJMRidGL0YyRklGaXItRjY2Jy1GIzYoRjpGYHNGZnFGL0ZARjJGL0YyRkNGRkZmcy1GLDYlUSJjRidGL0YyRklGaXJGX3RGZnNGXHRGSUZpci1GNjYnLUYjNihGYG9GYHNGZnFGL0ZARjJGL0YyRkNGRi1GSjYtUSI7RidGPkZNRmNzRlFGU0ZVRldGWUZlbkZfb0ZgcEZdcUZJLUYsNiVRIm5GJ0YvRjJGSUZpcUZJRmBvRklGXHJGSS1GLDYlUSJORidGL0YyLUZKNi5RKCZtaW51cztGJ0YvRjJGTUZPRlFGU0ZVRldGWUZpc0ZbdEZgb0ZJRmJyRmBwRmVyRmlyLUY2NictRiM2KkZddUZgc0ZmcUZmc0Zgb0YvRkBGMkYvRjJGQ0ZGRklGam5GSUZcdEZJRmlyLUY2NictRiM2KkZddUZjdUZgb0Zgc0ZmcUYvRkBGMkYvRjJGQ0ZGRmZzRmN0RklGaXItRjY2Jy1GIzYoRl11RmBzRmZxRi9GQEYyRi9GMkZDRkZGZnNGXHRGSUZpci1GNjYnLUYjNipGXXVGZnNGYG9GYHNGZnFGL0ZARjJGL0YyRkNGRkZqdEZgcC1GSjYvUSRlbmRGJ0ZgcUZicUZkcUZNRk9GUUZTRlVGV0ZZRmVuRmhuRklGYnJGXXBGYHBGZXJGaXItRjY2Jy1GIzYqRmB1RmBzRmZxRmZzRmBvRi9GQEYyRi9GMkZDRkZGSUZqbkZJRlx0RklGaXItRjY2Jy1GIzYqRmB1RmN1RmBvRmBzRmZxRi9GQEYyRi9GMkZDRkZGZnNGY3RGSUZpci1GNjYnLUYjNihGYHVGYHNGZnFGL0ZARjJGL0YyRkNGRkZmc0ZcdEZJLUY2NiUtRiM2Ky1GLDYlUSNoeEYnRi9GMkZJLUY2NiUtRiM2KkYrLUY2NictRiM2JkZmcUYvRkBGMkYvRjJGQ0ZGRmN1RmlyRmF3Ri9GQEYyRi9GMkZmc0ZpckZhd0YvRkBGMkYvRjJGanRGYHBGZXJGKy1GNjYnLUYjNihGZnFGZnNGYG9GL0ZARjJGL0YyRkNGRkZJRmpuRkktRiw2JVEickYnRi9GMkZJRitGYHhGZnMtRiw2JVEjaHRGJ0YvRjJGSUZpckZhd0ZqdEZgcEZlckZmdkZJRmJyRl1wRi9GQEYy Table 2 Code for finite-difference solution There are some 8811 nodal values. Figure 1 is a graph of the solution surface LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGQUZBRkE=, drawn using only 891 such values. From the graph, it appears that the solution rapidly approaches a limiting value of 0.1. 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Figure 1 Graph of solution surface LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGQUZBRkE= as determined by finite-difference solution Figure 2 is a graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==. It appears that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is rapidly asymptotic to a limiting value in the vicinity of 0.1. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVElcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUY2NiYtRiM2Ji1GLDYlUSRzZXFGJ0YvRjItRjY2JC1GIzYrLUY2NiYtRiM2Ki1GLDYlUSNzdEYnRi9GMi1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlQvJSlzdHJldGNoeUdGVC8lKnN5bW1ldHJpY0dGVC8lKGxhcmdlb3BHRlQvJS5tb3ZhYmxlbGltaXRzR0ZULyUnYWNjZW50R0ZULyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGXW8tRiw2JVEjaHRGJ0YvRjItRk02LVEiLEYnRlBGUi9GVkYxRldGWUZlbkZnbkZpbkZbby9GX29RLDAuMzMzMzMzM2VtRictRiw2JVEiZkYnRi9GMi1GNjYmLUYjNiVGSS8lK2V4ZWN1dGFibGVHRlRGUEZQLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmBwRlBGUEZicEZlcEZjb0ZJLUZNNi1RIj1GJ0ZQRlJGVUZXRllGZW5GZ25GaW4vRlxvUSwwLjI3Nzc3NzhlbUYnL0Zfb0ZccS1JI21uR0YkNiRRIjBGJ0ZQLUZNNi1RIy4uRidGUEZSRlVGV0ZZRmVuRmduRmluL0Zcb1EsMC4yMjIyMjIyZW1GJ0Zeby1GLDYlUSJNRidGL0YyRmBwRlBGUEZgcEZQRlBGYnBGZXBGYHBGUEZQLUYsNiNRIUYnRmBwRlA=

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Figure 2 Graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== from finite-difference solution JSFH (The code for Figures 1 and 2, and all other figures, is in the respective table holding the graphs. Likewise, cells tinted in yellow contain hidden input. To see such code, use the Table Properties dialog to select the Show Input option.)JSFH JSFH

JSFH To effect a solution of the BVP in Table 1 by the Laplace transform, transform the heat equation with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, obtaining 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 where subscripts indicate differentiation. Since 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, we can write this ordinary differential equation as 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 the general solution of which is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The homogeneous Neumann boundary condition on the left leads to 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, so that we can write the solution as 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The Robin condition on the right involves LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== so the IVP 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, must be solved first. Taking the Laplace transform of the ODE, we get 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Since 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, we have 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 The Laplace transform of the Robin condition 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 leads to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRInhGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2Ji1JI21uR0YkNiRRIjFGJy9GNlEnbm9ybWFsRictSSNtb0dGJDYtUSIsRidGSS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZRLyUqc3ltbWV0cmljR0ZRLyUobGFyZ2VvcEdGUS8lLm1vdmFibGVsaW1pdHNHRlEvJSdhY2NlbnRHRlEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLzYlUSJzRidGMkY1RklGSS1GTDYtUSI9RidGSUZPL0ZTRlFGVEZWRlhGWkZmbi9GaW5RLDAuMjc3Nzc3OGVtRicvRlxvRmZvLUYvNiVRIkZGJ0YyRjUtRkE2JC1GIzYkRl5vRklGSS1GTDYtUSgmbWludXM7RidGSUZPRmRvRlRGVkZYRlpGZm4vRmluUSwwLjIyMjIyMjJlbUYnL0Zcb0ZjcEYuRkBGSQ== and hence to 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Recalling that 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, we can then determine LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= from the equation 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyooSSJBRzYiIiIiKUkic0dGJSNGJiIiI0YmLCYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyokRidGJkYmLUYtNiMsJEYyISIiRjZGJiomLCYjRiYiIiZGJiooRiRGJkYoRiYsJkYsRiZGM0YmRiZGNkYmLCZGKEYmRiZGJkY2

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= (Maple code generating , and output in any other colored cell, is hidden in the cells. To see such code, use the Table Properties dialog to select the Show Input option.) JSFH The expression for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is surprisingly cumbersome, so we obtain it in Maple as 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0kiQUc2IiwkKiYjIiIiIiImRigsLiomKUkic0dGJCMiIiQiIiNGKC1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKiQpRi0jRihGMEYoRihGKComRjhGKEYxRihGKComRixGKC1GMjYjLCRGNyEiIkYoRj8qJkY4RihGPEYoRj8qJkYtRihGMUYoRigqJkYtRihGPEYoRihGP0Yo

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Hence, we now know LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInNGJ0YvRjJGQUZBRkE= and 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. For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInNGJ0YvRjJGQUZBRkE= we obtain 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkdHNiI2JEkieEdGJUkic0dGJSwkKigjIiIiIiImRiwsLiomKUYoIyIiJCIiI0YsLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJClGKCNGLEYzRixGLEYsKiZGO0YsRjRGLEYsKiZGMEYsLUY1NiMsJEY6ISIiRixGQiomRjtGLEY/RixGQiomRihGLEY0RixGLComRihGLEY/RixGLEZCLCYtRjU2IyomRjtGLEYnRixGLC1GNTYjLCRGSUZCRixGLEYs

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Fortunately, this can be written as 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkdHNiI2JEkieEdGJUkic0dGJSwkKigjIiIiIiImRiwtSSVjb3NoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJilGKCNGLCIiI0YsRidGLEYsLCYqJiwmKiQpRigjIiIkRjdGLEYsKiRGNUYsRixGLC1JJXNpbmhHRjA2I0Y/RixGLComRihGLC1GL0ZCRixGLCEiIkYs

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==, we obtain 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkZHNiI2I0kic0dGJSwkKigjIiIiIiImRissJi1JJWNvc2hHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyokKUYnI0YrIiIjRitGKyomRjVGKy1JJXNpbmhHRjBGM0YrRitGKywmKiYsJiokKUYnIyIiJEY3RitGK0Y0RitGK0Y5RitGKyomRidGK0YuRitGKyEiIkYr

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH Under suitable conditions on the functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGQUZBRkE=, it is possible to extract "initial" and "terminal" values for these function from their Laplace transforms. Theorems justifying these results are typically called Tauberian, in contrast to Abelian theorems that provide information in the opposite direction. First, if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is of exponential order, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is its Laplace transform, then 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. If we apply this to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tRiw2I1EhRidGPQ== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInNGJ0YvRjJGQUZBRkE=, we will have some evidence that these transforms are not a priori incorrect. Thus, we find 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JJkxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJkkic0dGKCIiIi1JIkZHRig2I0YrRiwvRitJKWluZmluaXR5R0YmI0YsIiIm

and 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JJkxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJkkic0dGKCIiIi1JIkdHRig2JEkieEdGKEYrRiwvRitJKWluZmluaXR5R0YmIiIh

JSFH Both initial values are recovered. Of greater importance is the "final value theorem" that gives the limiting value of a function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== in terms of the behavior of its transform as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=. In particular, if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is bounded and its limit at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmaW5maW47RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlNGOQ== exists, then that limit is given by 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. From the finite-difference numerical solution, we opined that both LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGQUZBRkE= tended to finite limits fairly rapidly. Since we don't know analytically that these functions are well behaved at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmaW5maW47RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlNGOQ==, the following two limits are informative, but not conclusive. JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JJkxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJkkic0dGKCIiIi1JIkdHRig2JEkieEdGKEYrRiwvRisiIiEjRiwiIzU=

Ly1JJkxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJkkic0dGKCIiIi1JIkZHRig2I0YrRiwvRisiIiEjRiwiIzU=

JSFH These limiting values are consistent with the finite-difference solution. JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Sufficient conditions for the Bromwich (line) integral 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= along 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, yielding the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== from its Laplace transform LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==, include LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== being analytic in some right-half plane LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEoJiM4NDc2O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI+RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUYsNiVRImNGJ0YvRjJGPQ== and being 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 with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= in that half plane. By the calculations at the end of the last section, both LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInNGJ0YvRjJGQUZBRkE= are 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. However, the Bromwich inversion integral will invert the transform LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21pR0YkNiVRInNGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGO0Y+LyUubGluZXRoaWNrbmVzc0dGMS8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZELyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjI= to 1, even though this transform does not satisfy all the sufficient conditions just stated. We implement this inversion integral with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= because the zeros of 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 the denominator of both LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInNGJ0YvRjJGQUZBRkE=, appear to fall on the nonpositive real axis. We determine this not only from Figure 3, a graph of the denominator of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== along the real line, but also by applying the Principle of the Argument to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== on the circles 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. The circles LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn are centered at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIitGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSJSRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8= on the real line, and have a radius of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Thus, as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= increases, more and more of the half-plane LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEoJiM4NDc2O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUS8mR3JlYXRlckVxdWFsO0YnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVC1JI21uR0YkNiRRIjFGJ0Y9Rj0= is enclosed. 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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

Figure 3 Zeros of the denominator of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== along the real line Numerically evaluating a succession of integrals 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 for increasing values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= shows that each such integral is zero. Since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the number of zeros inside LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the number of poles, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== has no poles inside LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIlJGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn, it then has no zeros inside either. JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Direct numerical evaluation of the Bromwich integral generally has unsatisfactory results. Along a vertical line, the integrand oscillates, making it difficult to integrate to infinity. This is true for 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, which, along the line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIitGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYzNi1RIn5GJ0YvRjZGOUY7Rj1GP0ZBRkMvRkZRJjAuMGVtRicvRklGVy1GSzYlUSJ5RidGTkZRRi8= has the form. JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqKiMiIiIiIiZGJSwmLUklY29zaEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiQpLCZGJUYlKiZeI0YlRiVJInlHRi1GJUYlI0YlIiIjRiVGJSomRjBGJS1JJXNpbmhHRipGLkYlRiVGJSwmKiYsJiokKUYxIyIiJEY2RiVGJUYvRiVGJUY4RiVGJSomRjFGJUYoRiVGJSEiIi1JJGV4cEdGKjYjKiZGMUYlSSJ0R0YtRiVGJUYl

JSFHJSFH The real part of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8=is an even function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, whereas the imaginary part is odd in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Thus, the Bromwich integral for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== on any interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtaUdGJDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYsNi1RJiZsZXE7RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRlMtRkc2JVEieUYnRkpGTUZPRkZGLw== can be reduced to 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 As might be imagined, even a numeric evaluation of this integral is a challenge. If, instead, we integrate to a "large" finite upper limit (we used 10,000), and use one of the NAG library methods (_d01akc, designed for oscillating integrands on a finite interval), we can obtain the graph shown in Figure 4. 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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

Figure 4 Numeric inversion of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== by direct numeric evaluation of the Bromwich integral JSFH Figure 4 uses ten computed points for the graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==, and this takes several minutes in Maple. But the apparent asymptotic behavior is in agreement with the prediction of the Tauberian theorem in the Laplace transform section. Similar results for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGQUZBRkE= are obtained in Figure 5 by numeric evaluation of the Bromwich integral on a uniform grid 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, where 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, and 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. Since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInNGJ0YvRjJGQUZBRkE= has the same symmetry properties along 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, the computations for Figure 5 are similar to those for Figure 4, except that the upper limit of integration is taken as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRJTc1MDBGJ0Y5Rjk=. It took more than two minutes to obtain Figure 4, and more than ten to obtain Figure 5. 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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

Figure 5 Numeric inversion of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInNGJ0YvRjJGQUZBRkE= by direct numeric evaluation of the Bromwich integral LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Like Figure 4, Figure 5 shows a limiting behavior for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGQUZBRkE= consistent with the Tauberian prediction in the Laplace transform section. JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The literature contains a number of methods for the numerical inversion of the Laplace transform. In [1], Talbot's approach of deforming the Bromwich contour is implemented in variable-precision software. The inversion is given by 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJm1mcmFjR0YkNigtRiM2Ji1JI21uR0YkNiRRIjJGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GZm4tRiw2JVEiTUYnRi9GMkY5LUYjNictRlU2JFEiNUYnRjlGWC1GLDYlUSJ0RidGL0YyRi9GMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGaG8vJSliZXZlbGxlZEdGPUY5 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the precision of the computing device. The relationship between LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiTUYnRi9GMkY5, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is empirical, and allows for an increase in device precision as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= increases. Our preliminary investigations suggest inverting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInNGJ0YvRjJGQUZBRkE= on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YyLUYsNiRRIjVGJ0YvRi8=, so with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5LUY2Ni1RIi9GJ0Y5RjtGPi9GQUYxRkJGREZGRkgvRktRLDAuMTY2NjY2N2VtRicvRk5GWC1GUDYkUSI1RidGOUY5, the default ten digits in Maple are more than enough for our purposes. This allows us to use Maple's built-in integrator rather than the trapezoid rule suggested in [1]. A graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== obtained by this method appears in Figure 6. Drawn in about 15 seconds with 20 computed points, it is consistent with our Tauberian results and Figure 4. 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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

Figure 6 Graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== obtained by numeric evaluation of the Bromwich integral using a deformed contour Figure 7 is obtained in about 80 seconds by applying the same algorithm to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInNGJ0YvRjJGQUZBRkE=. Again, the graph (drawn from 110 computed points) is consistent with our Tauberian results and with Figure 5. 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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Figure 7 Graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGQUZBRkE= obtained by numeric evaluation of the Bromwich integral using a deformed contour LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH [1] Abate J. and Valk\303\263 P. P., Multi-precision Laplace transform inversion, Int. J. Numer. Meth. Engng 2004, 60:979-993. JSFH

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