<Text-field style="Heading 1" layout="Heading 1">Initializations</Text-field>

<Text-field style="Heading 1" layout="Heading 1">Example 1: Parameter Estimation with Noisy Data</Text-field>

<Text-field style="Heading 2" layout="Heading 2">The Model</Text-field>

Consider the nonlinear damped oscillator in which the damping force is assumed proportional to the velocity, a system governed by the nonlinear differential equation

DE := diff(y(t),t,t) + a*diff(y(t),t)*abs(diff(y(t),t)) + b*y(t) = 0;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNERUdGKC8sKC1JJWRpZmZHRiY2JC1JInlHRig2I0kidEdGKC1JIiRHRiY2JEY2IiIjIiIiKihJImFHRihGOy1GMTYkRjNGNkY7LUkkYWJzR0YmNiNGPkY7RjsqJkkiYkdGKEY7RjNGO0Y7IiIhNyMvLCgtRjE2JEY+RjZGO0Y8RjtGQ0Y7RkU=

The unknown parameters are the damping coefficient LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn and the spring constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn.

To simulate observational data, we start with the model

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiJ0YnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRidGOC1GOTYuUScmcGx1cztGJ0YyRjxGPkZARkJGREZGRkhGSi9GTVEsMC4yMjIyMjIyZW1GJy9GUEZWLUkjbW5HRiQ2JVEiMkYnRjJGPC1GOTYtUTEmSW52aXNpYmxlVGltZXM7RidGPEY+RkBGQkZERkZGSEZKL0ZNRlFGT0YrRjhGZm4tSShtZmVuY2VkR0YkNiktRiM2JkYrRjhGMkY8RjJGPC9JK21zZW1hbnRpY3NHRiRRJGFic0YnLyUlb3BlbkdRKSZ2ZXJiYXI7RicvJSZjbG9zZUdGZG9GX29GUi1GWTYlUSI0RidGMkY8RmZuRistRjk2LlEiPUYnRjJGPEY+RkBGQkZERkZGSEZKL0ZNUSwwLjI3Nzc3NzhlbUYnL0ZQRl5wLUZZNiVRIjBGJ0YyRjxGMkY8

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

compute equi-spaced displacement 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, then add random noise to the observations.

<Text-field style="Heading 1" layout="Heading 1">Generating the Observations</Text-field>

Starting with the differential equation

q := eval(DE, [a=2, b=4]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJxR0YoLywoLUklZGlmZkdGJjYkLUkieUdGKDYjSSJ0R0YoLUkiJEdGJjYkRjYiIiMiIiIqJi1GMTYkRjNGNkY7LUkkYWJzR0YmNiNGPUY7RjpGMyIiJSIiITcjLywoLUYxNiRGPUY2RjtGPEY6RjNGQkZD

and the initial conditions

ic := y(0)=1/2, D(y)(0)=2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNpY0dGKDYkLy1JInlHRig2IyIiISMiIiIiIiMvLS1JIkRHRiU2I0YxRjJGNjckRi9GNw==

we obtain a numeric solution via

Q := dsolve({q,ic}, y(t), numeric, output=listprocedure):

The function

Yt := rhs(Q[2]):

is a Maple-generated procedure that provides 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 numerically. Choosing

N := 50; d := .2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJOR0YoIiNdNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJkR0YoJCIiIyEiIjcjRi4=

as the number of observations and the time interval between observations, we write

Y := [seq(Yt(d*k),k=0..N)]:

to obtain a list of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEnJnBsdXM7RidGMi9GNlEnbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JVEiMUYnRjJGPEYyRjw= observations, and

P := [seq([d*(k-1),Y[k]], k=1..N+1)]:

to obtain a list of corresponding points.

Figure 1 contains a graph of the numeric solution of the governing initial value problem, with the computed points superimposed.

p1 := odeplot(Q,[t,y(t)],0..10, numpoints=100, color=red): p2 := plot(P, style=point, color=black): display([p1,p2]);

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NiI=

Figure 1 Numeric solution of governing initial value problem

<Text-field style="Heading 1" layout="Heading 1">Adding Random Noise</Text-field>

We distinguish between two types of random noise appearing in experimental data. True experimental noise is relative to the data-collection process itself. External interference is additive.

A sample of size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEnJnBsdXM7RidGMi9GNlEnbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JVEiMUYnRjJGPEYyRjw= taken from a normal distribution with mean 1 and standard deviation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiw2JlEoJnNpZ21hO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjVGNw== provides factors normally distributed about 1. Multiplying each computed value of 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 by one such factor perturbes the computed values with noise of magnitude relative to that of the computed value.

Choosing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiw2JlEoJnNpZ21hO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEiPUYnRjVGNy8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk4tSSNtbkdGJDYlUSQwLjNGJ0Y1RjdGNUY3, we have

S := Sample(Normal(1,.3),N+1): Yn := zip(`*`, S, Vector(Y))^%T:

for our noisy data. Figure 2 shows the noisy observations as red dots, and the original computed displacements as black dots.

p3 := plot([seq([d*(k-1),Yn[k]],k=1..N+1)],style=point, color=red): display([p2,p3]);

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

Figure 2 Computed displacements (black) and noisy observations (red)

<Text-field style="Heading 1" layout="Heading 1">The Sum of Squares of Deviations</Text-field>

The following Maple procedure is a function of the two parameters LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn appearing in the differential equation of our nonlinear model. For each pairLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiUtRiM2Jy1GLDYmUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGOi9GPlEnbm9ybWFsRicvJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYmUSJiRidGN0Y6Rj1GOkZERjpGREYrRjpGRA==, the differential equation is integrated numerically so that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEnJnBsdXM7RidGMi9GNlEnbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JVEiMUYnRjJGPEYyRjw= equispaced nodes are computed. These nodes are compared to the noisy observations via the Euclidean norm of the difference between the vector of observations and the vector of computed solution values.

SS := proc(a,b) local F, V; if not type([a,b],[numeric,numeric]) then return 'SS'(a,b); elif a<0 or b<0 then return 1e100; end if; F := dsolve(eval({DE,ic},{:-a=a,:-b=b}), y(t), numeric, output=Array([seq(d*k,k=0..N)])); V := convert(Column(F[2,1],2),Vector); Norm(V-Yn,2); end proc:

(The penalty-function construct that keeps the parameters in the first quadrant was suggested by Dr. Allan Wittkopf of Maplesoft.)

Figure 3 contains a graph of the surface 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, which indicates that there is indeed at least one minimum for this function.

plot3d(SS,0..5, 2..6, axes=box, view=.3..3/2, labels=[a,b,""], orientation=[-65,70]);

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Figure 3 The surface 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 showing existence of a minimum

<Text-field style="Heading 1" layout="Heading 1">Minimization </Text-field>

What remains is to compute a minimum, a task Maple performs with its NLPSolve command from the Optimization package. By option, this command can implement the Nelder-Meade (simplex) algorithm that is Maple's only direct method - it does not differentiate the objective function. In two dimensions, the simplex is a triangle. Function values are obtained at the vertices of the triangle, and for a minimization, the triangle is reflected away from the vertex with the highest value. Various scaling and anti-cycling devices are included in the algorithm.

Using this command, we determine

R := NLPSolve(SS(a,b), method=nonlinearsimplex, initialpoint=[a=5,b=5], evaluationlimit=250);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJSR0YoNyQkIjMmKjRrN2pNLzJRISM9NyQvSSJhR0YoJCIwVTxqIilIai4jISM5L0kiYkdGKCQiMFxtMUI0WyslRjc3IzckRi83JC9GNF00MDAwNEE2NzQ5Njc5RkNCL0Y5XTQwMTAwNEVDQjYwMUI5RDY=

where the first number in the output list is the minimum function value, and the sublist contains the minimizing values of the parameters LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn. Surprisingly, they are very close toLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiUtRiM2Jy1JI21uR0YkNiVRIjJGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RIixGJ0Y4RjsvJSZmZW5jZUdGOi8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYlUSI0RidGOEY7RjhGO0Y4RjtGK0Y4Rjs= used to generate the noisy obserations.

The optimum values of the parameters, namely

A := eval(a,R[2]); B := eval(b,R[2]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJBR0YoJCIwVTxqIilIai4jISM5NyNdNDAwMDRBNjc0OTY3OUZDQg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJCR0YoJCIwXG0xQjRbKyUhIzk3I100MDEwMDRFQ0I2MDFCOUQ2

cause the differential equation governing our model to become

qq := eval(DE, [a=A, b=B]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxcUdGKC8sKC1JJWRpZmZHRiY2JC1JInlHRig2I0kidEdGKC1JIiRHRiY2JEY2IiIjIiIiKiYtRjE2JEYzRjZGOy1JJGFic0dGJjYjRj1GOyQiMFU8aiIpSGouIyEjOUYzJCIwXG0xQjRbKyVGRCIiITcjLywoLUYxNiRGPUY2RjtGPF00MDAwNEE2NzQ5Njc5RkNCRjNdNDAxMDA0RUNCNjAxQjlENkZH

A numeric integration of this equation results in Figure 4 where the noisy observations, and two numeric solutions of the model can be seen. In green, we have the solution for the derived parameters, and in red, the solution for the original parameters,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiUtRiM2Jy1JI21uR0YkNiVRIjJGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RIixGJ0Y4RjsvJSZmZW5jZUdGOi8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNTYlUSI0RidGOEY7RjhGO0Y4RjtGK0Y4Rjs=.

Q := dsolve({qq, ic},y(t),numeric): p4 := odeplot(Q,[t,y(t)],0..10, numpoints=100, color=green, thickness=3): display([p4,p1,p3]);

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Figure 4 The noisy observations, best-fit solution (green) and unperturbed solution (red)

<Text-field style="Heading 1" layout="Heading 1">Example 2: Determining Chemical Rate Constants</Text-field>

In a recent on-line forum, estimating chemical rate-reaction constants LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn for the system governed by the initial value problem

q1 := diff(x(t),t) = -a*x(t)*y(t) + b*z(t); q2 := diff(y(t),t) = -a*x(t)*y(t) + b*z(t); q3 := diff(z(t),t) = a*x(t)*y(t) - b*z(t); ic := x(0)=10,y(0)=10,z(0)=0;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMUdGKC8tSSVkaWZmR0YmNiQtSSJ4R0YoNiNJInRHRihGNSwmKihJImFHRigiIiJGMkY5LUkieUdGKEY0RjkhIiIqJkkiYkdGKEY5LUkiekdGKEY0RjlGOTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMkdGKC8tSSVkaWZmR0YmNiQtSSJ5R0YoNiNJInRHRihGNSwmKihJImFHRigiIiItSSJ4R0YoRjRGOUYyRjkhIiIqJkkiYkdGKEY5LUkiekdGKEY0RjlGOTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxM0dGKC8tSSVkaWZmR0YmNiQtSSJ6R0YoNiNJInRHRihGNSwmKihJImFHRigiIiItSSJ4R0YoRjRGOS1JInlHRihGNEY5RjkqJkkiYkdGKEY5RjJGOSEiIjcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNpY0dGKDYlLy1JInhHRig2IyIiISIjNS8tSSJ5R0YoRjJGNC8tSSJ6R0YoRjJGMzclRi9GNUY4

was proposed. As observed data, the 26 equispaced measurements of 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, captured in the vector

C := <0,4.981119825867072,6.619863704639184,7.424230767960948,7.895291258366989,8.200043893374819,8.410083696006469,8.561234239001111,8.673418582102649,8.75860469142499,8.824418203860477,8.875947188583103,8.916714710357871,8.949234971834077,8.975347266499856,8.996425002439704,9.01351139331151,9.027409882146703,9.038747104014094,9.048016122443334,9.055608432249834,9.061836841504377,9.066952814916256,9.071159284138764,9.074620916958418,9.07747159947345>:

were provided. Figure 5 displays these observations as black circles.

N := Dimension(C)-1: Pts := [seq([k-1,C[k]],k=1..N+1)]: p5 := plot(Pts, style=point, symbol=circle, color=black, labels=[t,"z"]): p5;

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Ig==

Figure 5 Equispaced observations of 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

Dr. Allan Wittkopf of Maplesoft showed that because of implied conservation laws, the given system can be reduced to the initial value problem

DE1 := diff(z(t),t)=a*(10-z(t))^2-b*z(t); IC := z(0) = 0;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRERTFHRigvLUklZGlmZkdGJjYkLUkiekdGKDYjSSJ0R0YoRjUsJiomSSJhR0YoIiIiLCYiIzVGOUYyISIiIiIjRjkqJkkiYkdGKEY5RjJGOUY8NyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNJQ0dGKC8tSSJ6R0YoNiMiIiFGMjcjRi4=

In fact, he showed that, in general, the conservation laws

C1 := x(t)+z(t)=x[0]+z[0]; C2 := y(t)+z(t)=y[0]+z[0];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNDMUdGKC8sJi1JInhHRig2I0kidEdGKCIiIi1JInpHRihGMkY0LCYmRjE2IyIiIUY0JkY2RjlGNDcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNDMkdGKC8sJi1JInlHRig2I0kidEdGKCIiIi1JInpHRihGMkY0LCYmRjE2IyIiIUY0JkY2RjlGNDcjRi4=

lead to the single equation

temp := rifsimp([q1,q2,q3,C1,C2],[x,y,z]): map(s -> collect(s,{a,b},factor), temp[Solved][1]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUklZGlmZkdGJjYkLUkiekdGKDYjSSJ0R0YoRjMsJiooLChGMCIiIiZGMTYjIiIhISIiJkkieUdGKEY5RjtGNywoRjBGN0Y4RjsmSSJ4R0YoRjlGO0Y3SSJhR0YoRjdGNyomSSJiR0YoRjdGMEY3RjtGKw==

Hence, the exact solution for 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 is

Zt := rhs(dsolve({DE1,IC},z(t)));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNadEdGKCwkKiYsKComLUkldGFuaEdGJTYjLCYqJkkidEdGKCIiIiomSSJiR0YoRjgsJkkiYUdGKCIjU0Y6RjhGOCNGOCIiI0Y+LUkoYXJjdGFuaEdGJTYjKiYsJkY8IiM/RjpGOEY4RjkjISIiRj9GOEY4RjlGPkY4RjwhIz9GOkZHRjhGPEZHRkY3I0Yu

Unfortunately, for positive a and b, the argument of the arctanh function is greater than 1, and hence, the value of the function is complex. This leads to complications with the fitting command next employed, so we modify this solution as follows.

Ztr := simplify(evalc(Zt)) assuming a>0, b>0;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRadHJHRigsJCooLCYqJiwmSSJhR0YoIiM/SSJiR0YoIiIiRjYtSSVzaW5oR0YlNiMsKCooSSJ0R0YoRjZGNSNGNiIiIywmRjMiI1NGNUY2Rj1GPS1JI2xuR0YlNiMsKEYzRjRGNUY2KiZGNUY9Rj9GPUY2Rj0tRkI2IywoRjNGNEY1RjZGRSEiIiNGSUY+RjZGNiooLUklY29zaEdGJUY5RjZGNUY9Rj9GPUZJRjZGM0ZJRjdGSUY9NyNGLg==

This function can be fit to the given observations via the NonlinearFit command from the Statistics package. In particular, it provides

Params := NonlinearFit(Ztr,<seq(k,k=0..N)>, C, [t], parameternames=[a,b], output=parametervalues, parameterranges=[a=0..infinity, b=0..infinity]): A := eval(a, Params): B := eval(b, Params): a = A; b = B;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvSSJhR0YoJCIwckxyMysrKyIhIzo3Iy9GLV0zRkI5OTk5OUJGMDYwRDYx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvSSJiR0YoJCIwMi0kKWY0NDQqISM8NyMvRi1dM0Y4MjlFNDEzQjUzMjYwNw==

as estimates of the rate constants LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn. Using these values, we obtain Figure 6, a graph of the numeric solution of the original IVP, along with the observed values of 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.

M := dsolve(eval({q1,q2,q3, ic},[a=.1,b=.00909090090909]), {x(t),y(t),z(t)},numeric): p6 := odeplot(M,[t,z(t)],0..25): display([p5,p6]);

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Ig==

Figure 6 Observations and numeric solution for 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

The parameter-estimation problem for differential equations is an important component of the modeling effort prevalent in the applied sciences. In particular, we point to the extensive work of the control-theory specialist H. T. Banks of North Carolina State University.

Parameters in a system of differential equations that can only be solved numerically can be estimated by the same technique used in Example 1. For each pair of parameteter valuesLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiUtRiM2Jy1GLDYmUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGOi9GPlEnbm9ybWFsRicvJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYmUSJiRidGN0Y6Rj1GOkZERjpGREYrRjpGRA==, the function

SS := proc(a,b) local F, V; if not type([a,b],['numeric','numeric']) then return 'SS'(a,b); elif a<0 or b<0 then return 1e100; end if; F := dsolve(eval({q1,q2,q3,ic},{:-a=a,:-b=b}),[x(t),y(t),z(t)],numeric, output=Array([seq(k,k=0..N)])); V := convert(Column(F[2,1],4),Vector); Norm(V-C,2); end proc:

integrates the given initial value problem numerically, then computes the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JlEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JVEiMkYnRjgvRjxRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGK0Y4RkI= norm of the difference between the vectors of computed and observed values of 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. Hence, 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, from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JVEiMkYnRjgvRjxRJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRjhGQg== to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn, represents a sum of squares of deviations.

Seen in Figure 7, the surface determined by 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 suggests the existence of at least one minimum.

plot3d(SS, .0995..0.1005, .009..0.0092, view=0..0.025, axes=box, labels=[a,b,"SS"], orientation=[-80,60]);

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Figure 7 Surface determined by the function 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

To find the minimum shown in Figure 7, we again invoke the Nelder-Meade (simplex) algorithm via the NLPSolve command, and obtain

params := NLPSolve(SS(a,b), method=nonlinearsimplex, initialpoint=[a=.2, b=.1],evaluationlimit=200): An := eval(a, params[2]): Bn := eval(b, params[2]): a = An; b = Bn;

L0kiYUc2IiQiMyVIIio9eEIrKysiISM9

L0kiYkc2IiQiMyFIWiEzYVwiNDQqISM/

as estimates for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn, with

params[1];

JCIzIyoqZnkleSVmPHkmISND

as the corresponding minimum value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEjU1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YyL0Y2USdub3JtYWxGJw==. By way of comparison, the values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn computed from the exact solution for 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 differ from these new values by

An-A; Bn-B;

JCIkXiIhIzU=

JCIlZmAhIzc=

respectively, and the minimum of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEjU1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YyL0Y2USdub3JtYWxGJw== at the former point is

SS(A,B);

JCIzbl5pLCd5UUdnKiEjQw==

a number marginally higher than that determined by our numeric method.