Is this a correct way of 3D graphing an equation with summation

Question

Alif 2024-1-7

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2067186-is-this-a-correct-way-of-3d-graphing-an-equation-with-summation

评论： Torsten 2024-1-7

x = 0:1:100;
t = 0:1:100;
[X,T] = meshgrid(x,t);
syms n;
u =  X + 20 + symsum((20./n.*pi).*(4+5.*-1.^n).*exp(-n^2.*pi.^2.*T./900).*sin(n.*pi.*X./30),n,1,10)
surf(X,T,u)
hold on
grid on

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Dyuman Joshi 2024-1-7

I suspect because the original sum has the upper limit of Infinity. And as an example, OP might have chosen to go upto 10 here.

Calculating the numerical "sum" for a larger value of upper limit will be computationally expensive and might be inaccurate, as sum() is unstable.

Torsten 2024-1-7

But you convert the symbolic sum to a function handle and evaluate this function handle. So what is different in your approach from just using "sum" from the very beginning ?

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Answer 1

Dyuman Joshi 2024-1-7

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2067186-is-this-a-correct-way-of-3d-graphing-an-equation-with-summation#answer_1384676

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It is better to let the expression be defined as a symbolic expression through-out calculations, then convert it to an anonymous functions and evaluate at last.

Converting to anonymous function and evaluating is much faster compared to evaluating the symbolic function/expression and converting to double.

%Declare symbolic variables

syms n x t

%Define the general term

term = (20/(n*pi)*(4+5*-1^n)*exp(-n^2*pi^2*t/900)*sin(n*pi*x/30));

%Define points

[X,T] = meshgrid(0:100);

%Call symsum(), u will be obtained as a function of x and t

u = 20 + x + symsum(term,n,1,10);

%Convert to anonymous function, specifying the order of variables

u = matlabFunction(u, 'Vars', {x t})

u = function_handle with value:

@(x,t)x-(exp(t.*(-1.096622711232151)).*sin((x.*pi)./3.0).*2.0)./pi-(exp(t.*(-2.741556778080377e-1)).*sin((x.*pi)./6.0).*4.0)./pi-(exp(t.*(-3.947841760435743e-1)).*sin((x.*pi)./5.0).*(1.0e+1./3.0))./pi-(exp(t.*(-9.869604401089358e-2)).*sin((x.*pi)./1.0e+1).*(2.0e+1./3.0))./pi-(exp(t.*(-4.386490844928604e-2)).*sin((x.*pi)./1.5e+1).*1.0e+1)./pi-(exp(t.*(-1.754596337971441e-1)).*sin(x.*pi.*(2.0./1.5e+1)).*5.0)./pi-(exp(t.*(-7.018385351885766e-1)).*sin(x.*pi.*(4.0./1.5e+1)).*(5.0./2.0))./pi-(exp(t.*(-1.096622711232151e-2)).*sin((x.*pi)./3.0e+1).*2.0e+1)./pi-(exp(t.*(-8.882643960980422e-1)).*sin(x.*pi.*(3.0./1.0e+1)).*(2.0e+1./9.0))./pi-(exp(t.*(-5.373451285037539e-1)).*sin(x.*pi.*(7.0./3.0e+1)).*(2.0e+1./7.0))./pi+2.0e+1

%Evaluate u at X, T

U = u(X, T);

%plot a 3D surface via surf()

surf(X,T,U)

hold on

grid on