How to calculate a composite triple integral with exponential function?

Question

Drawing 2024-4-11，11:05

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2105596-how-to-calculate-a-composite-triple-integral-with-exponential-function

评论： Drawing 2024-4-11，13:19

采纳的回答： Torsten

Hello! I have a composite triple integral, but i cant find a proper approach to calculate it.

,where the variables are "r","s","l".

r0 is a known constant.

I try to use integral3, but the exp() and the variable 'l' for the lower bound of the integral makes it hard to calculate.

Is there any other matlab function that is suitable to solve this composite triple integral? I would appreciate it if anyone could give me some advice.

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Drawing 2024-4-11，12:49

在 MATLAB Online 中打开

Another try is with "integral" funcion

r0=100;
int_r=@(s,d00) integral(@(r) 1+s.*r.^-2, d00, 100);
int_out=integral2(@(s,l) exp(-s)./s.*exp(int_r).*(1+s.*l.^-2).^-1,0,+Inf,0,r0);
Incorrect number or types of inputs or outputs for function exp.

Error in solution>@(s,l)exp(-s)./s.*exp(int_r).*(1+s.*l.^-2).^-1 (line 3)
int_out=integral2(@(s,l) exp(-s)./s.*exp(int_r).*(1+s.*l.^-2).^-1,0,+Inf,0,r0);

Error in integral2Calc>@(y)fun(xi*ones(size(y)),y) (line 18)
    @(y)fun(xi*ones(size(y)),y),y1i,y2i,opstruct.integralOptions), ...

Error in integralCalc>iterateScalarValued (line 334)
        fx = FUN(t);

Error in integralCalc>vadapt (line 148)
    [q,errbnd] = iterateScalarValued(u,tinterval,pathlen, ...

Error in integralCalc (line 77)
        [q,errbnd] = vadapt(vfunAB,interval, ...

Error in integral2Calc>@(xi,y1i,y2i)integralCalc(@(y)fun(xi*ones(size(y)),y),y1i,y2i,opstruct.integralOptions) (line 17)
innerintegral = @(x)arrayfun(@(xi,y1i,y2i)integralCalc( ...

Error in integral2Calc>@(x)arrayfun(@(xi,y1i,y2i)integralCalc(@(y)fun(xi*ones(size(y)),y),y1i,y2i,opstruct.integralOptions),x,ymin(x),ymax(x)) (line 17)
innerintegral = @(x)arrayfun(@(xi,y1i,y2i)integralCalc( ...

Error in integralCalc>iterateScalarValued (line 334)
        fx = FUN(t);

Error in integralCalc>vadapt (line 148)
    [q,errbnd] = iterateScalarValued(u,tinterval,pathlen, ...

Error in integralCalc (line 88)
        [q,errbnd] = vadapt(vfunA,interval, ...

Error in integral2Calc>integral2i (line 20)
[q,errbnd] = integralCalc(innerintegral,xmin,xmax,opstruct.integralOptions);

Error in integral2Calc (line 7)
    [q,errbnd] = integral2i(fun,xmin,xmax,ymin,ymax,optionstruct);

Error in integral2 (line 105)
    Q = integral2Calc(fun,xmin,xmax,yminfun,ymaxfun,opstruct);

Drawing 2024-4-11，12:52

Thank you Torsten! I have added that "r0 is a known constant."

Drawing 2024-4-11，12:53

Thank you Dyuman Joshi! I have added my tries, and i also have the Symbolic Math Toolbox.

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Answer 1

Torsten 2024-4-11，13:08

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2105596-how-to-calculate-a-composite-triple-integral-with-exponential-function#answer_1439691

编辑：Torsten 2024-4-11，13:15

在 MATLAB Online 中打开

Setting l, r and s to zero will cause problems because you divide by these variables.

But also changing

integral2(@(s,l)arrayfun(@(s,l)fun(s,l),s,l),0,Inf,0,r0)

to

integral2(@(s,l)arrayfun(@(s,l)fun(s,l),s,l),1e-4,Inf,1e-4,r0)

does not help.

Thus try to check theoretically whether your integral exists at all.

r0 = 100;
fun_inner = @(s,l)integral(@(r)1./(1+s./r.^2).^2.*r,l,r0)
fun_inner = function_handle with value:
    @(s,l)integral(@(r)1./(1+s./r.^2).^2.*r,l,r0)
fun = @(s,l)exp(-s)./s .*exp(fun_inner(s,l)).* 1./(1+s./l.^2)
fun = function_handle with value:
    @(s,l)exp(-s)./s.*exp(fun_inner(s,l)).*1./(1+s./l.^2)
value = integral2(@(s,l)arrayfun(@(s,l)fun(s,l),s,l),0,Inf,0,r0)
Warning: Inf or NaN value encountered.
Warning: The integration was unsuccessful.
value = NaN