MATLAB Answers

Consecutive nested integration (definite/indefinite)

8 views (last 30 days)
abouzar jafari
abouzar jafari on 28 Jan 2020
Answered: Alan Weiss on 29 Jan 2020
I am going to calculate a consecutive integration problem as I attached the picture here. I have tried to solve the integration using Mathematica and MATLAB even for the first two terms (F2, G2), but I did not find any meaningful results. Since the problem was solved for the first time in 1960 using a very simple computer, it can be solved using numerical integration. Actually, I have tried to do it using the code that I uploaded here, but the results obtained from the code does not match the solution. I hope someone can debug my code or give me some hints.
code:
clear all;
clc;
i = 1;
F{1} = @(x)(x./((1+x.^2).^2));
G{1} = @(x)((x.^3)./((1+x.^2).^2));
while i <= 20
i = i+1;
if mod(i, 2) == 0
% x is even [Sigma_y, x = 0]
FFny = @(x,y)(((y.*x.^2)./((y.^2+x.^2).^2)).*F{i-1}(x));
FGny = @(x,y)(((y.^3)./((y.^2+x.^2).^2)).*F{i-1}(x));
F{i} = @(y)integral(@(x)FFny(y,x),0,Inf);
G{i} = @(y)integral(@(x)FGny(y,x),0,Inf);
else
% x is odd [Sigma_x, y = 0]
FFnx = @(x,y)(((x.*y.^2)./((y.^2+x.^2).^2)).*F{i-1}(y));
FGnx = @(x,y)(((x.^3)./((y.^2+x.^2).^2)).*F{i-1}(y));
F{i} = @(x)integral(@(y)FFnx(y,x),0,Inf);
G{i} = @(x)integral(@(y)FGnx(y,x),0,Inf);
end
end

Answers (1)

Alan Weiss
Alan Weiss on 29 Jan 2020
I would try to answer this question by not integrating from 0 to infinity, but instead from 0 to a large value, and then use an asymptotic expression for the remainder of the integration (from the large value to infinity). Some of these integrals converge very quickly, some not, so I suspect that a bit of pre-analysis before trying numerical integration would be very helpful.
Alan Weiss
MATLAB mathematical toolbox documentation

Products


Release

R2018a

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Translated by