Solving integral equation with double ingegrals and singularities

In summary, the person is asking for help with solving an integral equation involving a complex number and a function. They have provided a code they have tried but are encountering some singularities. The expert suggests a brute-force search approach in the complex plane to find the solution.
  • #1
1
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Hello
I need help to solve the following integral equation:
f(x,y,w)=137.03.*y.^2./((0.238.*exp(0.067.*y.^2)+1).*(w-5.26.*x.*y-2.63).*(w-5.26.*x.*y+2.63))=1+8478./(10828-w.^2-1.13.*j.*w)
xmin=-1, xmax.=1, ymin=0, ymax=inf (nad can be taken 500 because the function decreases rapidly)
I want to find w which is complex number.
I tried the following code:
f=@(w) dblquad(@(x,y) 137.03.*y.^2./((0.238.*exp(0.067.*y.^2)+1).*(w-5.26.*y.*x-2.63).*(w-5.26.*y.*x+2.63)),-1,1,0,100)-1-8478./(10828-w.^2-j.*w.*1.13)
V=fsolve(f,160)
There are some singularities. How can I solve the problem
 
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  • #2
Looks like you have:

[tex]g(w)=\int\int f(w,x,y)dxdy[/tex]

for complex w. Then unless someone can suggest an analytical approach, I would recommend a brute-force search in the complex plane for w. Start with a circle about the origin. Say 100 points around it. Compute both sides. Find the minimum absolute value of the difference between the RHS and LHS. That the current point. Now take a small circle around that point, another 100 points, find the minimum again. Keep doing this and I think the trajectory will lead to the solution unless some problems are encountered.
 

1. What is an integral equation?

An integral equation is a mathematical equation that involves an unknown function and an integral. It is used to describe physical phenomena in which the desired function is defined as the integral of another function.

2. What is the purpose of solving integral equations with double integrals and singularities?

The purpose of solving integral equations with double integrals and singularities is to accurately model and predict the behavior of physical systems that have complex and non-linear relationships. These types of equations are commonly used in physics, engineering, and other scientific fields.

3. What are singularities in terms of integral equations?

Singularities refer to points in an integral equation where the function being integrated becomes infinite or undefined. These points are often critical in solving the equation and require special techniques to handle.

4. How do you handle singularities when solving integral equations with double integrals?

There are several methods for handling singularities in integral equations with double integrals. One approach is to transform the equation into a different form that does not have singularities, such as using a change of variables. Another method is to use numerical techniques, such as Monte Carlo integration, to approximate the solution.

5. What are some applications of solving integral equations with double integrals and singularities?

Solving integral equations with double integrals and singularities is used in a wide range of applications, including image processing, signal analysis, and computational physics. It is also commonly used in solving boundary value problems in differential equations and in solving inverse problems in various fields of science and engineering.

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