System of nonlinear integral equations

In summary, a system of nonlinear integral equations is a set of equations that involve both unknown functions and integrals, making them difficult to solve using traditional algebraic methods. They are commonly found in fields such as physics, engineering, and economics. These equations have many real-world applications, such as in fluid dynamics, heat transfer, and economics. To solve them, iterative methods such as the Picard and Newton iteration methods are used, but challenges include finding a suitable starting point, slow convergence, and high computational cost.
  • #1
jannyhuggy
5
0
Hello everybody!

While solving some physical problem I got stuck with some system of integral equations.
The problem is formulated in .pdf file below.

I will be over-satisfied with the following
1) to know whether and why this system has/doesn't have a solution
2) how it could be produced approximately
3) if there are any methods for exact solution of equations of such a type, perhaps with other V(k)

View attachment 11111.pdf

ANY comments are highly welcome!
 
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  • #2
I tried two iteration methods but perturbations seem to be divergent. Perhaps any regularization ideas?
 

What is a system of nonlinear integral equations?

A system of nonlinear integral equations is a set of equations that involve both unknown functions and integrals. These equations cannot be solved using traditional algebraic methods, and they often arise in fields such as physics, engineering, and economics.

What makes a system of nonlinear integral equations difficult to solve?

The presence of both unknown functions and integrals makes these equations difficult to solve. Additionally, the nonlinear nature of the equations means that they do not have a simple, linear relationship between their variables. This makes it challenging to find a closed-form solution, and numerical methods must be used instead.

What are some applications of systems of nonlinear integral equations?

Systems of nonlinear integral equations have many real-world applications, including the study of fluid dynamics, heat transfer, electromagnetic fields, and population dynamics. They are also used in economics to model complex systems such as market equilibrium and decision-making processes.

What are some methods for solving systems of nonlinear integral equations?

There are several methods for solving systems of nonlinear integral equations, including the Picard iteration method, the Newton iteration method, and the quadrature method. These methods involve iterative processes and numerical approximations to find a solution.

What are some challenges in solving systems of nonlinear integral equations?

One of the main challenges in solving these equations is finding a suitable starting point for the iterative methods. The convergence of these methods can also be slow or even fail in some cases. Additionally, the computational cost can be high, especially for systems with a large number of equations.

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