Dear Prof.Dr.

Gregorio Josué Martínez Sánchez

I think that you can follow the links

http://www.cfm.brown.edu/people/dobrush/am34/Mathematica/ch6/laplacem.html

https://cyberleninka.ru/article/n/the-solution-of-a-mixed-boundary-value-problem-for-the-laplace-equation-in-a-multiply-connected-domain

Conference Paper Numerical solution for Laplace equation with mixed boundary ...

Article Mixed boundary value problem of Laplace equation in a bounde...

https://www.researchgate.net/post/How_to_solve_laplace_equation_on_rectangle_domain_with_complex_mixed_boundary_conditions

with best wishes

Dear Dr A. M. Abdallah

Thank you for your answer.

I will look into it.

Is this what you mean? It is Laplace's equation solved using the Boundary Element Method with some constant values, zero normals, and zero tangentials. You can easily change the boundary conditions (see annulus.bem in the zip file). I cover this in my book, Differential Equations, which is free on 9/3 (today), 9/11, and 9/19. http://www.amazon.com/dp/B07DB9J2TK I also cover it in my book, Numerical Calculus, which is free on 9/15, 9/23, 10/1, 10/9, and 10/17. You can get the source code for the solver free anytime here: http://dudleybenton.altervista.org/software/index.html

The FEM code is in there also plus a free grid generation tool. I created, solved, and plotted both problems (boundary and triangular elements) in five minutes using numerical methods. You could use numerical integration (e.g., Gauss Quadrature) to find the Fourier series coefficients for the analytical solution, but that would seem a circuitous path. You could also check the form of your Fourier series for proper orthogonality over the domain using the same numerical means. The basic question is: Do you want to solve a problem or do you want to illustrate a particular solution technique?

Dear Dr. Dudley J Benton Initially, an analytical approach with the Fourier series expansion was that I was looking for in my problem. Reading in forums, this idea could be quite complex. So the numerical methods may be a good choice.

I'm interested in your software, I'll looking for on your website and your books, appreciate your suggestion.

Regards,

Do you see what I was suggesting about the Fourier series solution? The terms of a Fourier series are orthogonal over its domain. If you were to express these in terms of a matrix (for example least-squares regression), the off-diagonal elements will be zero. That's why you can write an expression for each term as an integration over the domain. With only diagonal elements, you don't need to perform a matrix inversion to solve the problem. You must come up with expressions that are orthogonal over the annular domain in order to obtain a series solution. Even if it's not practical to perform the integrals analytically with the complex boundary conditions, you could perform these numerically and thereby obtain the analytical solution in terms of an infinite series. You may find that your domain requires Bessel functions, which would be even more difficult to work with analytically, but not so numerically. I explain these principles in yet another book, The Many Uses of Orthogonal Functions. I think finding an analytical series solution to this problem would make a fascinating project. Please keep us informed as to your progress and let me know if I can help in any way.

In general, there is no closed-form solution of this problem. If you are interested in an analytical approach, then you should represent a solution in the form of a Fourier series and to satisfy the boundary condition on the inner circle. After this step, insert a series obtained in the conditions on the outer boundary and reformulate the problem as singular integral equations that should be solved numerically by various well-known methods. Maybe, there are analytical solutions in some special cases -it's dependent on geometry of the problem that isn't given accurately in your question.