A recursive numerical algorithm to computing Fourier series coefficients to find cylinder potential in electrodynamics

IIn this article, a new approach to find the Fourier expansion coefficients was carried out by a recursive algorithm, without computing their correspondent integral. Furthermore, in virtue of this new method, some partial differential equations were solved and compared with their exact solutions. After deriving the recursive relation, some differential equations along with the partial differentiation were solved and also compared with the numerical answers. The Fourier series coefficients were computed more accurately and swiftly with this method as compared to others and then cylinder potential in electrodynamics was calculated by this method. The results show that the algorithm proposed in this paper has achieved more optimal results. As we know, the cube problem, whose potential at its levels is defined as definite, also leads to a three-dimensional Fourier series, which can be used to obtain the potential inside it. Using these calculations, in the future we can further investigate the problems in the areas of ion capture inside the cylinder and the cube. Naturally, due to the use of the Fourier series in solving many physical issues, this technique can be used to improve computations in other sciences such as civil engineering and mechanics in such topics as vibrations and thermodynamics.