Investigation of an Exact Solution of a Mixed Boundary Value Problem Using the Residue Method

Year 2024, Volume: 2 Issue: 2, 98 - 105, 27.09.2024

Bahaddin Sinsoysal , Mahir A. Rasulov

Abstract

In this study, a finite differences method is proposed for solving a mixed problem, which represents phenomena in hydrodynamics. To evaluate the approximate solution of the problem, its analytical solution is also constructed by the residue method developed by M.L. Rasulov, which is applied to find the solution of the partial differential equations containing time-dependent derivatives at boundary conditions. The formula for expansion of an arbitrary function in a series of residues of the solution of the corresponding spectral problem is used to show that the solution of the mixed boundary-value problem can be represented by the given residue formula. The use of the residual method gives an exact solution for the mixed boundary value problem, represented as a rapidly decreasing series. The derived formula makes it possible to formally establish both the existence and the uniqueness of the solution. Moreover, the derived formula provides a framework for evaluation, allowing a comparative analysis between the exact solution and the numerical approach.

Keywords

Residue method, residue representation of the solution, expansion formula, boundary condition with higher order derivative

Ethical Statement

The authors have no relevant financial or non-financial interests to disclose.

Supporting Institution

No funding was received for conducting this study. No funds, grants, or other support was received. All authors did not receive support from any organization for the submitted work.

Thanks

We would like to thank the referees and editor for reviewing the paper carefully and their valuable comments to improve the quality of the paper.

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