Work

An Orthogonal Polynomial Solution to the Confluent-Type Heun’s Differential Equation

Year: 2025

Type: article

Abstract: In this work, we present both analytical and numerical solutions to a seven-parameter confluent Heun-type differential equation. This second-order linear differential equation features three singularities: two regular singularities and one irregular singularity at infinity. First, employing the tridiagonal representation method (TRA), we derive an analytical solution expressed in terms of Jacobi polynomials. The expansion coefficients of the series are determined as solutions to a three-term recurrence relation, which is satisfied by a modified form of continuous Hahn orthogonal polynomials. Second, we develop a numerical scheme based on the basis functions used in the TRA procedure, enabling the numerical solution of the seven-parameter confluent Heun-type differential equation. Through numerical experiments, we demonstrate the robustness of this approach near singularities and establish its superiority over the finite difference method.

Source: Mathematics

DOI: 10.3390/math13081233

Authors: Saiful R. Mondal, Varun Kumar

Institutions: King Faisal University, Alliance University

Language: English

Topic: Mathematical functions and polynomials

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funder: Deanship of Scientific Research, King Faisal University