In steady-state heat conduction through a sphere, when the temperature depends on the angle, the part of the solution that deals with the angle is usually described using Legendre polynomials. Rodrigue’s formula is presented in Leibniz form, and the solution to Legendre’s equation is obtained using the Laplace transform and the quadratic formula. As an alternative form derived from Legendre’s equation, Rodrigue’s formula provides a basis for this approach. In this study, the Laplace transform is utilized as the main tool for solving Legendre’s equation. The results obtained can also be achieved with other transforms. The research outcomes offer a more detailed solution compared to previous studies.

Legendre’s equation, quadratic formula, integral transform, spherical heat conduction.

Received: September 13, 2024; Accepted: November 22, 2024; Published: January 4, 2025

How to cite this article: Eunyoung Lim, Arjun K. Rathie, Sunyoung Yeun and Hwajoon Kim, Solution to Legendre’s equation using Laplace transform and the quadratic formula, JP Journal of Heat and Mass Transfer 38(1) (2025), 95-102. https://doi.org/10.17654/0973576325005

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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