A non-uniform approach to approximate the fractional Sturm–Liouville problem with generalized kernel

Abstract

Eigenvalues and eigenfunctions are of importance due to its significance in the system’s state and its associated energy. Eigenvalues represent the possible energy levels a system can possess when it is in a state with a clearly defined energy. Each eigenfunction reflects the system’s state when its energy aligns with the respective eigenvalue. This paper presents a numerical algorithm for estimating the energies (eigenvalues) of the fractional Sturm–Liouville problem (FSLP) formulated using Caputo fractional derivative with generalized kernel (CFDGK). Firstly, the well-posedness of the considered fractional Sturm–Liouville problem with generalized kernel (FSLPGK) is discussed. Further, we have divided the provided interval into non-uniform node points and varied the weight and scale functions to calculate the estimated eigenvalues and their associated eigenfunctions. The eigenvalues obtained are real, and the eigenfunctions are orthogonal to each other. Furthermore, we have found the bound for the solution and discussed the convergence order numerically. After this, utilizing the estimated energy and the eigenfunctions, an approximate solution is developed for the fractional diffusion model with a generalized kernel. © The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2025.