December 2, 2025Open Access

An Explicit Shifted Legendre Petrov–Galerkin Technique for the Time Fractional Cable Problem

Key Points

The numerical technique achieves stability in solving the time fractional cable problem with Petrov–Galerkin framework.
Error bounds indicate effective truncation estimation when applying inequalities on shifted Legendre polynomials.
Numerical experiments validate the approach through comparisons against established numerical techniques in literature.
This method may enable new understandings in fractional differential equations and their applications.

Abstract

This paper focuses on analyzing and implementing a numerical technique using the Petrov–Galerkin technique (PGT) to solve the time fractional cable problem (TFCP). The trial functions are a modified set of shifted Legendre polynomials (LPs). An appropriate numerical approach can be used to solve the linear algebraic equations resulting from the application of the PGT. With error bounds, we discuss the truncation estimation and stability in the L2 norm. We apply some inequalities on the modified set of shifted LPs to this research. Numerical experiments include benchmark issues for which exact solutions are presented to show how efficient and accurate the method is. Comparisons with different techniques in the literature are used to support our examples.

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