Chebyshev Petrov-Galerkin procedure for the time-fractional heat equation with nonlocal conditions

Abstract

In this research paper, we address the time-fractional heat conduction equation in one spatial dimension, subject to nonlocal conditions in the temporal domain. To tackle this challenging problem, we propose a novel numerical approach, the ‘Rectified Chebyshev Petrov-Galerkin Procedure,’ which extends the classical Petrov-Galerkin method to efficiently handle the fractional temporal derivatives involved. Our method is characterized by several key contributions; We introduce a set of basis functions that inherently satisfy the homogeneous boundary conditions of the problem, simplifying the numerical treatment. Through careful mathematical derivations, we provide explicit expressions for the matrices involved in the Petrov-Galerkin method. These matrices are shown to be efficiently invertible, leading to a computationally tractable scheme. A comprehensive convergence analysis is presented, ensuring the reliability and accuracy of our approach. We demonstrate that our method converges to the true solution as the spatial and temporal discretization parameters are refined. The proposed Rectified Chebyshev Petrov-Galerkin Procedure is found to be robust, and capable of handling a wide range of problems with nonlocal temporal conditions. To illustrate the effectiveness of our method, we provide a series of numerical examples, including comparisons with existing techniques. These examples showcase the superiority of our approach in terms of accuracy and computational efficiency.