Efficient Algorithms for the Solution of Linear and Nonlinear Caputo-Tempered Variable Order Partial Differential Equations

Abstract

This study introduces a new wavelet framework, referred to as the tempered fractional Gegenbauer wavelet (TFGW), for the numerical solution of tempered variable-order differential equations. We construct the TFGW operational matrices for tempered variable-order integration and develop the TFGW method to efficiently solve Caputo-tempered variable-order ordinary and boundary value problems. To further enhance computational efficiency for partial differential equations involving both time-fractional and spatial variableorder derivatives, we propose the L1-TFGW method based on the L1 approximation and the fast TFGW method, which incorporates a fast algorithm for fractional time derivatives in combination with the TFGW approach for spatial operators. For nonlinear problems, a fast-quasi TFGW method is devised by coupling quasilinearization with the fast TFGW strategy. The orthonormality of TFGW is established, and corresponding operational matrices are derived, including those tailored for boundary value problems. Error analyses are provided, and extensive numerical simulations demonstrate the accuracy, efficiency, and robustness of the proposed methods. The results confirm that the TFGW-based techniques offer a reliable and effective computational framework for linear and nonlinear Caputo-tempered variable-order models. To the best of our knowledge, this is the first work to introduce such wavelet-based fast algorithms for tempered fractional and spatial variable-order differential equations, providing a valuable tool for scientists and engineers dealing with complex multiscale fractional dynamics.