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

The primary objective of this study is to introduce a novel wavelet, termed as the tempered fractional Gegen-bauer wavelet (TFGW), aimed at solving tempered variable order differential equations. Additionally, the study seeks to develop the L1- TFGW and the fast TFGW technique for numerically solving Caputo-tempered partial differential equations featuring both time fractional and spatial variable order derivatives. Lastly, the study aims to devise the fast-quasi TFGW method for handling nonlinear Caputo-tempered partial differential equations with time fractional and spatial variable order derivatives. We have introduced a method, the TFGW method, for the solution of Caputo-tempered variable order ordinary differential equations. For the method, we have constructed the TFGW operational matrices of tempered variable order integration. The purpose of utilizing the operational matrices is to reduce the computational cost of the TFGW method. We proposed the L1- TFGW and the fast TFGW method by utilizing the L1 approximation and the fast algorithm for time fractional derivative, respectively, and the TFGW method for space variable order derivatives, for the numerical solutions of partial differential equations with time fractional and space variable order derivatives. The aim of proposing these methods are to increase the efficiency of the method and to further reduce the computational cost of the TFGW method. For nonlinear Caputo-tempered variable order partial differential equations, we have proposed the fast-quasi TFGW method that utilizes the quasilinearization technique alongside the fast TFGW method. We have focused on establishing the orthonormality of the TFGW and have developed several key components: the TFGW matrix, the TFGW operational matrix of tempered variable order integration and the TFGW operational matrix of tempered variable order integration for boundary value problem. We develop the methodology of the TFGW method for solving Caputo-tempered variable order boundary value problems with variable coefficients. Additionally, we establish the methodology of the L1- TFGW and the fast TFGW method for numerically solving both linear and nonlinear partial differential equations with time fractional and spatial variable order derivatives. Furthermore, we conducted an error analysis for the presented methods. Numerical simulations are presented to illustrate the reliability, efficiency, and accuracy of the TFGW method, the fast TFGW method and the fastquasi TFGW method. Theoretical analysis is corroborated by numerical findings. Engineers and scientists can make use of the insights provided in this study to rapidly evaluate their tempered variable order differential models. To the author’s knowledge, this current study is novel and has not been previously proposed or applied to the numerical solution of linear and nonlinear partial differential equations involving tempered time fractional and spatial variable order derivatives.