A new method for one- and higher-dimensional linear and nonlinear tempered Caputo fractional diffusion-type equations

Purpose The objectives of this study are threefold: (1) to introduce the tempered Chebyshev wavelet (TCW), (2) to propose the TCW method for solving linear tempered Caputo fractional diffusion-type equations and (3) to develop a fast tempered Chebyshev wavelet (fTCW) method for solving one- and two-dimensional nonlinear tempered Caputo fractional diffusion-type equations. Design/methodology/approach The fTCW method integrates the TCW method with L2−1σ and sum-of-exponentials approximations of the tempered Caputo fractional time derivative. For this purpose, we introduce the TCW and derive its new operational matrices for tempered fractional integration by using the hypergeometric function. For nonlinear problems, we employ the interpolation technique in conjunction with operational matrices and fast approximations. The efficiency of the fTCW method is demonstrated through comparisons with the exact solution and the solution obtained using the TCW method. Findings We have derived the TCW operational matrix of tempered fractional integration and the TCW operational matrix of tempered fractional integration for boundary value problems. These matrices, in conjunction with the fast approximations of the tempered Caputo fractional time derivative and the interpolation technique, form the basis for the construction of the fTCW method. We present a detailed methodology for solving linear tempered Caputo fractional equations using the TCW method. Additionally, we provide a comprehensive methodology for solving both one- and two-dimensional nonlinear tempered Caputo fractional diffusion-type equations using the fTCW method. The convergence and error analysis of both methods are thoroughly discussed. Numerical simulations are presented to validate and illustrate the theoretical results. These simulations demonstrate the effectiveness and accuracy of the proposed methods by solving three test problems, including one linear and two nonlinear tempered Caputo fractional diffusion-type equations. The results are compared with analytical solutions or with each other to highlight the efficiency and precision of the TCW and fTCW methods. Originality/value Many engineers and scientists can utilize the presented methods for solving their linear and nonlinear tempered Caputo fractional diffusion-type models.