The objective of this study is to introduce a more efficient method than classical wavelet methods, for solving linear and nonlinear Caputo fractional variable-order diffusion-type equations.
We first construct the Haar wavelet operational matrices for Caputo variable-order integration. Next, we integrate these matrices with the L2 − 1σ approximations to solve linear Caputo fractional variable-order diffusion-type equations. For nonlinear problems, we employ the quasilinearization technique in conjunction with the operational matrices and L2 − 1σapproximations. The proposed method is named “the modified Haar wavelet (mHw) method.” The efficiency of the mHw method is demonstrated through comparisons with the exact solution and the solution obtained using the classical Haar wavelet method.
We have derived the Haar wavelet operational matrix of variable-order integration and the variable-order integration matrix of Haar wavelet for boundary value problems. These matrices, in conjunction with the L2 − 1σapproximations and the quasilinearization technique, form the basis for the construction of the mHw method. We also provide the theoretical analysis of the mHw method. Additionally, the mHw method is shown to be second-order accurate in both time and space domains. We also performed the comparison with the classical Haar wavelet method. Numerical simulations are presented to validate and illustrate the theoretical results.
Many engineers and scientists can utilize the presented method for solving their linear and nonlinear Caputo fractional variable-order models.
