A method for solving Caputo–Hadamard fractional initial and boundary value problems

Abstract

In this article, we proposed a method by generalizing the classical CAS wavelets for the approximate solutions of nonlinear fractional Caputo–Hadamard initial and boundary value problems. We have generated the new operational matrices for the generalized CAS (gCAS) wavelets, and these matrices are successfully utilized for the solution of nonlinear Caputo–Hadamard fractional initial and boundary value problems. The method that we have proposed in the present study is the combination of gCAS wavelets (based on new operational matrices) and the quasilinearization technique. We have derived and constructed the gCAS wavelets, the gCAS wavelets operational matrix of Hadamard fractional integral of order , and the gCAS wavelets operational matrix of Hadamard fractional integral for boundary value problems. We have also derived the orthonormality condition for the gCAS wavelets. The purpose of these operational matrices is to make the calculations faster. Furthermore, we worked out the error analysis of the proposed method. We presented the procedure of implementation for both nonlinear Caputo–Hadamard fractional initial and boundary value problems. Numerical simulations are provided to illustrate the reliability and accuracy of the method. Since the Hadamard differential equation is a new and emerging field, many engineers can utilize the proposed method for the numerical simulations of their linear/nonlinear Caputo–Hadamard fractional differential models.