A NEW NUMERICAL TECHNIQUE FOR INVESTIGATING BOUNDARY VALUE PROBLEMS WITH Ψ-CAPUTO FRACTIONAL OPERATOR

Abstract

This article introduces a new numerical approach for solving linear and non-linear boundary value problems for Ψ-fractional differential equations (Ψ-FDEs). This approach relies on the Ψ-Haar wavelet operational integration matrices. The Ψ-operational matrices (Ψ-OMs) are used to convert the Ψ-FDE to an algebraic system of equations. The non-linear fractional boundary value problems are first linearized using the quasi-linearization technique, and then the Ψ-Haar wavelet technique is applied to the linearized problem. The solution is updated by the Ψ-Haar wavelet method in each iteration of the quasi-linearization technique. The proposed method is a good and simple mathematical technique for numerically solving non-linear Ψ-FDEs. The operational matrix (OM) method is computationally more efficient. Several linear and non-linear boundary value problems are discussed to demonstrate the applicability, efficiency, and simplicity of the method. Moreover, the error analysis is carried out resulting a rigorous error bound for the proposed method.