Solution of Fractional Differential Equations Involving Hilfer-Hadamard Fractional Derivatives

Abstract

Objectives: The aim is to establish prerequisite properties for the Hilfer-Hadamard fractional derivatives and address boundary value problems related to fractional polar Laplace and fractional Sturm-Liouville equations involving Hilfer-Hadamard fractional derivatives. Methods: Existing definitions and findings are utilized to obtain the properties for fractional derivatives, and the Adomian decomposition method is employed to solve the fractional differential equations. Findings: Validity conditions for the law of exponents are determined, and the study investigates the fractional differential equations and their corresponding solutions, possessing the capacity to replace the traditional polar Laplace and Sturm-Liouville boundary value problems to effectively represent real-world phenomena. Novelty: The study introduces the substitution of two consecutively operated Hilfer-Hadamard fractional derivatives with a corresponding single Hilfer-Hadamard fractional derivative using the law of exponents. Additionally, the polar Laplace and Sturm-Liouville boundary value problems are extended to their respective fractional counterparts, expressed in a concise format using HilferHadamard fractional derivatives. Keywords: Adomian decomposition method, Hilfer-Hadamard fractional derivative, Fractional polar Laplace equation, Fractional Sturm-Liouville boundary value problem