TITLE:
Treating Fractional Boundary Value Problems via Inverse Fractional Operator and Decomposition Method
AUTHORS:
Mona Al-Sulami, Asmaa Al-Masoudi
KEYWORDS:
Fractional Derivatives, Fractional Inversion Operator, Decomposition Methods, Optimal Closed-Form Solutions, Initial-Boundary Value Problems
JOURNAL NAME:
Applied Mathematics,
Vol.17 No.8,
August
31,
2026
ABSTRACT: This paper aims to apply the standard Adomian decomposition method and its reliable modification to numerically treat the class of two-point boundary value problems (BVPs) and two-point initial-boundary value problems (IBVPs) using fractional partial differential equations (FPDEs). A special fractional inversion operator is used to simplify the decomposition process and further construct an optimal closed-form solution of the governing IBVP after exhausting the imposed two-point Dirichlet boundary conditions. Numerous descriptive testing fractional problems are presented to exhibit the efficacy of the devised approach, all of which lead to exact analytical solutions in most cases or optimal closed-form solutions when exact solutions are not achievable. The obtained solutions demonstrate that the proposed operator is accurate and suitable for solving FPDEs amidst the prescription of suitable initial and boundary conditions. Thus, several comparison tables are reported to assess the effectiveness of the devised method over other open methods via absolute error analysis.