A Novel Double Mohand-Generalized ARA Transform Coupled with Adomian Decomposition Method for Multi-Dimensional Fractional Partial Differential Equations

The present research aims to introduce a brand new theoretical framework for solving multi-dimensional fractional partial differential equations (FPDEs) by developing a novel integral transform tool called the Double Mohand-Generalized ARA Transform (DM-GART). The DM-GART is a triple-integral operator that applies the Mohand transform twice—once in each spatial variable x and y and the ARA transform once in the temporal variable t; the adjective “Double” refers specifically to the double spatial application of the Mohand transform. The theoretical properties and existence/uniqueness results of this newly developed integral transform are rigorously established in a Banach fixed-point theorem setting. The newly developed integral transform tool is then synergistically combined with the Adomian Decomposition Method (ADM) to produce a novel technique called the Coupled Double Mohand-Generalized ARA Decomposition Method (CDM-GADM). The CDM-GADM is applied for solving generalised fractional biological population equations. The technique is assessed by comparing exact solutions with N-term series solutions for N = 4, 6, and 8. From the results obtained in Tables 3–10, it can be noted that with an increase in the terms from N = 4 to N = 8, the absolute errors decrease several orders of magnitude; the absolute errors for N = 8 are as low as 10⁻¹⁰ for α = 1.0 at smaller values of time. The results are obtained in the form of convergent series characterized by the Mittag-Leffler function, validating the efficiency of the proposed method. A tolerance of ε = 10⁻⁶ is used as the practical stopping criterion.