Generalized Laplace Transform Approach to Modelling a Caputo Fractional Differential Equation Involving the Appell Function $F_3$

This study introduces a new concept of fractional modelling for multivariable special functions and fractional differential equations within the framework of Caputo fractional calculus. The modelling structure is constructed using the Appell  function $F_3$ together with a generalized Laplace-type integral transform, denoted by $\mathfrak{L}_n$. This approach provides a unified analytical setting for representing systems governed by fractional dynamics and multivariable functional structures. Within this modelling framework, the  $\mathfrak{L}_n$-transform is applied to the Appell function $F_3$ under fractional-power scaling of its variables. A new transform representation is derived in the form of a double-series expansion involving Gamma functions and Pochhammer symbols, which extends classical Laplace-transform techniques to a broader fractional modelling context. The developed modelling approach is further employed to construct and solve a nonhomogeneous Caputo fractional differential equation with an Appell-type forcing term. The obtained solution is expressed as a coupled Appell–Mittag-Leffler series, capturing both the hypergeometric structure of the forcing term and the memory effects inherent in fractional-order systems. Existence and uniqueness of the solution are established via a Volterra integral formulation and the Banach fixed-point theorem. Finally, numerical simulations are performed for different values of the fractional order $\alpha$. The results demonstrate a clear transition from strong memory-dominated dynamics in the fractional regime to classical behavior in the integer-order limit, highlighting the effectiveness of the proposed modelling framework.