Hybrid Method for Fractional Partial Differential Equations: Analytical Solution of Linear and Nonlinear Systems

Year 2025, Volume: 13 Issue: 4, 224 - 239, 15.12.2025

Süleyman Çetinkaya

https://doi.org/10.36753/mathenot.1810214

Abstract

In this study, we obtain analytical and semi-analytical solutions of systems of fractional partial differential equations using a hybrid approach we call the Laplace-Daftardar-Gejji Method (LDGM). The Laplace transform is used to simplify the nonlinear system with fractional derivatives. The solution components are then obtained using the Daftardar-Gejji iterative technique. In a final step, to demonstrate the applicability and effectiveness of the method, analytical solutions of two linear and nonlinear systems modeled with Caputo fractional derivatives are obtained. It is observed that the analytical solutions approach the classical solutions as the order of the fractional derivative, $\alpha$, approaches $1$. This observation is supported by tables and graphs. In summary, since analytical solutions are obtained using the proposed method, LDGM is an effective and useful tool for solving nonlinear systems involving Caputo fractional differentials.

Keywords

Caputo fractional derivative , Daftardar-Gejji method , Laplace transform , Mittag-Leffler function , Nonlinear systems

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