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

Süleyman Çetinkaya

doi:10.36753/mathenot.1810214

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

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

References

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Details

Primary Language

English

Subjects

Mathematical Methods and Special Functions

Journal Section

Research Article

Authors

Süleyman Çetinkaya ^*
0000-0002-8214-5099
Türkiye

Early Pub Date

December 15, 2025

Publication Date

December 15, 2025

Submission Date

October 24, 2025

Acceptance Date

December 14, 2025

Published in Issue

Year 2025 Volume: 13 Number: 4

DOI

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

IZ

https://izlik.org/JA58RE54YF

APA

Çetinkaya, S. (2025). Hybrid Method for Fractional Partial Differential Equations: Analytical Solution of Linear and Nonlinear Systems. Mathematical Sciences and Applications E-Notes, 13(4), 224-239. https://doi.org/10.36753/mathenot.1810214

AMA

1.Çetinkaya S. Hybrid Method for Fractional Partial Differential Equations: Analytical Solution of Linear and Nonlinear Systems. Math. Sci. Appl. E-Notes. 2025;13(4):224-239. doi:10.36753/mathenot.1810214

Chicago

Çetinkaya, Süleyman. 2025. “Hybrid Method for Fractional Partial Differential Equations: Analytical Solution of Linear and Nonlinear Systems”. Mathematical Sciences and Applications E-Notes 13 (4): 224-39. https://doi.org/10.36753/mathenot.1810214.

EndNote

Çetinkaya S (December 1, 2025) Hybrid Method for Fractional Partial Differential Equations: Analytical Solution of Linear and Nonlinear Systems. Mathematical Sciences and Applications E-Notes 13 4 224–239.

IEEE

[1]S. Çetinkaya, “Hybrid Method for Fractional Partial Differential Equations: Analytical Solution of Linear and Nonlinear Systems”, Math. Sci. Appl. E-Notes, vol. 13, no. 4, pp. 224–239, Dec. 2025, doi: 10.36753/mathenot.1810214.

ISNAD

Çetinkaya, Süleyman. “Hybrid Method for Fractional Partial Differential Equations: Analytical Solution of Linear and Nonlinear Systems”. Mathematical Sciences and Applications E-Notes 13/4 (December 1, 2025): 224-239. https://doi.org/10.36753/mathenot.1810214.

JAMA

1.Çetinkaya S. Hybrid Method for Fractional Partial Differential Equations: Analytical Solution of Linear and Nonlinear Systems. Math. Sci. Appl. E-Notes. 2025;13:224–239.

MLA

Çetinkaya, Süleyman. “Hybrid Method for Fractional Partial Differential Equations: Analytical Solution of Linear and Nonlinear Systems”. Mathematical Sciences and Applications E-Notes, vol. 13, no. 4, Dec. 2025, pp. 224-39, doi:10.36753/mathenot.1810214.

Vancouver

1.Süleyman Çetinkaya. Hybrid Method for Fractional Partial Differential Equations: Analytical Solution of Linear and Nonlinear Systems. Math. Sci. Appl. E-Notes. 2025 Dec. 1;13(4):224-39. doi:10.36753/mathenot.1810214