Using the Laplace Transform Technique to Solve Hilfer Fractional Partial Differential Equations

Mehmet Merdan; Şeyma Şişman

doi:10.33187/jmsm.1699169

Using the Laplace Transform Technique to Solve Hilfer Fractional Partial Differential Equations

Abstract

This work derives analytical solutions to fractional partial differential equations utilizing the Hilfer fractional derivative through the use of the Laplace transform method. The suggested method offers a cohesive solution framework that concurrently retrieves the Liouville-Caputo and Riemann-Liouville formulations as particular instances for designated selections of the Hilfer parameters. Exact novel solution representations are derived for the fractional KdV, modified KdV, K(2,2), and Klein-Gordon equations under appropriate initial conditions, emphasizing the impact of fractional order and type parameters on wave propagation characteristics. The approach circumvents linearization or perturbation assumptions and demonstrates a rapid convergence rate. Numerical simulations and graphical representations produced with the Maple program corroborate the analytical findings and illustrate the efficacy and universality of the proposed method.

Keywords

Hilfer fractional derivative, Riemann fractional derivative, Caputo fractional derivative, Laplace transform, Klein-Gordon, Mittag-Leffler

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Details

Primary Language

English

Subjects

Applied Mathematics (Other)

Journal Section

Research Article

Authors

Mehmet Merdan ^*
0000-0002-8509-3044
Türkiye

Şeyma Şişman
0000-0003-0756-4872
Türkiye

Early Pub Date

March 14, 2026

Publication Date

March 14, 2026

Submission Date

May 14, 2025

Acceptance Date

March 13, 2026

Published in Issue

Year 2026 Volume: 9 Number: 1

DOI

https://doi.org/10.33187/jmsm.1699169

IZ

https://izlik.org/JA73GB89CK

APA

Merdan, M., & Şişman, Ş. (2026). Using the Laplace Transform Technique to Solve Hilfer Fractional Partial Differential Equations. Journal of Mathematical Sciences and Modelling, 9(1), 55-76. https://doi.org/10.33187/jmsm.1699169

AMA

1.Merdan M, Şişman Ş. Using the Laplace Transform Technique to Solve Hilfer Fractional Partial Differential Equations. Journal of Mathematical Sciences and Modelling. 2026;9(1):55-76. doi:10.33187/jmsm.1699169

Chicago

Merdan, Mehmet, and Şeyma Şişman. 2026. “Using the Laplace Transform Technique to Solve Hilfer Fractional Partial Differential Equations”. Journal of Mathematical Sciences and Modelling 9 (1): 55-76. https://doi.org/10.33187/jmsm.1699169.

EndNote

Merdan M, Şişman Ş (March 1, 2026) Using the Laplace Transform Technique to Solve Hilfer Fractional Partial Differential Equations. Journal of Mathematical Sciences and Modelling 9 1 55–76.

IEEE

[1]M. Merdan and Ş. Şişman, “Using the Laplace Transform Technique to Solve Hilfer Fractional Partial Differential Equations”, Journal of Mathematical Sciences and Modelling, vol. 9, no. 1, pp. 55–76, Mar. 2026, doi: 10.33187/jmsm.1699169.

ISNAD

Merdan, Mehmet - Şişman, Şeyma. “Using the Laplace Transform Technique to Solve Hilfer Fractional Partial Differential Equations”. Journal of Mathematical Sciences and Modelling 9/1 (March 1, 2026): 55-76. https://doi.org/10.33187/jmsm.1699169.

JAMA

1.Merdan M, Şişman Ş. Using the Laplace Transform Technique to Solve Hilfer Fractional Partial Differential Equations. Journal of Mathematical Sciences and Modelling. 2026;9:55–76.

MLA

Merdan, Mehmet, and Şeyma Şişman. “Using the Laplace Transform Technique to Solve Hilfer Fractional Partial Differential Equations”. Journal of Mathematical Sciences and Modelling, vol. 9, no. 1, Mar. 2026, pp. 55-76, doi:10.33187/jmsm.1699169.

Vancouver

1.Mehmet Merdan, Şeyma Şişman. Using the Laplace Transform Technique to Solve Hilfer Fractional Partial Differential Equations. Journal of Mathematical Sciences and Modelling. 2026 Mar. 1;9(1):55-76. doi:10.33187/jmsm.1699169