Caputo Fractional Operator and Aboodh Transform Technique of WBK Equations: Some Comparisons of Analytical and Numerical Solutions with Other Techniques

Faten Damag; Mohammed Alsharafi; Amin Saif

Caputo Fractional Operator and Aboodh Transform Technique of WBK Equations: Some Comparisons of Analytical and Numerical Solutions with Other Techniques

Abstract

Among the most important equations in the theory of shallow water waves are the Whitham–Broer–Kaup equations (WBK equations), which form a coupled system of nonlinear partial differential equations describing the propagation of shallow water waves with different dispersion relations and diffusion effects. In this paper, we combine the Aboodh transform with the residual power series method to develop a new technique called the Aboodh Residual Power Series Technique (ARPST). This technique, together with the Caputo fractional operator, is used to obtain analytical and numerical solutions of the WBK equations by calculating the coefficients of the corresponding power series. The proposed ARPST is applied to solve the WBK equations in two cases: exact initial conditions and approximate initial conditions. To demonstrate the capability, efficiency, and reliability of the proposed technique, the numerical results of the analytical solutions are presented graphically and compared with results obtained by other established methods, including the Adomian decomposition method, the optimal homotopy asymptotic method, and the Sumudu decomposition method.

Keywords

References

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Details

Primary Language

English

Subjects

Numerical Solution of Differential and Integral Equations

Journal Section

Research Article

Authors

Faten Damag ^* This is me
0009-0009-4773-7046
Saudi Arabia

Mohammed Alsharafi This is me
0000-0001-6252-8968
Yemen

Amin Saif This is me
Yemen

Publication Date

June 30, 2026

Submission Date

April 7, 2026

Acceptance Date

May 22, 2026

Published in Issue

Year 2026 Volume: 3 Number: 3(1)

IZ

https://izlik.org/JA96RE82GL

Cite

RIS / Bibtex

APA

Damag, F., Alsharafi, M., & Saif, A. (2026). Caputo Fractional Operator and Aboodh Transform Technique of WBK Equations: Some Comparisons of Analytical and Numerical Solutions with Other Techniques. Türkiye Mathematical Sciences, 3(3(1), 17-41. https://izlik.org/JA96RE82GL

AMA

1.Damag F, Alsharafi M, Saif A. Caputo Fractional Operator and Aboodh Transform Technique of WBK Equations: Some Comparisons of Analytical and Numerical Solutions with Other Techniques. TMS. 2026;3(3(1):17-41. https://izlik.org/JA96RE82GL

Chicago

Damag, Faten, Mohammed Alsharafi, and Amin Saif. 2026. “Caputo Fractional Operator and Aboodh Transform Technique of WBK Equations: Some Comparisons of Analytical and Numerical Solutions With Other Techniques”. Türkiye Mathematical Sciences 3 (3(1): 17-41. https://izlik.org/JA96RE82GL.

EndNote

Damag F, Alsharafi M, Saif A (June 1, 2026) Caputo Fractional Operator and Aboodh Transform Technique of WBK Equations: Some Comparisons of Analytical and Numerical Solutions with Other Techniques. Türkiye Mathematical Sciences 3 3(1) 17–41.

IEEE

[1]F. Damag, M. Alsharafi, and A. Saif, “Caputo Fractional Operator and Aboodh Transform Technique of WBK Equations: Some Comparisons of Analytical and Numerical Solutions with Other Techniques”, TMS, vol. 3, no. 3(1), pp. 17–41, June 2026, [Online]. Available: https://izlik.org/JA96RE82GL

ISNAD

Damag, Faten - Alsharafi, Mohammed - Saif, Amin. “Caputo Fractional Operator and Aboodh Transform Technique of WBK Equations: Some Comparisons of Analytical and Numerical Solutions With Other Techniques”. Türkiye Mathematical Sciences 3/3(1) (June 1, 2026): 17-41. https://izlik.org/JA96RE82GL.

JAMA

1.Damag F, Alsharafi M, Saif A. Caputo Fractional Operator and Aboodh Transform Technique of WBK Equations: Some Comparisons of Analytical and Numerical Solutions with Other Techniques. TMS. 2026;3:17–41.

MLA

Damag, Faten, et al. “Caputo Fractional Operator and Aboodh Transform Technique of WBK Equations: Some Comparisons of Analytical and Numerical Solutions With Other Techniques”. Türkiye Mathematical Sciences, vol. 3, no. 3(1), June 2026, pp. 17-41, https://izlik.org/JA96RE82GL.

Vancouver

1.Faten Damag, Mohammed Alsharafi, Amin Saif. Caputo Fractional Operator and Aboodh Transform Technique of WBK Equations: Some Comparisons of Analytical and Numerical Solutions with Other Techniques. TMS [Internet]. 2026 Jun. 1;3(3(1):17-41. Available from: https://izlik.org/JA96RE82GL