Page 41 - IJOCTA-15-1

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.1, pp.35-49 (2025)
https://doi.org/10.36922/ijocta.1554

RESEARCH ARTICLE

Approximate analytical solutions of fractional coupled
Whitham-Broer-Kaup equations via novel transform

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1
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Lokesh Kumar Yadav , Murli Manohar Gour , Vikash Kumar Meena , Ebenezer Bonyah ,
Sunil Dutt Purohit 3,4*
1 Department of Mathematics, Vivekananda Global University Jaipur, India
2 Department of Mathematics and Statistics, Kumasi Polytechnic, Kumasi, Ghana
3 Department of HEAS (Mathematics), Rajasthan Technical University, Kota, India
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Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
lokesh.yadav@vgu.ac.in; murlimanohar.gaur@vgu.ac.in; meenavikashkumar343@gmail.com;
ebbonya@yahoo.com; sunil a purohit@yahoo.com

ARTICLE INFO ABSTRACT

Article History: In this study, the approximated analytical solution for the time-fractional
Received 2 March 2024 coupled Whitham-Broer-Kaup (WBK) equations describing the propagation
Accepted 23 July 2024 of shallow water waves are obtained with the aid of an efficient computa-
Available Online 21 January 2025 tional technique called, homotopy analysis Shehu transform methodm(briefly,
Keywords: HASTM). The Caputo operator is utilized to describe fractional-order deriva-
Coupled WBK equations tives. Our proposed approach combines the Shehu transformation with the
Homotopy analysis method homotopy analysis method, employing homotopy polynomials to handle non-
Shehu transform linear terms. To validate the correctness of our method, we offer a comparison
Caputo fractional derivative of obtained and exact solutions with different fractional order values. Given
its novelty and straightforward implementation, our method is considered a re-
AMS Classification 2010:
liable and efficient analytical technique for solving both linear and non-linear
26A33; 34A08; 35R11
fractional partial differential equations.

1. Introduction and intrinsic consequences in material properties.
The concept of fractional calculus has been ex-
The contemporary and broadly considered con- tensively examined and elucidated by esteemed
cept of fractional calculus originated from a ques- scholars, who have formulated revolutionary def-
tion posed by L’Hospital to G.W. Leibniz in 1695. initions, laying the groundwork for the field of
L’Hospital was persistent in seeking information fractional calculus.
about the outcome of the derivative of order
α = 1 , thereby establishing the groundwork for Fractional partial differential equations (FPDEs)
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a potent fractional calculus. The exploration of have gained popularity in developing procedures
fractional calculus associated with symmetry has for non-linear models and investigating dynami-
recently captivated numerous researchers across cal systems. The solutions of nonlinear FPDEs of
diverse disciplines, allowing them to present their arbitrary order play a vital role in describing the
perspectives while addressing real-world prob- nature and characteristics of complex problems
lems. In recent years, numerous authors have em- arising in Mathematical science and technology.
barked on the study of fractional calculus due to However, obtaining analytical solutions for these
its capability to precisely describe various types of differential equations is highly challenging. Over
nonlinear phenomena. Fractional-order differen- the past three decades, there has been a signifi-
tial equations represent a generalization of tradi- cant focus on initiating and studying a variety of
tional differential equations, exhibiting non-local numerical techniques. 1–8
*Corresponding Author
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