Analytical Solution of Time Fractional Kawahara and Modified Kawahara Equations by Homotopy Analysis Method

Zafar, Husna; Ali, Amir; Khan, Khalid Ali; Sadiq, Muhammad Noveel

doi:10.1007/s40819-022-01296-3

Cited by 2 publications

(1 citation statement)

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“…Recently, a number of scholars have looked at the TFKE and TFMKE utilizing a variety of approaches and methods, including the new iterative method [44], the homotopy analysis method [45], the residual power series method [46], and the Laplace iterative transform method [47]. To the best of our knowledge, this is the first time to apply the HPTM and ETDM to examine the Kawahara equations within the Caputo operator.…”

Section: Introductionmentioning

confidence: 99%

Application of Analytical Techniques for Solving Fractional Physical Models Arising in Applied Sciences

AlBaidani,

Ganie,

Aljuaydi

et al. 2023

Fractal Fract

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In this paper, we examined the approximations to the time-fractional Kawahara equation and modified Kawahara equation, which model the creation of nonlinear water waves in the long wavelength area and the transmission of signals. We implemented two novel techniques, namely the homotopy perturbation transform method and the Elzaki transform decomposition method. The derivative having fractional-order is taken in Caputo sense. The Adomian and He’s polynomials make it simple to handle the nonlinear terms. To illustrate the adaptability and effectiveness of derivatives with fractional order to represent the water waves in long wavelength regions, numerical data have been given graphically. A key component of the Kawahara equation is the symmetry pattern, and the symmetrical nature of the solution may be observed in the graphs. The importance of our suggested methods is illustrated by the convergence of analytical solutions to the precise solutions. The techniques currently in use are straightforward and effective for solving fractional-order issues. The offered methods reduced computational time is their main advantage. It will be possible to solve fractional partial differential equations using the study’s findings as a tool.

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Section: Introductionmentioning

confidence: 99%

Application of Analytical Techniques for Solving Fractional Physical Models Arising in Applied Sciences

AlBaidani,

Ganie,

Aljuaydi

et al. 2023

Fractal Fract

View full text Add to dashboard Cite

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An Efficient Technique to Solve Time-Fractional Kawahara and Modified Kawahara Equations

Koppala

Raghavendar

2022

Symmetry

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In this article, we analysed the approximate solutions of the time-fractional Kawahara equation and modified Kawahara equation, which describe the propagation of signals in transmission lines and the formation of nonlinear water waves in the long wavelength region. An efficient technique, namely the natural transform decomposition method, is used in the present study. Fractional derivatives are considered in Caputo, Caputo–Fabrizio, and Atangana–Baleanu operative in the Caputo manner. We have presented numerical results graphically to demonstrate the applicability and efficiency of derivatives with fractional order to depict the water waves in long wavelength regions. The symmetry pattern is a fundamental feature of the Kawahara equation and the symmetrical aspect of the solution can be seen from the graphical representations. The obtained outcomes of the proposed method are compared to those of other well-known numerical techniques, such as the homotopy analysis method and residual power series method. Numerical solutions converge to the exact solution of the Kawahara equations, demonstrating the significance of our proposed method.

show abstract

Analytical Solution of Time Fractional Kawahara and Modified Kawahara Equations by Homotopy Analysis Method

Cited by 2 publications

References 32 publications

Application of Analytical Techniques for Solving Fractional Physical Models Arising in Applied Sciences

Application of Analytical Techniques for Solving Fractional Physical Models Arising in Applied Sciences

An Efficient Technique to Solve Time-Fractional Kawahara and Modified Kawahara Equations

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