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18 pages, 2505 KiB

Open AccessArticle

Analyzing the Harry Dym System Using the Laplace Residual Power Series Technique and New Iterative Technique with Caputo Derivative

by Muhammad Nasir, Shuobing Yang, Hijaz Ahmad and Taha Radwan

Symmetry 2025, 17(6), 882; https://doi.org/10.3390/sym17060882 - 5 Jun 2025

Viewed by 315

Abstract

Fractional-order differential equations are prevalent in many scientific fields; hence, their study has seen a renaissance in recent years. The fascinating realm of fractional calculus is explored in this research study, with particular emphasis on the Harry Dym equation. To solve this problem, we use the Laplace Residual Power Series Method (LRPSM) and introduce the New Iterative Method (NIM). Both the mathematical complexity of the Harry Dym problem and the viability of the Caputo operator in this setting are investigated in our work. We go beyond the limitations of traditional mathematical methods to provide novel insights into the results of fractional-order differential equations via careful analysis and cutting-edge procedures. In this paper, we combine theory and practice to provide a novel perspective to the results of high-order fractional differential equations. Our efforts pay off by expanding our knowledge of mathematics and revealing the latent potential of the Harry Dym equation. This study expands researchers’ and mathematicians’ perspectives, bringing in a new and exciting period of progress in the field of fractional calculus. Full article

(This article belongs to the Special Issue Symmetries and Fractional Differential Equations: Theory and Applications)

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42 pages, 2461 KiB

Open AccessArticle

Soliton Solution of the Nonlinear Time Fractional Equations: Comprehensive Methods to Solve Physical Models

by Donal O’Regan, Safoura Rezaei Aderyani, Reza Saadati and Mustafa Inc

Axioms 2024, 13(2), 92; https://doi.org/10.3390/axioms13020092 - 30 Jan 2024

Cited by 5 | Viewed by 1348

Abstract

In this paper, we apply two different methods, namely, the

\frac{G^{'}}{G}

-expansion method and the

\frac{G^{'}}{G^{2}}

-expansion method to investigate the nonlinear time fractional Harry Dym equation in the Caputo sense and the symmetric regularized long wave equation [...] Read more.

In this paper, we apply two different methods, namely, the

\frac{G^{'}}{G}

-expansion method and the

\frac{G^{'}}{G^{2}}

-expansion method to investigate the nonlinear time fractional Harry Dym equation in the Caputo sense and the symmetric regularized long wave equation in the conformable sense. The mentioned nonlinear partial differential equations (NPDEs) arise in diverse physical applications such as ion sound waves in plasma and waves on shallow water surfaces. There exist multiple wave solutions to many NPDEs and researchers are interested in analytical approaches to obtain these multiple wave solutions. The multi-exp-function method (MEFM) formulates a solution algorithm for calculating multiple wave solutions to NPDEs and at the end of paper, we apply the MEFM for calculating multiple wave solutions to the (2 + 1)-dimensional equation. Full article

(This article belongs to the Special Issue New Trends in Fractional Operators with Applications in Mathematical Physics)

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10 pages, 2324 KiB

Open AccessArticle

Exact Traveling Wave Solutions in a Generalized Harry Dym Type Equation

by Rong Wu and Yan Zhou

Symmetry 2022, 14(7), 1480; https://doi.org/10.3390/sym14071480 - 20 Jul 2022

Cited by 3 | Viewed by 2235

Abstract

The traveling wave solutions of a generalized HD type equation are investigated in this study. The traveling wave system is a singular system of the first class with given parameter conditions. From the standpoint of dynamical systems, the bifurcations of traveling wave solutions in parameter space are examined. It is demonstrated that solitary wave solutions, periodic peakons, pseudo-peakons, and compacton solutions exist. All conceivable exact explicit parametric representations of various solutions are presented. Full article

(This article belongs to the Topic Dynamical Systems: Theory and Applications)

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11 pages, 239 KiB

Open AccessArticle

Self-Adjointness and Conservation Laws of Frobenius Type Equations

by Haifeng Wang and Yufeng Zhang

Symmetry 2020, 12(12), 1987; https://doi.org/10.3390/sym12121987 - 1 Dec 2020

Cited by 1 | Viewed by 1499

Abstract

The Frobenius KDV equation and the Frobenius KP equation are introduced, and the Frobenius Kompaneets equation, Frobenius Burgers equation and Frobenius Harry Dym equation are constructed by taking values in a commutative subalgebra

Z_{2}^{ε}

in the paper. The five equations are [...] Read more.

Z_{2}^{ε}

in the paper. The five equations are selected as examples to help us study the self-adjointness of Frobenius type equations, and we show that the first two equations are quasi self-adjoint and the last three equations are nonlinear self-adjointness. It follows that we give the symmetries of the Frobenius KDV and the Frobenius KP equation in order to construct the corresponding conservation laws. Full article

(This article belongs to the Section Mathematics)

7 pages, 241 KiB

Open AccessArticle

Nonclassical Symmetries of a Power Law Harry Dym Equation

by Daniel J. Arrigo and Andrea N. Weaver

Symmetry 2018, 10(4), 100; https://doi.org/10.3390/sym10040100 - 6 Apr 2018

Cited by 2 | Viewed by 3355

Abstract

It is generally known that classical point and potential Lie symmetries of differential equations can be different. In a recent paper, we were able to show for a class of nonlinear diffusion equation that the nonclassical potential symmetries possess all nonclassical symmetries of the original equation. We question whether this is true for the power law Harry Dym equation. In this paper, we show that the nonclassical symmetries of the power law Harry Dym equation and an equivalent system still possess special separate symmetries. However, we will show that we can extend the nonclassical method so that all nonclassical symmetries of the original power law Harry Dym equation can be obtained through the equivalent system. Full article

(This article belongs to the Special Issue Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models, II)

19 pages, 252 KiB

Open AccessArticle

Bäcklund Transformations for Integrable Geometric Curve Flows

by Changzheng Qu, Jingwei Han and Jing Kang

Symmetry 2015, 7(3), 1376-1394; https://doi.org/10.3390/sym7031376 - 3 Aug 2015

Cited by 7 | Viewed by 5052

Abstract

We study the Bäcklund transformations of integrable geometric curve flows in certain geometries. These curve flows include the KdV and Camassa-Holm flows in the two-dimensional centro-equiaffine geometry, the mKdV and modified Camassa-Holm flows in the two-dimensional Euclidean geometry, the Schrödinger and extended Harry-Dym flows in the three-dimensional Euclidean geometry and the Sawada-Kotera flow in the affine geometry, etc. Using the fact that two different curves in a given geometry are governed by the same integrable equation, we obtain Bäcklund transformations relating to these two integrable geometric flows. Some special solutions of the integrable systems are used to obtain the explicit Bäcklund transformations. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

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