• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.847-858
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• 2011

We make use of the simplified Hirota's bilinear method with computer symbolic computation to study a variety of coupled potential KdV (pKdV) equations. Each coupled equation is completely integrable and gives multiple soliton solutions and multiple singular soliton solutions. The phase shifts for all coupled pKdV equations are identical whereas the coefficients of the obtained solitons are not identical. The four coupled pKdV equations are resonance free.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.2
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• pp.359-370
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• 2011

In this work, we obtain new solitary wave solutions for some nonlinear partial differential equations. The Jacobi elliptic function rational expansion method is used to establish new solitary wave solutions for the combined KdV-mKdV and Klein-Gordon equations. The results reveal that Jacobi elliptic function rational expansion method is very effective and powerful tool for solving nonlinear evolution equations arising in mathematical physics.

• Honam Mathematical Journal
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• v.35 no.4
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• pp.683-699
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• 2013

In this paper, an improved ($\frac{G^{\prime}}{G}$)-expansion method is proposed for obtaining travelling wave solutions of nonlinear evolution equations. The proposed technique called ($\frac{F}{G}$)-expansion method is more powerful than the method ($\frac{G^{\prime}}{G}$)-expansion method. The efficiency of the method is demonstrated on a variety of nonlinear partial differential equations such as KdV equation, mKd equation and Boussinesq equations. As a result, more travelling wave solutions are obtained including not only all the known solutions but also the computation burden is greatly decreased compared with the existing method. The travelling wave solutions are expressed by the hyperbolic functions and the trigonometric functions. The result reveals that the proposed method is simple and effective, and can be used for many other nonlinear evolutions equations arising in mathematical physics.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.383-395
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• 2010

In the present paper, we construct the traveling wave solutions involving parameters of nonlinear evolution equations in the mathematical physics via the (3+1)- dimensional potential- YTSF equation, the (3+1)- dimensional generalized shallow water equation, the (3+1)- dimensional Kadomtsev- Petviashvili equation, the (3+1)- dimensional modified KdV-Zakharov- Kuznetsev equation and the (3+1)- dimensional Jimbo-Miwa equation by using a simple method which is called the ($\frac{G'}{G}$)- expansion method, where $G\;=\;G(\xi)$ satisfies a second order linear ordinary differential equation. When the parameters are taken special values, the solitary waves are derived from the travelling waves. The travelling wave solutions are expressed by hyperbolic, trigonometric and rational functions.

• Honam Mathematical Journal
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• v.43 no.3
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• pp.373-383
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• 2021

In this study, a numerical method deals with the Chebyshev wavelet collocation and Adomian decomposition methods are proposed for solving Korteweg-de Vries equation. Integration of the Chebyshev wavelets operational matrices is derived. This problem is reduced to a system of non-linear algebraic equations by using their operational matrix. Thus, it becomes easier to solve KdV problem. The error estimation for the Chebyshev wavelet collocation method and ADM is investigated. The proposed method's validity and accuracy are demonstrated by numerical results. When the exact and approximate solutions are compared, for non-linear or linear partial differential equations, the Chebyshev wavelet collocation method is shown to be acceptable, efficient and accurate.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.557-573
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• 1996

When we attempt to describe the propagation of nonlinear dispersive waves, it is frequently necessary to take account of diffipative effects.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.167-178
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• 2007

This paper deals with the solutions of linear inhomogeneous time-fractional partial differential equations in applied mathematics and fluid mechanics. The fractional derivatives are described in the Caputo sense. The fractional Green function method is used to obtain solutions for time-fractional wave equation, linearized time-fractional Burgers equation, and linear time-fractional KdV equation. The new approach introduces a promising tool for solving fractional partial differential equations.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1583-1602
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• 2011

A special system of partial differential equations (PDEs) occur in a natural way while studying a class of irrotational inviscid fluid flow problems involving infinite channels. Certain aspects of solutions of such PDEs are analyzed in the context of flow problems involving multiple layers of fluids of different constant densities in a channel associated with arbitrary bottom topography. The whole analysis is divided into two parts-part A and part B. In part A the linearized theory is employed along with the standard Fourier analysis to understand such flow problems and physical quantities of interest are derived analytically. In part B, the same set of problems handled in part A are examined in the light of a weakly non-linear theory involving perturbation in terms of a small parameter and it is shown that the original problems can be cast into KdV type of nonlinear PDEs involving the bottom topography occurring in one of the coefficients of these equations. Special cases of bottom topography are worked out in detail and expressions for quantities of physical importance are derived.

• Journal of the Korean Mathematical Society
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• v.38 no.6
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• pp.1205-1234
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• 2001

We consider the time local well-posedness of the Benjamin equation. Like the result due to Keing-Ponce-Vega [10], [12], we show that the initial value problem is time locally well posed in the Sobolev space H$^{s}$ (R) for s>-3/4.

• Nonlinear Functional Analysis and Applications
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• v.28 no.4
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• pp.1017-1034
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• 2023

In this paper, we work two different problems. First, we investigate a new class of fractional differential equations involving Caputo sequential derivative with some (e₁, e₂, θ)-periodic conditions. The existence and uniqueness of solutions are proven. The stability of solutions is also discussed. The second part includes studying traveling wave solutions of a conformable fractional Korteweg-de Vries-Burger (KdV Burger) equation through the Tanh method. Graphs of some of the waves are plotted and discussed, and a conclusion follows.