Browsing by Author "Kumar D."

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A Variety of Novel Exact Solutions for Different Models With the Conformable Derivative in Shallow Water
(2020-06-16) Kumar D.; Kaplan M.; Haque M.R.; Osman M.S.; Baleanu D.
For different nonlinear time-conformable derivative models, a versatile built-in gadget, namely the generalized exp(−φ(ξ))-expansion (GEE) method, is devoted to retrieving different categories of new explicit solutions. These models include the time-fractional approximate long-wave equations, the time-fractional variant-Boussinesq equations, and the time-fractional Wu-Zhang system of equations. The GEE technique is investigated with the help of fractional complex transform and conformable derivative. As a result, we found four types of exact solutions involving hyperbolic function, periodic function, rational functional, and exponential function solutions. The physical significance of the explored solutions depends on the choice of arbitrary parameter values. Finally, we conclude that the GEE method is more effective in establishing the explicit new exact solutions than the exp(−φ(ξ))-expansion method.
An effective computational approach and sensitivity analysis to pseudo-parabolic-type equations
(2021-01-01) Kaplan M.; Butt A.R.; Thabet H.; Akbulut A.; Raza N.; Kumar D.
The researchers have developed numerous analytical and numerical techniques for solving fractional partial differential equations most of which provide approximate solutions. Exact solutions, however, are vitally important in a convenient conception of the qualitative properties of the concerned phenomena and processes. In this paper, the pseudo-parabolic-type equations with conformable fractional derivatives are reduced to conformable fractional nonlinear ordinary differential equations by implementing a simple wave transformation. An important benefit of the proposed transformation is that it yields analytical solutions of the conformable pseudo-parabolic type equations by applying the exponential rational function strategy. The sensitivity behaviour of the model has been mentioned thoroughly.
Application of the modified Kudryashov method to the generalized Schrödinger–Boussinesq equations
(2018-09-01) Kumar D.; Kaplan M.
In the paper, the modified Kudryashov method is applied to find new exact solutions for the generalized Schrödinger–Boussinesq equation with the help of symbolic computation package Maple through the complex transform. The obtained solutions have been checked by substituting back into its corresponding equation with the aid of Maple package program.
New analytical solutions of (2 + 1)-dimensional conformable time fractional Zoomeron equation via two distinct techniques
(2018-10-01) Kumar D.; Kaplan M.
In this paper, the new exact solutions for the (2 + 1) dimensional time fractional Zoomeron equation have been derived via two efficient analytical techniques, which are the extended exp(−Φ(ξ))-expansion technique and the novel exponential rational function technique. The fractional derivative is designated based on the conformable derivative sense. Consequently, many new closed form solutions of this equation are obtained including hyperbolic function solutions, trigonometric function solutions and exponential function solutions by using these techniques. The obtained results show that the applied methods are very effective, reliable and simple for solving other nonlinear fractional differential equations in mathematical physics and nonlinear optics.
New Exact Traveling Wave Solutions of the Unstable Nonlinear Schrö dinger Equations
(2017-12-01) Hosseini K.; Kumar D.; Kaplan M.; Bejarbaneh E.Y.
The present paper studies the unstable nonlinear Schrödinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schrödinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schrödinger equation and its modified form are formally obtained.
On some novel solution solutions to the generalized Schrödinger-Boussinesq equations for the interaction between complex short wave and real long wave envelope
(2022-08-01) Kumar D.; Hosseini K.; Kaabar M.K.A.; Kaplan M.; Salahshour S.
This paper explores some novel solutions to the generalized Schrödinger-Boussinesq (gSBq) equations, which describe the interaction between complex short wave and real long wave envelope. In order to derive some novel complex hyperbolic and complex trigonometric function solutions, the sine-Gordon equation method (sGEM) is applied to the gSBq equations. Novel complex hyperbolic and trigonometric function solutions are expressed by the dark, bright, combo dark-bright, W-shaped, M-shaped, singular, combo singular, and periodic wave solutions. The accuracy of the explored solitons is examined under the back substitution to the corresponding equations via the symbolic computation software Maple. It is found from the back substitution outcomes that all soliton solutions satisfy the original equations. The proper significance of the explored outcomes is demonstrated by the three-dimensional (3D) and two-dimensional (2D) graphs, which are presented under the selection of particular values of the free parameters. All the combo-soliton (W-shaped, M-shaped, and periodic wave) solutions are found to be new for the interaction between complex short wave and real long wave envelope in laser physics that show the novelty of the study. Moreover, the applied method provides an efficient tool for exploring novel soliton solutions, and it overcomes the complexities of the solitary wave ansatz method.
The analysis of conservation laws, symmetries and solitary wave solutions of Burgers-Fisher equation
(2021-09-10) Akbulut A.; Kaplan M.; Kumar D.; Taşcan F.
In this paper, the conservation laws, significant symmetries' application, and traveling wave solutions are obtained for Burger-Fisher equation (BFE). Conservation laws have a great importance for partial and fractional differential equations and their solutions, especially in physics implementations. The conservation theorem and partial Noether approach are implemented for conservation laws for this equation, and the extended sinh-Gordon expansion method (esGEM) is presented for new solitary wave solutions. All obtained conservation laws are trivial conservation laws. The new and comprehensive solitary wave solutions of the equation by the esGEM are also obtained.