Nonlinear PDEs In Mathematical Physics Research Articles

New investigation of the analytical behaviors for some nonlinear PDEs in mathematical physics and modern engineering

Numerous complex situations arising from science and engineering are modeled by integral-differential equations (IDEs). Examples of these situations include mathematical modeling of infectious diseases, epidemics, circuit analysis, voltage drop equations, computational neuroscience, intergalactic modeling, and many more. To explain the mathematical structure and the behavior of various phenomena in nature, two equations with simple mathematical structure namely: the (1 + 1)-dimensional integro-differential Ito equation (IDIE) and the (2 + 1)-dimensional integro-differential Sawada-Kotera equation (IDSKE) are considered. In this analytical investigation, we have conducted an in-depth enquiry to trace the distinctive closed form progressive wave solutions of these proposed models using the (G′G′+G+A)-expansion method. The graphical forms of the acquired solutions have been displayed in the bell-shaped soliton, singular periodic, and anti-kink shape soliton solutions after the values for the free parameters were specified. The two non-algebraic solutions, named, exponential and trigonometric function solutions, have emerged by applying our proposed method. Based on the general findings of our investigation for different parametric values, the attained wave solutions revealed can keep a vital role for natural balances and can be engaged in modern physical applications. The parameters have a visible effect on the wave amplitude and the swiftness of the traveling wave. The executed outcomes are innovative and have monumental applications in the present research, especially in the fields of theoretical physics, astrophysics, plasma physics, particle physics, theory of relativity, advance quantum mechanics, string theory, and geotechnical engineering.

Extended generalized $$(Zakh\frac{G^{\prime }}{G})$$ ( Z a k h G ′ G ) -expansion method for solving the nonlinear quantum Zakharov–Kuznetsov equation

In this article, we apply the extended generalized \((\frac{G^{\prime }}{G})\)-expansion method combined with the Jacobi elliptic equation to find new exact solutions of the nonlinear quantum Zakharov–Kuznetsov (QZK) equation with the aid of computer algebraic system Maple. Soliton solutions, periodic solutions, rational functions solutions and Jacobi elliptic functions solutions are obtained. Based on reductive perturbation technique and a series of transformation, the nonlinear QZK had been derived by many authors which can be reduced to a nonlinear ordinary differential equation (ODE) using the wave transformation. The extended generalized \((\frac{G^{\prime }}{G})\)-expansion method is straightforward and concise, and it can also be applied to other nonlinear PDEs in mathematical physics.

The [formula omitted]-expansion method and its applications to two nonlinear Schrödinger equations describing the propagation of femtosecond pulses in nonlinear optical fibers

The propagation of the optical solitons is usually governed by the nonlinear Schrödinger equations. In this article, the two variable G′G,1G-expansion method is employed to construct exact traveling wave solutions with parameters of two higher order nonlinear Schrödinger equations. When the parameters are replaced by special values, the well-known solitary wave solutions of these equations rediscovered from the traveling waves. This method can be thought of as the generalization of well-known original G′G-expansion method proposed by Wang et al. It is shown that the two variable G′G,1G-expansion method provides a more powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics.

Solutions of some class of nonlinear PDEs in mathematical physics

Abstract In this work, the modified simple equation (MSE) method is applied to some class of nonlinear PDEs, namely, a system of nonlinear PDEs, a (2 + 1)-dimensional nonlinear model generated by the Jaulent–Miodek hierarchy, and a generalized KdV equation with two power nonlinearities. As a result, exact traveling wave solutions involving parameters have been obtained for the considered nonlinear equations in a concise manner. When these parameters are chosen as special values, the solitary wave solutions are derived. It is shown that the proposed technique provides a more powerful mathematical tool for constructing exact solutions for a broad variety of nonlinear PDEs in mathematical physics.

On solving two higher-order nonlinear PDEs describing the propagation of optical pulses in optic fibers using the $$\left( \frac{G^{\prime }}{G},\frac{1}{G}\right) $$ G ′ G , 1 G -expansion method

The propagation of the optical solitons is usually governed by the nonlinear Schrodinger equations. In this article, the two variable \((\frac{ G^{\prime }}{G},\frac{1}{G})\)-expansion method is employed to construct the exact traveling wave solutions with parameters of two nonlinear PDEs namely, the (2\(+\)1)-dimensional nonlinear cubic–quintic Ginzburg–Landau equation and the (1\(+\)1)-dimensional resonant nonlinear Schrodinger’s equation with dual-power law nonlinearity which describe the propagation of optical pulses in optic fibers. When the parameters are replaced by special values, the well-known solitary wave solutions of these equations rediscovered from the traveling waves. This method can be thought of as the generalization of well-known original \((\frac{G^{\prime }}{G})\)-expansion method proposed by M. Wang et al. It is shown that the two variable \((\frac{G^{\prime }}{G}, \frac{1}{G})\)-expansion method provides a more powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics.

The (G′/G,1/G)-Expansion Method and Its Applications to Find the Exact Solutions of Nonlinear PDEs for Nanobiosciences

The two-variable (G′/G,1/G)-expansion method is employed to construct exact traveling wave solutions with parameters of nanobiosciences partial differential equation. When the parameters are replaced by special values, the solitary wave solutions and the periodic wave solutions of this equation have been obtained from the traveling waves. This method can be thought of as the generalization of well-known originalG′/G-expansion method proposed by M. Wang et al. It is shown that the two-variable (G′/G,1/G)-expansion method provides a more powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics. Comparison between our results and the well-known results is given.

The(G'/G,1/G)-Expansion Method and Its Applications for Solving Two Higher Order Nonlinear Evolution Equations

The two variable(G'/G,1/G)-expansion method is employed to construct exact traveling wave solutions with parameters of two higher order nonlinear evolution equations, namely, the nonlinear Klein-Gordon equations and the nonlinear Pochhammer-Chree equations. When the parameters are replaced by special values, the well-known solitary wave solutions of these equations are rediscovered from the traveling waves. This method can be thought of as the generalization of well-known original(G'/G)-expansion method proposed by Wang et al. It is shown that the two variable(G'/G,1/G)-expansion method provides a more powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics.

Application of the new approach of generalized (G' /G) -expansion method to find exact solutions of nonlinear PDEs in mathematical physics

The exact solutions of nonlinear evolution equations (NLEEs) play a crucial role to make known the internal mechanism of complex physical phenomena. In this article, we construct the traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation by means of the new approach of generalized (G′ /G) -expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, trigonometric, and rational functions. It is shown that the new approach of generalized (G′ /G) -expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations. BIBECHANA 10 (2014) 58-70 DOI: http://dx.doi.org/10.3126/bibechana.v10i0.9312

Exact Traveling Wave Solutions of Nonlinear PDEs in Mathematical Physics

In the present article, we construct the exact traveling wave solutions of nonlinear PDEs in mathematical physics via the variant Boussinesq equations and the coupled KdV equations by using the extended mapping method and auxiliary equation method. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

The modified alternative -expansion method for finding the exact solutions of nonlinear PDEs in mathematical physics

In this article, we construct new exact traveling wave solutions of some nonlinear evolution equations in mathematical physics with the aid of the modified alternative –expansion method. The nonlinear equations under investigation include, the (1+1)-dimensional Drinfeld-Sokolov-Wilson (DSW) equation, the travelling regularised long wave (TRLW) equation, the Caudrey-Dodd-Gibbon (CDG) equation and the (2+1)-dimensional Nizhnik-Novikov-Vesselov (NNV) equation. In the alternative –expansion method, satisfies the Riccati equation where and are arbitrary constants. The obtained solutions are expressed in terms of hyperbolic function, trigonometric function and rational functions. Key words: The -expansion method, the Riccati equation, the traveling wave solutions, the Drinfeld-Sokolov-Wilson equation, the travelling regularised long wave equation, the Caudrey-Dodd-Gibbon equation, the Nizhnik-Novikov-Vesselov equation.

The Traveling Wave Solutions for Some Nonlinear PDEs in Mathematical Physics

In the present article, we construct the exact traveling wave solutions of some nonlinear PDEs in the mathematical physics via (1 + 1) dimensional Kaup Kupershmit equation, the (1 + 1) dimensional seventh order KdV equation and (1 + 1) dimensional Kersten-Krasil Shchik equations by using the modified truncated expansion method. New exact solutions of these equations are found.

Applications of an Extended (G′/G)‐Expansion Method to Find Exact Solutions of Nonlinear PDEs in Mathematical Physics

We construct the traveling wave solutions of the (1+1)‐dimensional modified Benjamin‐Bona‐Mahony equation, the (2+1)‐dimensional typical breaking soliton equation, the (1+1)‐dimensional classical Boussinesq equations, and the (2+1)‐dimensional Broer‐Kaup‐Kuperschmidt equations by using an extended (G′/G)‐expansion method, where G satisfies the second‐order linear ordinary differential equation. By using this method, new exact solutions involving parameters, expressed by three types of functions which are hyperbolic, trigonometric and rational function solutions, are obtained. When the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions.