Analytical solutions of linear diffusion and wave equations in semi-infinite domains by using a new integral transform

In this article, the extended multiple Riccati equations expansion method has been used to construct a series of double soliton- like solutions, double triangular function solutions and complexiton soliton solutions for some nonlinear partial differential equations in mathematical physics via the (2+1) dimensional breaking soliton equations, (2+1) dimensional painleve integrable Burger's equations and (2+1) dimensional Nizhnik- Novikov- Vesselov equations. With the help of symbolic computation as Maple or Mathematica, we obtain many new types of complexiton soliton solutions, for example, various combination of trigonometric periodic function and hyperbolic function solutions, various combination of trigonometric periodic function and rational function solutions, various combination of hyperbolic function and rational function solutions. Key words: The extended multiple Riccati equations expansion method, double soliton-like solutions, double triangular function solutions, complexiton soliton solutions, the (2+1) dimensional breaking soliton equations, the (2+1) dimensional painleve integrable Burger's equations, the (2+1)-dimensional Nizhnik-Novikov- Vesselov equations.

New solitons and periodic solutions for nonlinear physical models in mathematical physics

In this article, many new exact solutions of the (2+1)-dimensional nonlinear Boussinesq-Kadomtsev-Petviashvili equation and the (1+1)-dimensional nonlinear heat conduction equation are constructed using the Riccati equation mapping method. By means of this method, many new exact solutions are successfully obtained. This method can be applied to many other nonlinear evolution equations in mathematical physics. doi: 10.14456/WJST.2014.14

In this paper, we study the application of a version of the method of simplest equation for obtaining exact traveling wave solutions of the Zakharov–Kuznetsov equation, the modified Zakharov–Kuznetsov equation, and their generalized forms. The Duffing-type equation is used as simplest auxiliary equation. In the meantime, the proposed method is proved to be a powerful mathematical tool for obtaining exact solutions of nonlinear partial differential equations in mathematical physics.

We apply the Exp-function method (EFM) to the Biswas-Milovic equation and derive the exact solutions. This paper studies the Biswas-Milovic equation with power law, parabolic law and dual parabolic law nonlinearities by the aid of the Exp-function method. The obtained solutions not only constitute a novel analytical viewpoint in nonlinear complex phenomena, but they also form a new stand alone basis from which physical applications in this arena can be comprehended further, and, moreover, investigated. Furthermore, to concretely enrich this research production, we explain all cases, namely $m=1$ and $ m\geq 2$ . This method is developed for searching exact travelling-wave solutions of nonlinear partial differential equations. It is shown that this methods, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear partial differential equations in mathematical physics.

Analytical approach for space–time fractional Klein–Gordon equation

Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres

In this paper, we study an extended modified Korteweg-de Vries equation, which contains the relevant higher-order nonlinear terms and fifth-order dispersion. This equation is the extension of the modified Korteweg-de Vries equation and described by the Ablowitz-Kaup-Newell-Segur hierarchy. The standard Korteweg-de Vries equation is the pioneer integrable model in solitary waves theory, which gives rise to multiple soliton solutions. The Korteweg-de Vries equation arises naturally from shallow water, plasma physics, and other fields of science. To obtain exact solutions the sine-cosine method is applied. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics. Traveling wave solutions are determined for extended modified Korteweg-de Vries equation. The study shows that the sine–cosine method is quite efficient and practically well suited for use in calculating traveling wave solutions for extended modified Korteweg-de Vries equation.

О разноснтых схемах для многомерных дифференциальных уравнений математической физики

A generalized F-expansion method and new exact solutions of Konopelchenko–Dubrovsky equations

The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equations

In this paper, the generalized Jacobi elliptic functions expansion method with computerized symbolic computation are employed to investigate explicitly analytic solutions of the (N + 1)-dimensional generalized Boussinesq equation. The exact solutions to the equation are constructed analytically under certain circumstances, some of these solutions are degenerated to soliton-like solutions and trigonometric function solutions in the limit cases when the modulus of the Jacobi elliptic function solutions tends to 0 and 1, which shows that the applied method is more powerful and will be used in further works to establish more entirely new exact solutions for other kinds of higher-dimensional nonlinear partial differential equations in mathematical physics.

Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as a “Guide to the Functions”. This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to “new” classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag–Leffler type functions, among them the “Queen function of fractional calculus”, the Mittag–Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author’s works for more than 30 years, and support the wide spreading and important role of these functions by several examples.

In this paper, we propose a new method called exp(−I•(ξ)) fractional expansion method to seek traveling wave solutions of the nonlinear fractional Sharma-Tasso-Olver equation. The result reveals that the method together with the new fractional ordinary differential equation is a very ini¬‚uential and effective tool for solving nonlinear fractional partial differential equations in mathematical physics and engineering. The obtained solutions have been articulated by the hyperbolic functions, trigonometric functions and rational functions with arbitrary constants

New analytical solutions of (2 + 1)-dimensional conformable time fractional Zoomeron equation via two distinct techniques