Partial Differential Equations In Mathematical Physics Research Articles

A new technique of reduce differential transform method to solve local fractional PDEs in mathematical physics

In this manuscript, we investigate solutions of the partial differential equations (PDEs) arising in mathematical physics with local fractional derivative operators (LFDOs). To get approximate solutions of these equations, we utilize the reduce differential transform method (RDTM) which is based upon the LFDOs. Illustrative examples are given to show the accuracy and reliable results. The obtained solutions show that the present method is an efficient and simple tool for solving the linear and nonlinear PDEs within the LFDOs.

Symmetries of partial differential equations and stochastic processes in mathematical physics and in finance

In 1971, B. Kent Harrison and Frank B. Estabrook introduced a method to determine the symmetries of partial differential equations (PDEs). These last years, the determination of the symmetries of PDEs in Mathematical Physics, in Mathematical Biology and in Financial Mathematics has proved useful. The computations effected in all these cases let appear a remarkable degree of similarity between them. So with the same aim in mind, we develop a general framework for the computation of the symmetries with this method, we give properties of isovectors for a rather general type of PDE’s and some results on the Lie algebra itself. Finally we present three examples for which all the results we exposed hold.

Solitons and Other Exact Solutions for Two Nonlinear PDEs in Mathematical Physics Using the Generalized Projective Riccati Equations Method

We apply the generalized projective Riccati equations method with the aid of Maple software to construct many new soliton and periodic solutions with parameters for two higher-order nonlinear partial differential equations (PDEs), namely, the nonlinear Schrödinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity and the nonlinear quantum Zakharov-Kuznetsov (QZK) equation. The obtained exact solutions include kink and antikink solitons, bell (bright) and antibell (dark) solitary wave solutions, and periodic solutions. The given nonlinear PDEs have been derived and can be reduced to nonlinear ordinary differential equations (ODEs) using a simple transformation. A comparison of our new results with the well-known results is made. Also, we drew some graphs of the exact solutions using Maple. The given method in this article is straightforward and concise, and it can also be applied to other nonlinear PDEs in mathematical physics.

Solitons and other exact solutions for a class of nonlinear Schrödinger-type equations

Recently, Zayed and Alurrfi have applied an extended auxiliary equation method for constructing the exact solutions of three nonlinear partial differential equations (PDEs) via the (2+1)-dimensional nonlinear cubic-quintic Ginzburg–Landau equation, the (1+1)-dimensional resonant nonlinear Schrödinger's equation with parabolic law nonlinearity and the (1+1)-dimensional nonlinear generalized Zakharov system of equations. In this article we apply a different method called a new mapping method proposed by Zeng and Yong for finding many other new exact solutions for the same three nonlinear PDEs mentioned above. Comparing the solutions resulting from these two different methods are given. The obtained results confirm that both the extended auxiliary equation method and the new mapping method are efficient techniques for analytic treatments of a wide variety of nonlinear PDEs in mathematical physics.

Exact traveling wave solutions for nonlinear PDEs in mathematical physics using the generalized Kudryashov method

The generalized Kudryashov method is applied in this article for finding the exact solutions of nonlinear partial differential equations (PDEs) in mathematical physics. Solitons and other solutions are given. To illustrate the validity of this method, we apply it to three nonlinear PDEs, namely, the diffusive predator-prey system, the nonlinear Bogoyavlenskii equations and the nonlinear telegraph equation. These equations are related to signal analysis for transmission and propagation of electrical signals. As a result, many analytical exact solutions of these equations are obtained including symmetrical Fibonacci function solutions and hyperbolic function solutions. Physical explanations for some solutions of the given three nonlinear PDEs are obtained. Comparison our new results with the well-known results are given.

A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines

The first elliptic function equation is used in this article to find a new kind of solutions of nonlinear partial differential equations (PDEs) based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method. New exact solutions to the Jacobi elliptic functions of a nonlinear PDE describing the nonlinear low-pass electrical lines are obtained with the aid of computer algebraic system Maple. Based on Kirchhoff’s law, the given nonlinear PDE has been derived and can be reduced to a nonlinear ordinary differential equation (ODE) using a simple transformation. The given method in this article is straightforward and concise, and it can also be applied to other nonlinear PDEs in mathematical physics. Further results may be obtained.

A New Jacobi Elliptic Function Expansion Method for Solvinga Nonlinear PDE Describing Pulse Narrowing Nonlinear Transmission Lines

In this article, we apply the first elliptic function equation to find a new kind of solutions of nonlinear partial differential equations (PDEs) based on the ho- mogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method. New exact solutions to the Jacobi elliptic functions of a nonlinear PDE describing pulse narrowing nonlinear transmission lines are given with the aid of computer program, e.g. Maple or Mathematica. Based on Kirchhoff's current law and Kirchhoff's voltage law, the given nonlinear PDE has been derived and can be reduced to a nonlinear ordinary differentialequation (ODE) using asimple transformation. The given method in this article is straightforward and concise, and can be applied to other nonlinear PDEs in mathematical physics. Further results may be obtained.

An efficient algorithm for some highly nonlinear fractional PDEs in mathematical physics.

In this paper, a fractional complex transform (FCT) is used to convert the given fractional partial differential equations (FPDEs) into corresponding partial differential equations (PDEs) and subsequently Reduced Differential Transform Method (RDTM) is applied on the transformed system of linear and nonlinear time-fractional PDEs. The results so obtained are re-stated by making use of inverse transformation which yields it in terms of original variables. It is observed that the proposed algorithm is highly efficient and appropriate for fractional PDEs and hence can be extended to other complex problems of diversified nonlinear nature.

English

In this paper we apply the improved Riccati equation mapping method to construct many families of exact solutions of a nonlinear partial differential equation involving parameters of a special interest in nanobiosciences and biophysics which describe a model of microtubules as nonlinear RLC transmission lines. As results, we can successfully recover the previously known results that have been found using other methods. This method is straightforward and concise, and it can be applied to other nonlinear PDEs in mathematical physics. Comparison between our new results and the well-known results are given. Some comments on the well-known results are also presented at the end of this article. Key words: Improved Riccati equation mapping method, exact traveling wave solutions, nonlinear partial differential equations (PDEs) of microtubules, Nonlinear RLC transmission lines. PACS: 02.30.Jr, 05.45.Yv, 02.30.Ik