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A method to construct the exact solution of the PDE is presents, which combines the two kind methods(the nonlinear transformation and RQ(Reduction the PDE to a Quadrature problem) method).The nonlinear diffusion equation is chosen to…
This report provides an investigation into solving the Kuramoto-Sivashinsky equation in two spatial dimensions (2DKS) using a pseudo-spectral method on various rectangular periodic domains. The Kuramoto-Sivashinsky equation is a fluid…
The Conjugate Gradient method (CGM) is known to be the fastest generic iterative method for solving linear systems with symmetric sign definite matrices. In this paper, we modify this method so that it could find fundamental solitary waves…
In this paper, we study the existence, regularity, and approximation of the solution for a class of nonlinear fractional differential equations. {In order to do this}, suitable variational formulations are defined for a nonlinear boundary…
The variable separated ODE method is extended by choosing the additional variable separated equation as the general elliptic equation. More exact traveling wave solutions of nonlinear equations are obtained by using the method of comparison…
The Neumann--Neumann method is a commonly employed domain decomposition method for linear elliptic equations. However, the method exhibits slow convergence when applied to semilinear equations and does not seem to converge at all for…
It is well recognized that in auxiliary equation methods, the exact solutions of different types of auxiliary equations may produce new types of exact travelling wave solutions to nonlinear partial differential equations in hand. In this…
A non-linear differential equation arising from a stochastic process known as branching Brownian motion is considered. We find an explicit solution and show the uniqueness of the solution under some boundedness conditions using…
The iterative problem of solving nonlinear equations is studied. A new Newton like iterative method with adjustable parameters is designed based on the dynamic system theory. In order to avoid the derivative function in the iterative…
A multiple exp-function method to exact multiple wave solutions of nonlinear partial differential equations is proposed. The method is oriented towards ease of use and capability of computer algebra systems, and provides a direct and…
We show how to reduce the problem of solving members of a certain family of nonlinear differential equations to that of solving some corresponding linear differential equations.
The approach allowing is considered to represent the solutions such as stationary lonely waves of various nonlinear wave the equations as system of the ordinary differential equations in variable action - angle.
The Kudryashov-Sinelshchikov equation for describing the pressure waves in liquid with gas bubbles is studied. New exact solution of this equation are found. Modification of truncated expansion method is used for obtaining exact solution of…
We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the…
In this paper we study about the existence of solutions of certain kind of non-linear differential and differential-difference equations. We give partial answer to a problem which was asked by chen et al. in [13].
We introduce a direct numerical treatment of nonlinear higher-index differential-algebraic equations by means of overdetermined polynomial least-squares collocation. The procedure is not much more computationally expensive than standard…
An explicit analytic solution to the nonlinear differential equation d^k y (--) ^n = y^l dx^kk is obtained for arbitrary integer values of k, l and n.
In this paper, we introduce the new optimal perturbation iteration method based on the perturbation iteration algorithms for the approximate solutions of nonlinear differential equations of many types. The proposed method is illustrated by…
We present a numerical algorithm for finding real non-negative solutions to polynomial equations. Our methods are based on the expectation maximization and iterative proportional fitting algorithms, which are used in statistics to find…
In this paper, two kinds of the exact singular solutions are obtained by the improved homogeneous balance (HB) method and a nonlinear transformation. The two exact solutions show that special singular wave patterns exists in the classical…