Partial differential equations with variable coefficients involving discontinuous case play an important part in engineering, physics and ecology. In this paper, we will study nonlinear partial differential equations with variable coefficients arised from population models. Generally speaking, it is hard to analyze the behavior of nonlinear partial differential equations, therefore we usually r...

Partial Differential Equations (PDEs) have become an important tool in image processing and analysis. A PDE mode for image zooming is introduced in this paper. This model exploits a higher order nonlinear partial differential equation. The resulted nonlinear equation is solved by an explicit finite difference schemes. Numerical results on real digital images are given to show effectiveness and ...

The multilevel iterative technique is a powerful technique for solving the systems of equations associated with discretized partial differential equations. We describe how this technique can be combined with a globally convergent approximate Newton method to solve nonlinear partial differential equations. We show that asymptotically only one Newton iteration per level is required; thus the comp...

In this paper, we explore more applications of the hyperbolic function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations (PDES), to special discrete nonlinear equations. Some exact travelling wave solution of the discrete sine-Gordon equation are obtained in terms of hyperbolic fun...

Abstract: In this article, we propose a reliable combination of Homotopy analysis method (HAM) and Laplace decomposition method (LDM) to solve linear and nonlinear partial differential equations effectively with less computation. The proposed method is called homotopy analysis transform method (HATM). This study represents significant features of HATM and its capability of handling linear and n...

Specific competencies for the project 25 years of research experience in nonlinear partial differential equations; Extensive photographic skills (see http://pbase.com/markowich); Wittgenstein Award of the Austrian Research Fund (FWF) 2000; Corresponding Member of the Austrian Academy of Sciences since 2005; Coordination and participation in Austrian and international grants and network projects...

Laboratory experiments have shown that when nonlinear, dispersive waves are forced periodically from one end of an undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. It is our purpose here to establish this as a fact at least in the context of a damped Korteweg-de Vries equation. Thus, consideration is given to the initial-...

We study the Whitham equations for the defocusing complex modified KdV (mKdV) equation. These Whitham equations are quasilinear hyperbolic equations and they describe the averaged dynamics of the rapid oscillations which appear in the solution of the mKdV equation when the dispersive parameter is small. The oscillations are referred to as dispersive shocks. The Whitham equations for the mKdV eq...

Approved: Thesis Supervisor Title and Department

We use the improved general mapping deformationmethod based on the generalized Jacobi elliptic functions expansionmethod to construct some of the generalized Jacobi elliptic solutions for some nonlinear partial differential equations in mathematical physics via the generalized nonlinear Klein-Gordon equation and the classical Boussinesq equations. As a result, some new generalized Jacobi ellipt...