Finite difference method

Abstract: . A conservative difference scheme is presented for the initial-boundary value problem for generalized Zakharov equations. The scheme canbe implicit or semiexplicit depending on the choice of a parameter. On thebasis of a priori estimates and an inequality about norms, convergence of thedifference solution is proved in order 0(h2 + t2) , which is better than previous results. IntroductionThe Zakharov equations [20](1.1) iEt + Exx-NE = 0,(1.2) ^Ntt-{N+\E\2)xx = 0describe the propagation of Langmuir waves in plasmas. Here the complexunknown function E is the slowly varying envelope of the highly oscillatoryelectric field, and the unknown real function N denotes the fluctuation of the ion density about its equilibrium value.The global existence of a weak solution for the Zakharov equations in one dimension is proved in [19], and existence and uniqueness of a smooth solutionfor the equations are obtained provided smooth initial data are prescribed.Numerical methods for the Zakharov equations are studied only in [5, 9, 10,

Abstract: In this paper, an efficient numerical method is introduced for solving the fractional (Caputo sense) Fisher equation. This equation presents the problem of biological invasion and occurs, e.g., in ecology, physiology, and in general phase transition problems and others. We use the spectral collocation method which is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce the proposed problem to a system of ODEs, which is solved by using finite difference method (FDM). Some theorems about the convergence analysis are stated. A numerical simulation and a comparison with the previous work are presented. We can apply the proposed method to solve other problems in engineering and physics.