Numerical Analysis For Some Delay Differential And Difference Equations

In this paper, we study numerical methods for solving free boundary value prob-lem and two-point boundary value problem of one-dimensional p-Laplacian differential equations with delay. We study the multiplicity of positive solutions for a class of p-Laplacian difference equations with delay. We also study the variational iteration method for solving delay differential equations.Delay differential equations is a fundamental and simple kind of functional differen-tial equations [23]. In many applications, one assumes the system under consideration is governed by an equation involving the past and current states of the system. In the real world, some processes are reasonable to be described as delay differential equations. The reason is that the differential of the unknown solutions depends not only on the values of the unknown solutions at the current time, but also on the values prior to that. Such equations play an important role in the mathematical modeling of real world phenomena, such as mechanical, biology, economics, epidemiology and so on [79].The equation with p-Laplacian operator is generalization of the second order or-dinary differential equation. The p-Laplacian differential equations have been vastly applied in many fields such as non-Newtonian mechanic and nonlinear flow laws, etc. [19]. There exists a large number of papers devoted to studying the existence of solutions or positive solutions for this kind of problem [10,39,41,65,72,74]. The main purpose of this paper is to study the numerical method for some delay differential equations.The first part of this paper is to study numerical method for solving free boundary value problem of one-dimensional p-Laplacian differential equations with delay. Consider with the boundary conditions We construct the following difference scheme for equation (1) When p≥2. we analyze the truncation error of the scheme (3).The boundary condition is dealt as The error of (4) is O(h).Theorem 1 The difference scheme to approximate differential equation (1), (2) is When p≥2, then the truncation error of the scheme is O?(h).We prove the existence of positive solutions for difference equation, with the boundary conditions△u(N)=0, u(n)=0, n∈Z(-n0,0). (6)Finally, the error estimate of the difference scheme is also studied for concrete problems. Some numerical experiments are carried out to illustrate our results.The second part of this paper is to study numerical method for solving two-point boundary value problem of one-dimensional p-Laplacian differential equations with de-lay. Consider with the boundary conditionsWe construct the difference scheme for equation (7), (8). Furthermore, we analyze the truncation error of the difference scheme with p≥2.Theorem 2 The difference scheme to approximate differential equation (7), (8) is When p> 2, the truncation error of the scheme is O(h). When p>3 or p=2, the truncation error of the scheme is O(h2).We prove the existence of positive solutions for following difference equation, with the boundary conditions u(N+1)=0, u(n)= 0, n∈Z(-n0,0).Finally, the error estimate of the difference scheme is also studied for concrete problems. Some numerical experiments are carried out to illustrate our results.Differential equations and difference equations can be used as two difference de-scriptions of the system. Difference equations usually describe the evolution of certain phenomena over the course of time. Difference equations play an important role in the mathematical modeling of real world phenomena, such as computer science, engineer-ing mechanics, cybernetic system, non-Newtonian mechanic, neural networks, economics and so on [3,14]. Thus, difference equations have received a lot of attention[6,8,45,46].In this paper, we are concerned with the following p-Laplacian difference equation with delay with the boundary conditions△u(N)=0,u(n)=0, n∈Z(-n0,0). (10)We prove the multiplicity of positive solutions to equation (9), (10).We assume that(H1)f:Z(1, N+l)xR→R+is a continuous function;(H2) a is a positive function defined on Z(1, N+1).Assumefsatisfies the following conditionsTheorem 3 Assume (H1)-(H5), let 0