Studies Of Exact Solutions To Several Kinds Of The Ordinary Difference Equations

The study of difference equations is main about two problems : exact solutions of the equations and qualitative analysis of the solutions. Qualitative analysis of the solutions generally bases on difference models and concretely talks about some problems of the equations, such as stability, boundedness, vibratility, asymptotic property, periodicity and almost periodicity. People have got many important results related above problems, but researching for exact solutions of the equations comparatively lag. In spite of some people have got along in this way, there are still many immature results. For examples: some argumentations are not consummate and the anti-difference of general equation can not be got easily in linear difference equations; the independent linearity between the multi-ply latent roots of homogenous linear equations with constant coefficients and their responded special solutions isn't apparent.; the main results of difference equations with variable coefficients are still on the construction and form of the solutions, there still isn't a universal method; it is hard to find the exact solutions of the non-linear equations even an order discrete Riccalti equation.In the article, methods of finding the exact solution of several kinds of difference equations which are defined in set of integral Z or it's subset D and the apparent expression of the solutions are mostly talked about. Main results are listed as follow::â… ) For linear homogenous difference equations :We get the solution of equations (1) and point out the general solution can be expressed as y(k) = c₁y₁(k) + c₂y₂(k)+...+ c_ny_n(k),and we can say c₁,c₂,...,c_n are arbitraryconstants, and y₁(k), y₂(k),...,y_n(k) is group of base solutions. We can prove theindependent linearity between the multi-ply latent roots of homogenous linear equations with constant coefficients and their responded special solutions and deduce the expression of the general solution of the difference equations by using method of undetermined parameters (Ruan J., 2002).â…¡) For linear non-homogenous equations:we have proved the general solution of equation (2) equals to the sum of one special solution and the general solution of equation (1). If a group of base solutions of a homogenous equation are known, the form of a special solution of non-homogenous equation can be got by method of variation of constant and computing anti-difference.1. For linear non-homogenous difference equation with constant coefficients, we can use method of eigenfunotion (Li Zi-zhen and Gong Dong-shan, 2009) to get the formulaspecial solution when the non-homogenous item f(k) is polynomial functions,exponential functions, trigonometric function, product of polynomial functions and exponential functions, logarithm functions or their linear combination. The method is simple and feasible and the formation of the special solution is very intuitionistic, without the defect of the old method which compute excessively, such as variation of constant (Paul Mason Batchelder, 1927), coefficient comparison (Saber Elaydi, 2005), Laplace transformation (Zhang G. and Zhang G Y., 2001).2. For several kind of linear difference equations with variable coefficients which can be solved exactly, we can use method of constructing function to change some equations with variation coefficients into equation with constant coefficients and find the general solution of one order linear difference equations with variation coefficients by importing definite anti-difference of variable upper limit. Then use method of function integral to get the general solution when coefficient is a linear function and observe a special solution which is the base to get the general solution of the two order linear homogenous difference equations with variation coefficients. Last by dint of depression of order we translate the compound difference equation into some one order linear difference equations with variation coefficients to find the solution when coefficient functions satisfy some conditions.â…¢) For linear difference equation group:We get the sufficient and necessary condition under which equation group (3) has solution. The result strengthen the sufficient conclusion about the same solution of one order linear difference equations and high order linear difference equations, made by (Wang L., 1991) .Also, this result consummates the theory system of finding exact solutions of linear difference equations and deduce from it out the apparent expression of the exact solution of linear difference equation group with constant coefficients by theory of linear algebra and matrix.â…£) For two kinds of non-linear difference equationsThe general solution of the homogenous equation responded to (4) and (5) can be got by importing independent general solution (Gong Dong-shan, 2008). Research proves that the general solution of the two kinds of homogenous difference equations is formed by some independent general solution whose number isn't related to the order of the difference equation. It is also proved that the linear correlation of some provided special solution whose number is related to the equation degree when non-homogenous item is some special functions.