Approximate solution of general high-order linear nonhomogeneous difference equations by means of Taylor collocation method

https://doi.org/10.1016/j.amc.2005.04.048Get rights and content

Abstract

In this paper, a Taylor collocation method is developed to find an approximate solution of general high-order linear nonhomogenous difference equations with variable coefficients under the mixed conditions. The solution is obtained in terms of Taylor polynomials about any point. Also, examples are presented which illustrate the pertinent features of the method.

Introduction

Taylor and Chebyshev collocation methods for the approximate solutions of differential, integral and integrodifferential equations have been presented in many papers [1], [2], [3], [4].

On the other hand, a Taylor expansions approach to solve high-order linear difference equations has been presented by Gülsu and Sezer [5]. In this paper, these methods are developed and applied to the mth-order linear nonhomogenous difference equation with variable coefficients, which is given in [6, p. 174] and [7, p. 176],k=0mPk(x)y(x+k)=f(x),k0,kN+with the mixed conditionsj=0paijy(cj)=μi,acj,xb;i=0,1,,m-1and the solution is expressed as the Taylor polynomialy(x)=n=0Ny(n)(c)n!(x-c)n,ax,cbso that y(n)(c), n = 0, 1,  , N are the coefficients to be determined. Here Pk(x) and f(x) are functions defined on a  x  b; the real coefficients aij, cj and μi are appropriate constants.

Section snippets

Fundamental matrix relations

Let us first consider the desired solution y(x) of Eq. (1) defined by a truncated Taylor series (3). Then the solutions y(x) can be expressed in the matrix form[y(x)]=XM0A,whereX=[1(x-c)(x-c)2(x-c)N],A=[y(0)(c)y(1)(c)y(N)(c)]TandM0=10!00...0011!0...00012!...0............000...1N!.To obtain such a solution, we can use the following matrix method, which is a Taylor collocation method [1]. This method is based on computing the Taylor coefficients by means of the Taylor collocation points [1] and

Method of solution

The fundamental matrix Eq. (11) for Eq. (1) corresponds to a system of (N + 1) algebraic equations for the (N + 1) unknown coefficients y(0)(c), y(1)(c),  , y(N)(c).

Briefly we can write (11)WA=For[W;F]so thatW=[wnh]=k=0mPkCMk;h,n=0,1,,N.We can write the corresponding matrix form (13) for the mixed conditions (2) in the augmented matrix form as[ui;μi]=[ui0ui1uiN;μi],i=0,1,,m-1.To obtain the approximate solution of Eq. (1) under the mixed conditions (2) in terms of Taylor polynomials, by replacing

Illustration

The method of this study is useful in finding the solutions of linear difference equations with variable coefficients in terms of Taylor polynomials. We illustrate it by the following examples.

Example 1

Let us first consider the linear second-order difference equation(x-1)y(x+2)+(2-3x)y(x+1)+2xy(x)=1with y(0) = 2 and y(1) = 2, 0  x  2 and approximate the solution y(x) by the Taylor polynomialy(x)=n=05y(n)(0)n!xn,where a = 0, b = 2, c = 0, P0(x) = 2x, P1(x) = (2  3x), P2(x) = x  1, f(x) = 1. Then, for N = 5, the collocation

Conclusions

High order difference equations are usually difficult to solve analytically. Then it is required to obtain the approximate solutions. For this reason, the present method has been proposed for approximate solution and also analytical solution.

The method presented in this study is a method for computing the coefficients in the Taylor expansion of the solution of a linear difference equation, and is valid when the functions Pk(x) and f(x) are defined [a, b].

The Taylor collocation method is an

References (7)

There are more references available in the full text version of this article.

Cited by (5)

  • Generalized Taylor polynomials for axisymmetric plates and shells

    2016, Applied Mathematics and Computation
    Citation Excerpt :

    Higher-order linear complex differential equations in the elliptic domains have been solved by Sezer et al. [18], using a computer program written in Maple9, based on the Taylor collocation method and its validity tested using some illustrative examples. Solution of high-order nonhomogeneous difference equations has been obtained by Gülsu et al. [19] and by Gökmen and Sezer [20]. The obtained solutions, in terms of Taylor polynomials about any point, are for the equations having variable coefficients in both the aforementioned two references.

  • The solution of the Bagley-Torvik equation with the generalized Taylor collocation method

    2010, Journal of the Franklin Institute
    Citation Excerpt :

    Therefore, this method can be used to solve many important fractional differential equations. The basic idea of the Taylor collocation method in [25,26] is developed and applied to the mαth-order linear fractional differential equations with variable coefficients. We adopt Caputo's fractional derivative definition, which is a modification of the Riemann–Liouville definition and has an advantage of dealing properly with initial value problems, for the concept of the fractional derivative.

View full text