Approximate solution of general high-order linear nonhomogeneous difference equations by means of Taylor collocation 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],with the mixed conditionsand the solution is expressed as the Taylor polynomialso 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 formwhereandTo 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)so thatWe can write the corresponding matrix form (13) for the mixed conditions (2) in the augmented matrix form asTo 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 equationwith y(0) = 2 and y(1) = 2, 0 ⩽ x ⩽ 2 and approximate the solution y(x) by the Taylor polynomialwhere 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)
- et al.
Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients
Appl. Math. Comput.
(2003) - et al.
A Taylor collocation method for the solution of linear integro-differential equations
Int. J. Comput. Math.
(2002) - M. Sezer, A. Karamete, M. Gülsu, Solutions of systems of linear differential equations with variable coefficients, Int....
Cited by (5)
Generalized Taylor polynomials for axisymmetric plates and shells
2016, Applied Mathematics and ComputationCitation 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 InstituteCitation 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.
A matrix method for solving high-order linear difference equations with mixed argument using hybrid legendre and taylor polynomials
2006, Journal of the Franklin InstituteThe maple program procedures at solution systems of differential equation with Taylor collocation method
2014, Springer Proceedings in Mathematics and StatisticsSolving high-order linear differential equations by a legendre matrix method based on hybrid legendre and taylor polynomials
2010, Numerical Methods for Partial Differential Equations