## Exact finite-difference schemes for two-dimensional linear systems with constant coefficients.(English)Zbl 1151.65066

The autor presents an exact finite-difference method that can be applied to any two-dimensional linear system of first-order differential equations with constant coefficients. Table 1 gives the six possible cases of an exact finite-difference method $(1/\varphi)(x_{k+1}-x_k)=A(\theta x_{k+1}+(1-\theta)x_k)$ for the differential system $$x'(t)=A x(t)$$, where the functions $$\varphi$$ and $$\theta$$ are determined by the two eigenvalues of $$A$$, the step size $$h$$ and each of the six Jordan canonical forms of $$A$$. There is an alternative method in one case, and three special cases in another case of the Jordan canonical form.

### MSC:

 65L12 Finite difference and finite volume methods for ordinary differential equations 34A30 Linear ordinary differential equations and systems 65L05 Numerical methods for initial value problems involving ordinary differential equations
Full Text:

### References:

 [1] Mickens, R.E., Nonstandard finite difference models of differential equations, (1994), World Scientific New Jersey · Zbl 0925.70016 [2] Mickens, R.E., Applications of nonstandard finite difference schemes, (2000), World Scientific New Jersey · Zbl 1237.76105 [3] Mickens, R.E., Advances in the applications of nonstandard finite difference schemes, (2005), World Scientific New Jersey · Zbl 1079.65005 [4] Mickens, R.E.; Oyedejib, K.; Rucker, S., Exact finite difference scheme for second-order, linear ODEs having constant coefficients, J. sound vibration, 287, 1052-1056, (2005) · Zbl 1243.65097 [5] Patidar, K.C., On the use of nonstandard finite difference methods, J. difference equations appl., 11, 735-758, (2005) · Zbl 1073.65545 [6] L.-I.W. Roeger, Exact nonstandard finite-difference methods for a linear system—the case of a center, J. Difference Equations Appl., to appear. [7] L.-I. W. Roeger, Exact finite-difference method for second-order linear equations with constant coefficients, under review.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.