一种基于完全线性变换法则的差分方程解析解算法

摘要

It is difficult to get the analytic solution of difference equation by using traditional methods with computer. A linear method based on complete linear transformation for getting the analytic solution of difference equation is proposed in this paper. The idea of this new method is dimensionality reduction.The orders of the high-order difference equation are reduced through determinant calculation, and this process is repeated till the last analytic expression of the solution of the difference equation appears. In essence, this algorithm is a transformation of Z-transform method combined with linear algebra method.The linear property of this algorithm enables one to get analytic solution other than numerical solution of difference equations using computer.%利用传统方法很难在计算机上实现差分方程的解析解求解,提出了一种获得差分方程解析解的线性算法,该算法的基础是完全线形变化法.其核心操作为降维处理,对高阶差分方程进行逐次降阶运算,直至获得其解析解表达式.本质上,该算法属于Z变换法的一种矩阵法变形.算法的线性特征使得其容易移植到计算机上实现差分方程的解析解运算,而非传统的数值迭代解.