本文主要在探討非線性橢圓方程的多重週期解。欲求並延拓非線性偏微分方程組之週期解是一種很困難的工作，本文提供一種演算法來求整個解路徑，並延拓隨著一個參數變動的非線性偏微分方程組之週期解，我們稱之為虛擬弧長延拓法。我們將虛擬弧長延拓法應用在非線性橢圓方程式上，而週期解路徑之虛擬弧長延拓法是基於打靶法、牛頓法、Crank-Nicolson法、猜測及解法及隱函數定理等數值方法，且我們將利用其來探討非線性橢圓方程式週期解路徑。

This paper’s main purpose is to confer the multiple period solution of the nonlinear elliptic equation. Secondly, It is this paper’s aim to solve and continue the period solution of partial differential equations. This paper
will provide the method to calculate whole solution paths and continue the period solution of partial differential equations which variate a parameter.
We call this is the pseudo- arclength continuation method. It is applied to the nonlinear elliptic equation with the pseudo-arclength continuation method, and the pseudo-arclength continu- ation method of the period solution paths on the basis of numerical methods of shooting method、Newton’s method、Crank-Nicolson’s method、predictor-solver and implicit function theorem. These solutions will be used to confer the period solution paths of the non-linear elliptic equation.

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