本研究以微分方程式最大值原理為主要觀念，建構出完整的理論架構，並求解微分方程式之近似解，為正確解之較大或較小近似值。依據所求得之平均近似解進行誤差分析，有效解決傳統數值方法上常無謂地增加格點數或近似函數數目之缺點。此理論具備誤差分析之特性，將節省計算時間與運算空間，並避免不必要反覆式地數值精度測試，預期將成為極具學術價值與實用性之數值研究方法。
在微分方程式最大值原理應用於殘差修正法前，必須判斷該方程式是否具備單調之性質。除運用傳統單調性基礎理論之單調疊代的觀念外，本研究提出漸進式疊代法，讓最大值原理適用於更複雜的方程式中。在具有精準度高及應用範圍廣泛的樣線函數離散微分方程式後，利用殘差修正法之觀念，將單調性所推導出不等式拘束條件的數學規劃問題，轉換為簡單的等式疊代問題。
因此，本研究在確認微分方程式具備單調性質後，由殘差修正法得到較大及較小近似解及其的最大可能誤差範圍。除此，更可有效且準確地求解各類複雜的微分方程式。

The maximum principle of the differential equation is used as the main concept to form the complete structure in this study. Based on this, the upper and lower approximate solutions of the exact solution can be obtained. Analyzing the error according to the mean value of the upper and lower approximate solutions can effectively deal with the defects resulting from increasing the numbers of grids or approximate functions when using traditional numerical methods. This methodology can reduce the computing time, save the memory, and promote the numerical accuracy outstandingly. Predictably, it will achieve high academic value and practicability in numerical research in the future.
Before applying the maximum principle of differential equations to residual correction method, the existence of the monotonicity of the certain equation must be determined. Except the monotonic iteration method used in conventional fundamental theory of monotonicity, the evolutionary iteration method is applied in this study and makes the maximum principle more suitable to complicated equations. Moreover, based on that the properties of the spline functions which the solutions are more accurate and are applied in wider field, the mathematic programming problems with inequality constraints derived by the monotonicity of the maximum theory could be transform to the simple iteration problems with equality constraints by the residual correction method after the discretization of the differential equations.
Hence, after making sure the differential equations have the monotonicity. By the residual correction method, the upper and lower approximate solutions and maximum error range of solutions can be acquired. Besides, many kinds of complicated differential equations are able to be solved with high accuracy and efficiency by this methodology.

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