在科學研究及工程應用上，經常遇到各式各樣定解問題；即在已知的初始值條件或邊界條件下，求解問題的統御方程式。這些統御方程式，不僅有微分方程式，亦有積分方程式；既有線性方程式，也有非線性方程式；既有單一的方程式，也有耦合的方程式。然而，目前關於微分方程式或積分方程式的研究，在較複雜邊界問題中能得到精確解者仍極為少數。因此在某些問題的求解上，往往只能透過近似解來實現。本文使用的殘差修正法(Residual Correction Method)就是諸多近似方法中之一種。
隨著科技的進步，在雷射熱源或微波的應用上，往往牽涉極短時間、極高頻率、或非常高溫度梯度的情形，此時非傅立葉熱傳變得愈來愈引人注目。基於微分方程最大值原理的觀念，在本文中，分別應用殘差修正法結合有限差分法，求解具有球面等向性的連續體之非傅立葉熱傳及熱彈性問題的較大近似解及較小近似解。在微分方程式最大值原理下，殘差關於解之單調性可被推論出，而本研究所探討之殘差修正法恰為一建構於此殘差與解單調關係上之數值方法。數值求解以有限差分法來離散微分方程式，然後配合本文所提之「殘差修正法」在離散的格點上加入殘差修正量，使得原本複雜的不等式拘束數學規劃(mathematical programming)問題得以轉換成簡單的等式迭代問題。
數值結果發現，本方法在修正過程中具有將較大與較小近似解修正成大致對稱於正確解兩側的特性，因此發現即使在格點數相當少的情況下所得的平均近似解依然相當接近正確解。此外，有限差分法為早期便開始使用的數值方法，其具有運算簡單，使用於低階的方程式又具有良好的準確性，而應用殘差修正法的特點為可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，可節省計算時間與運算空間，此理論還具備誤差分析之特性，可避免不必要反覆式地數值精度測試，整體來說，殘差修正法為便利且效率高又具有良好精度的數值方法，預期將成為極具學術價值與實用性之數值研究方法。

In scientific research and engineering applications, there exist various kinds of problems of finding solutions are faced, that is, solving governing equations of the desired problems under the given initial conditions and boundary conditions. These governing equations have a wide range of types, including either differential equations or integral equations; either linear equations or nonlinear equations; either a single equation or a set of coupled equations. However, to date, there are many investigators that still pose a challenge for solving some differential equations with complex boundary conditions by either numerical or theoretical methods. As for some problems, the analytic exact solutions are impossible to find, and only their approximate solutions can be obtained by some kinds of methods. The residual correction method (RCM) is one promising scheme of them.
With the progress of technology, the effect of non-Fourier heat conduction becomes more and more attractive in practical engineering problems for situations involving a heat source such as laser or microwave with extremely short times or very high frequency, or very high temperature gradient. Based on the maximum principle of differential equations, this dissertation deals with application of the residual correction method in combination with the finite difference method to seek the upper and lower approximate solutions of the non-Fourier heat conduction and thermoelastic problems of a spherical region when the continuum medium pos-sesses spherically isotropic properties. Under the fundamental of the maximum principle of differential equations, the monotonicity of residual versus solution can be concluded and the residual correction method is just a numerical method which is constructed on the basis of this monotonic relation. To obtain numerical solutions, the finite difference method is utilized to discretize differential equations, followed by adoption of the “Residual Correction Method” proposed in this article to add residual correction quantity in discretized grid points to convert constraint mathematical programming problems of inequalities that were complex originally into simpler equational iteration problems.
Numerical results show that this approach is characterized by the advantage in correcting the acquired upper and lower approximate solutions to enable them to be distributed on two sides of the exact solutions roughly in a symmetrical way, thus making the mean approximate solutions remain considerably close to the exact solutions, even if the number of calculation grid points is rather small. In addition, the finite difference method is simple to operate and has good accuracy in low-order equations. One characteristic of residual correction method is analyzing the error according to the mean value of the upper and lower approximate solutions. It 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.

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