以指數試函數解奇異常微分方程及橢圓偏微分方程

二階線性奇異攝動常微分方程式轉化為奇異線性型拋物線偏微分方程式。由格林第二恆等式所推導的邊界積分方法得到伴隨Trefftz試函數，導出符合伴隨性質的控制方程式和伴隨邊界條件的譜函數後，我們使用弱形式積分方程式方法求解奇異解。該方法與指數擬合試函數是為了用來自動滿足邊界條件。對於在解線性和非線性變係數奇異問題之內部邊界層也非常準確。奇異對流擴散方程和橢圓型奇異反應擴散方程式若使用一般數值方法會使得問題變病態。於是我們使用配點法來提高精準度。當奇異解以一組二維指數試函數方程式形式表示，並滿足邊界條件與控制方程式我們可以得到一個小尺度的線性系統來求展開係數。在數值算例中證實了配點法求解高度奇異橢圓型問題非常精確有效。

關鍵字

奇異攝動問題；弱形式積分方程式方法；伴隨Trefftz試函數；譜函數；指數擬合試函數；奇異橢圓型方程式；配點法

並列摘要

Second-order linear singularly perturbed ordinary differential equation is transformed into a singular linear parabolic type partial differential equation. Then, the Green’s second identity is employed to derive a boundary integral equation in terms of the adjoint Trefftz test functions. After deriving the closed-form spectral functions, we develop a weak-form integral equation method (WFIEM) to find the singular solution. The WFIEM together with the exponentially fitted trial functions, which are designed to satisfy the boundary conditions automatically, can provide accurate solutions of the highly singular problem, even for the time-varying equation with the internal boundary layer. We develop a collocation method (CM) to solve the singular convection- diffusion equation and singular reaction-diffusion equation of elliptic type, which are too ill-posed to be solved by the conventional numerical method. However, when the singular solution is expressed in terms of 2D exponential trial functions, and after collocating points to satisfy the boundary conditions and the governing equation, we can obtain a small scale linear system, which is solved to determine the expansion coefficients. The numerical algorithm CM is effective and accurate in solutions of highly singular elliptic type problems, and the numerical examples confirm these assessments.

並列關鍵字

Singularly perturbed problem ； Weak-form integral equation method ； Adjoint Trefftz test functions ； Spectral functions ； Exponentially fitted trial functions ； Singular elliptic equation ； Collocation method

參考文獻

[19] Dogan N, Erturk VS, Akin O. Numerical treatment of singularly perturbed two-point boundary value problems by using differential transformation method, Disc. Dyna. Nat. Soc., Article ID 579431 (2012) 10 pages.

[3] Shampine LF, Gear CW. A user's view of solving stiff ordinary diferential equations.SIAM Rev. 21 (1979) 1-17.

[4] Nayfeh AH. Introduction to Perturbation Techniques. Wiley, New York, 1981.

[5] Kevorkian J, Cole JD. Perturbation Methods in Applied Mathematics. Springer-Verlag,New York, 1981.

[7] O'Malley RE. Singular Perturbation Methods for Ordinary Differential Equations. Springer-Verlag, New York, 1991.

國際替代計量

以指數試函數解奇異常微分方程及橢圓偏微分方程

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