微分方程奇异摄动边界层问题的数值逼近解

On the Numerical Approach Solution to the Problem of Boundary Layer of Singular Perturbation for the Differential Equation

通过对抛物型偏微分方程和一阶双曲型偏微分方程奇异摄动问题的讨论,提出了在使边界层的特性不至于丧失的前提下的边界层格式。对一类在Ω1和Ω2上的抛物型奇异摄动的初、边值问题进行了进一步研究,利用渐近方法、差分方法和常微分方程的二点边值问题的方法,求得了偏微分方程边界层问题的数值解。得到了当步长可取中等大小,h→0,τ→0,ε→0时,且当自由项函数和初、边值条件函数均为给定的充分光滑的函数,含有小参数ε(0<ε1)的一类偏微分方程奇异摄动问题的一致数值逼近解。并将此结论应用于实际问题中。

Through discussing the singular perturbed problem of parabolic type and first-order hyperbolic type in partial differential equations, this paper puts forward a boundary layer format without losing its characteristics. Furthermore, initial and boundary values problems of parabolic-type singular perturbation Ω1 and Ω2 are researched. In the use of asymptotic method, difference method and two-point boundary value problem methods in ordinary differential equation, we have gained a numerical solution to boundary layer in partial differential equation. Moreover, on the premise of step size middling and h, τ and 8 each approaching to zero, when free functions and initial, boundary values condition functions are smooth functions given, we obtain a uniformly numerical approach solution to the singular perturbed problem of one partial different equation with a small parameter ε（0 〈 ε≤1）. Finally, we apply the conclusions to practice.