一类两参数非线性反应扩散方程奇摄动问题的广义解

Generalized Solution to a Class of Singularly Perturbed Problem of Nonlinear Reaction Diffusion Equation With Two Parameters

利用奇异摄动方法讨论了一类两参数广义奇摄动反应扩散方程问题.首先,在适当的条件下,对两个小参数进行幂级数展开,构造了问题的形式外部解.其次,在区域边界邻近,建立局部坐标系,利用多重尺度变量方法分别构造了问题解的第一、第二边界层校正项.最后,利用合成展开理论,得到了问题广义解的渐近表示式,并用泛函分析不动点原理,估计了渐近展开式的精度.该文得到问题的广义解在重叠区域内具有两个不同厚度的校正函数.它们分别对边界条件起着校正的作用,扩展了问题研究范围,同时还提供了构造这类在重叠区域上不同厚度的校正项的方法,因此具有广泛的研究前景.

A class of generalized singularly perturbed problems of reaction diffusion equations with two parameters were considered with the singular perturbation method. ~rstly, under suitable conditions, the outer solution to the problem was found. Next, the power series of the two small parameters were developed, and the first and second boundary layer corrective terms for the solution to the problem were constructed with the multiscale variable method, respec- tively. Finally, based on the composite expansion method, the asymptotic expression of the generalized solution to the problem was obtained, and according to the fixed point theory for tractional analysis, the precision of the asymptotic expansion was estimated. Two corrective functions with different thicknesses were obtained for the generalized solution in the overlap- ping area, and they take effects on the boundary conditions respectively and expand the range of study; moreover, the work provides a costruction method for this kind of corrective terms with different thicknesses in the overlapping area, thus has a wide study foreground.