Abstract

Solutions to a nonlinear parabolic convection–diffusion equation are constructed in the form of a diffusion wave that propagates over a zero background at a finite velocity. The theorem of existence and uniqueness of the solution is proven. The solution is constructed in the form of a characteristic series whose coefficients are determined using a recurrent procedure. Exact solutions of the considered type and their characteristics, including the domain of existence, are determined, and the behavior of these solutions on the boundaries of this domain of existence is studied. The boundary element method and the dual reciprocity method are used to develop, implement, and test an algorithm for constructing approximate solutions.

中文翻译：

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对流扩散方程奇异性问题的精确解

摘要

非线性抛物线对流扩散方程的解以扩散波的形式构造，该扩散波以有限的速度在零背景上传播。证明了该解的存在性和唯一性定理。该解决方案以特征序列的形式构建，其特征系数使用递归过程确定。确定所考虑类型及其特征（包括存在域）的精确解，并研究这些解在此存在域边界上的行为。边界元方法和对等互易方法用于开发，实现和测试构造近似解的算法。