The paper deals with two-dimensional boundary-value problems for the degenerate nonlinear parabolic equation with a source term, which describes the process of heat conduction in the case of the power-law temperature dependence of the heat conductivity coefficient. We consider a heat wave propagation problem with a specified zero front in the case of two spatial variables. The solution existence and uniqueness theorem is proved in the class of analytic functions. The solution is constructed as a power series with coefficients to be calculated by a proposed constructive recurrent procedure. An algorithm based on the boundary element method using the dual reciprocity method is developed to solve the problem numerically. The efficiency of the application of the dual reciprocity method for various systems of radial basis functions is analyzed. An approach to constructing invariant solutions of the problem in the case of central symmetry is proposed. The constructed solutions are used to verify the developed numerical algorithm. The test calculations have shown the high efficiency of the algorithm.

中文翻译：

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带源项的二维非线性热方程的解析与数值研究

本文研究了具有源项的退化非线性抛物线方程的二维边值问题，描述了在导热系数的幂律温度依赖性情况下的热传导过程。我们考虑在两个空间变量的情况下具有指定零前沿的热浪传播问题。在解析函数类中证明了解存在唯一性定理。该解决方案被构造为一个幂级数，其系数由一个提议的构造循环程序计算。开发了一种基于边界元法使用对偶互易法的算法来数值求解该问题。分析了对偶互易法在各种径向基函数系统中的应用效率。提出了一种在中心对称情况下构造问题不变解的方法。构建的解决方案用于验证开发的数值算法。测试计算表明了该算法的高效性。