任意非线性热方程的解析和数值径向对称解

A. L. Kazakov, L. Spevak
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引用次数: 0

摘要

本文研究了在运动流形上非零边界条件下具有任意形式非线性的热传导方程的径向对称波的构造。本文所研究的边值问题是我们以前所解决的边值问题的推广。首先,对所考虑的抛物型方程进行了扩展;其次,在任意维空间中产生热浪的边界条件具有更一般的形式。证明了该问题的热浪型解析解的存在唯一性定理。提出了一种基于径向基函数展开与配点法相结合的构造所需形式解的近似方法。在每个时间步骤中,解决方案分为两个阶段构造。第一阶段是在给定的运动流形和热浪锋面交界的区域内求解一个先验未知的问题。这里使用了一种特殊的替换,即所需要的函数和空间变量改变了它们的作用。在第二阶段,在当前步长和初始时间的指定移动流形的位置所限定的区域内完成求解。边界条件由第一步解定义。在测试实例中,将该算法构造的解与已知的精确解进行了比较。计算表明,在不同数值参数值下,包括空间维数,数值解具有良好的精度。对时间步长的数值收敛表明了所提计算方法的正确性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Analytical and numerical radially symmetric solutions to a heat equation with arbitrary nonlinearity
The paper deals with the construction of radially symmetric heat waves, which are solutions to the heat conduction equation with an arbitrary form of nonlinearity under nonzero boundary condition specified on a moving manifold. The boundary value problem under study is a generalization of those solved by us earlier. Firstly, the class of the considered parabolic equations is extended; secondly, the boundary condition generating a heat wave in a space of arbitrary dimensionality has a more general form. A new theorem of the existence and uniqueness of the heat-wave-type analytical solution is proved for this problem. An approximate method of constructing solutions of the required form is proposed, which is based on expansion in radial basis functions combined with the collocation method. At each time step, the solution is constructed in two stages. The first stage is solving a problem in the region bounded by a specified moving manifold and a heat wave front, which is a priori unknown and evaluated during solving. Herewith, a special substitution is used, i.e. the required function and the spatial variable change their roles. In the second stage, the solution is completed in the region bounded by the positions of the specified moving manifold on a current step and at the initial time. The boundary conditions are defined from the first-step solution. In the test example, the solutions constructed by the developed algorithm are compared with the known exact solution. Calculations show a good accuracy of the numerical solutions at various values of the numerical parameters, including space dimensionality. The observed numerical convergence with respect to the time step shows the correctness of the proposed computational procedure.
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