Research of Thermal Explosion Conditions in Nonlinear Heat Conduction Problems with a Nonlinear Heat Source

IF 0.8 Q2 MATHEMATICS Lobachevskii Journal of Mathematics Pub Date : 2024-08-28 DOI:10.1134/s1995080224602443

V. A. Kudinov, K. V. Trubitsyn, K. V. Kolotilkina, S. V. Zaytsev, T. E. Gavrilova

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Abstract

Using an additional sought function (ASF) and additional boundary conditions (ABCs), an analytical solution of the nonlinear heat conduction problem for a symmetric plate with a nonlinear heat source has been obtained. ASF characterizes the temperature change over time at the center of the plate. Its usage enables the solution reduction of the original partial differential equation to the integration of a temporal ordinary differential equation (ODE). From its solution, exact eigenvalues are found, determined by classical methods from solving the Sturm–Liouville boundary value problem specified in the spatial coordinate domain. Hence, this study considers another direction of their determination, based on solving the temporal ODE with respect to ASF. ABCs are formulated in such a way that their fulfillment by the sought solution is equivalent to satisfying the original differential equation at the boundary points. It leads to its fulfillment within the considered domain, bypassing the direct integration process over the spatial variable and confining it only to the execution of the heat balance integral—the averaged original differential equation.

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带非线性热源的非线性热传导问题中的热爆炸条件研究

摘要利用附加求函数 (ASF) 和附加边界条件 (ABC)，获得了带有非线性热源的对称板非线性热传导问题的解析解。ASF 描述了板中心温度随时间的变化。使用 ASF 可以将原始偏微分方程简化为时间常微分方程（ODE）的积分。通过求解空间坐标域中指定的 Sturm-Liouville 边界值问题的经典方法，可以找到精确的特征值。因此，本研究考虑了确定特征值的另一个方向，即在求解与 ASF 有关的时间 ODE 的基础上确定特征值。在制定 ABC 时，所寻求的解决方案满足 ABC 等价于满足边界点上的原始微分方程。这将导致在所考虑的域内满足其要求，绕过空间变量的直接积分过程，而只限于执行热平衡积分--平均原始微分方程。

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来源期刊

Lobachevskii Journal of Mathematics MATHEMATICS-

CiteScore

1.50

自引率

42.90%

发文量

127

期刊介绍： Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.

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