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首页> 外文期刊>International journal of non-linear mechanics >Submodels of model of nonlinear diffusion with non-stationary absorption
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Submodels of model of nonlinear diffusion with non-stationary absorption

机译：具有非平稳吸收的非线性扩散模型的子模型

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We study the model, describing a nonlinear diffusion process (or a heat propagation process) in an inhomogeneous medium with non-stationary absorption (or source). We found tree submodels of the original model of the nonlinear diffusion process (or the heat propagation process), having different symmetry properties. We found all invariant submodels. All essentially distinct invariant solutions describing these invariant submodels are found either explicitly, or their search is reduced to the solution of the nonlinear integral equations. For example, we obtained the invariant solution describing the nonlinear diffusion process (or the heat distribution process) with two fixed "black holes", and the invariant solution describing the nonlinear diffusion process (or the heat distribution process) with the fixed "black hole" and the moving "black hole". The presence of the arbitrary constants in the integral equations, that determine these solutions provides a new opportunities for analytical and numerical study of the boundary value problems for the received submodels, and, thus, for the original model of the nonlinear diffusion process (or the heat distribution process). For the received invariant submodels we are studied diffusion processes (or heat distribution process) for which at the initial moment of the time at a fixed point are specified or a concentration (a temperature) and its gradient, or a concentration (a temperature) and its rate of change. Solving of boundary value problems describing these processes are reduced to the solving of nonlinear integral equations. We are established the existence and uniqueness of solutions of these boundary value problems under some additional conditions. The obtained results can be used to study the diffusion of substances, diffusion of conduction electrons and other particles, diffusion of physical fields, propagation of heat in inhomogeneous medium.
机译：我们研究该模型，描述了在具有非平稳吸收（或源）的非均匀介质中的非线性扩散过程（或热传播过程）。我们发现非线性扩散过程（或热传播过程）的原始模型的树子模型具有不同的对称性。我们找到了所有不变的子模型。可以明确找到描述这些不变子模型的所有本质上不同的不变解，或者将其搜索简化为非线性积分方程的解。例如，我们获得描述具有两个固定“黑洞”的非线性扩散过程（或热分布过程）的不变解，以及获得描述具有固定“黑洞”的非线性扩散过程（或热分布过程）的不变解。和移动的“黑洞”。积分方程中任意常数的存在，决定了这些解决方案，为接收子模型的边界值问题的解析和数值研究提供了新的机会，从而为非线性扩散过程的原始模型（或热分布过程）。对于接收的不变子模型，我们研究了扩散过程（或热分布过程），在该过程中指定了固定时间的初始时刻，浓度（温度）及其梯度或浓度（温度）和它的变化率。描述这些过程的边值问题的求解被简化为非线性积分方程的求解。我们确定了在某些附加条件下这些边值问题解的存在性和唯一性。所得结果可用于研究物质的扩散，传导电子和其他粒子的扩散，物理场的扩散，热在不均匀介质中的传播。

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