广义Cattaneo方程解的积分分解
Integral decomposition for the solutions of the generalized Cattaneo equation
摘要：
本文给出了两个时间导数都被模糊的广义Cattaneo方程的基本解的积分分解，并将其与记忆核卷积。对于幂律核$t^{-α}$、$α\in(0,1]$，该方程变为由最高阶为2的Caputo导数支配的时间分数阶方程。为了将从傅里叶-拉普拉斯域到时空域的解反转，我们使用基于Efross定理的分析方法，并发现所寻找的解由积分分解表示，该积分分解使标准Cattaneo方程的基本解与非负和可归一化函数唯一地依赖于记忆核。此外，使用的方法所产生的理论完整的伯恩斯坦函数，使我们能够分配这种构造的积分分解的解释从属。这一事实在广义Cattaneo方程中的两种极限情况下得到了保留，即，扩散方程或波动方程。我们指出，应用Efross定理使我们能够超越标准的方法，这通常会导致积分分解涉及高斯分布描述的布朗运动。我们的方法澄清了令人困惑的情况下发生的幂律核$t^{-α}$的从属布朗运动的基础上不工作，如果$α\in(1/2,1]$。
英文摘要：
We present the integral decomposition for the fundamental solution of the generalized Cattaneo equation with both time derivatives smeared through convoluting them with some memory kernels. For power-law kernels $t^{-α}$, $α\in(0,1]$ this equation becomes the time fractional one governed by the Caputo derivatives which highest order is 2. To invert the solutions from the Fourier-Laplace domain to the space-time domain we use analytic methods based on the Efross theorem and find out that solutions looked for are represented by integral decompositions which tangle the fundamental solution of the standard Cattaneo equation with non-negative and normalizable functions being uniquely dependent on the memory kernels. Furthermore, the use of methodology arising from the theory of complete Bernstein functions allows us to assign such constructed integral decompositions the interpretation of subordination. This fact is preserved in two limit cases built into the generalized Cattaneo equations, i.e., either the diffusion or the wave equations. We point out that applying the Efross theorem enables us to go beyond the standard approach which usually leads to the integral decompositions involving the Gaussian distribution describing the Brownian motion. Our approach clarifies puzzling situation which takes place for the power-law kernels $t^{-α}$ for which the subordination based on the Brownian motion does not work if $α\in(1/2,1]$.
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