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On the Functions Related With the Analytic Solutions of the Cauchy Problem for Wave and Heat Equations

Year: 2024
Type: article
Abstract: We investigate properties of entire solutions of the Cauchy problem for one-dimensional homogeneous hyperbolic equation. Considering analytic continuation of the solutions given by the D'Alambert formula we have found some conditions providing $L$-index boundedness in the direction for some functions related with the solutions. In particular, for homogomogeneous wave equation $c^2\frac{\partial^2 }{\partial x^2}u(x,t)=\frac{\partial^2 }{\partial t^2}u(x,t)$ with initial conditions $u(x,0)=\varphi (x),$ $u_t(x,0)=\psi (x)$ its solution has the form $u(x,t)=\frac{\varphi(x+ct)+\varphi(x-ct)}{2}+\frac{1}{2c}\int\limits^{x+ct}_{x-ct}{\psi(\alpha)d \alpha}.$ We study the functions $\mathfrak{H}(x,t)=\frac{\varphi(x+ct)+\varphi(x-ct)}{2}+ \frac{\mathfrak{E}}{2c}\int\limits^{x+ct}_{x-ct}{\psi(\alpha)d \alpha},$ where $x,$ $t,$ $c\in\mathbb{C},$ $\mathfrak{E}$ is a positive constant which is determined with some conditions by the functions $\varphi$ and $\psi$. Our main result gives sufficient conditions of boundedness of $\mathfrak{L}$-index in a direction $\mathbf{b}$ for the functions $\mathfrak{H}$. Its proof uses known sufficent conditions for the sum of entire functions. At the end, we pose open problems concerning conditions of the directional $L$-index boundedness for analytic solutions of the Cauchy problem of the heat equation. The conditions will allow a qualitative description of local and asymptotic behavior of the parabolic equation analytic solutions presenting the temperature distribution in the process of plasma electrolytic oxidation.
DOI: 10.31471/2304-7399-2024-19(73)-9-17
Language: English
Open Access status: bronze
OpenAlex ID: W4406214609
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