Construction of solutions to parabolic and hyperbolic initial-boundary value problems

TL;DR

This paper demonstrates that solutions to parabolic and hyperbolic initial-boundary value problems with analytical right-hand sides are also analytical in time, and provides a method to construct such solutions using Taylor expansions and recursive relations.

Contribution

It introduces a novel approach to construct solutions as Taylor series with recursive coefficients, addressing boundary conditions through approximation methods.

Findings

Solutions are analytical in time at each fixed spatial point.

Solutions can be constructed via Taylor expansions with recursive coefficients.

The method applies to both regular and weak solutions of parabolic and hyperbolic equations.

Abstract

We show that infinitely differentiable solutions to parabolic and hyperbolic equations, whose right-hand sides are analytical in time, are also analytical in time at each fixed point of the space. These solutions are given in the form of the Taylor expansion with respect to time $t$ with coefficients depending on $x$. The coefficients of the expansion are defined by recursion relations, which are obtained from the condition of compatibility of order $k=\infty$. The value of the solution on the boundary is defined by the right-hand side and initial data, so that it is not prescribed. We show that exact regular and weak solutions to the initial-boundary value problems for parabolic and hyperbolic equations can be determined as the sum of a function that satisfies the boundary conditions and the limit of the infinitely differentiable solutions for smooth approximations of the data of the corresponding problem with zero boundary conditions. These solutions are represented in the form of the Taylor expansion with respect to $t$. The suggested me