The Cauchy problem for higher-order linear partial differential equation

TL;DR

This paper derives explicit solutions for higher-order linear PDEs with constant coefficients using abstract operators and integral representations, providing a clear method for solving the Cauchy problem.

Contribution

It introduces a novel approach to solving higher-order linear PDEs via abstract operators and integral forms, extending existing methods.

Findings

Explicit integral solutions for specific higher-order PDEs.

Use of abstract operators to represent solution operators.

General framework applicable to a class of linear PDEs.

Abstract

For the linear partial differential equation $P(\partial_x,\partial_t)u=f(x,t)$, where $x\in\mathbb{R}^n,\;t\in\mathbb{R}^1$, with $P(\partial_x,\partial_t)$ is $\prod^m_{i=1}(\frac{\partial}{\partial{t}}-a_iP(\partial_x))$ or $\prod^m_{i=1}(\frac{\partial^2}{\partial{t^2}}-a_i^2P(\partial_x))$, the authors give the analytic solution of the cauchy problem using the abstract operators $e^{tP(\partial_x)}$ and $\frac{\sinh(tP(\partial_x)^{1/2})}{P(\partial_x)^{1/2}}$. By representing the operators with integrals, explicit solutions are obtained with an integral form of a given function.