The Cauchy problem for higher-order linear partial differential equation

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 ${\prod }_{i=1}^m(\frac{\partial }{\partial t}-a_{i}P({\partial }_x))$ or ${\prod }_{i=1}^m(\frac{{\partial }^2}{\partial t^2}-a_i^{2}P({\partial }_x))$, 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.