The D’Alembert Type Solution of the Cauchy Problem for the Homogeneous with Respect to Fourth Order Derivatives for Hyperbolic Equation

Abstract

In this paper the D’Alembert type solution for the following

$$\begin{aligned} \sum ^4_{i=0}{a_i\frac{{\partial }^4u\left( x,t\right) }{\partial t^{4-i}\partial x^i}=0}, \ \ \ {\left. \frac{{\partial }^ku(x,t)}{\partial t^k}\right| }_{t=0}=\phi _k\left( x\right) ,\ \ \ (k=0,1,2,3) \end{aligned}$$

Cauchy problem is constructed. Here, $a_i$, $(i=1,2,3,4)$ and $\phi _k\left( x\right)$, $(k=1,2,3,4)$ are given constants and functions, respectively. The cases where the roots of the characteristic equation are folded are examined and compact expressions for the solutions are obtained. The obtained solutions allow proving the uniqueness and existence of the solutions of the considered problem.