IJPAM: Volume 81, No. 4 (2012)

ON THE NONLINEAR WAVE EQUATION

Wanchak Satsanit
Department of Mathematics
Faculty of Science
Maejo University
Chiang Mai, 50290, THAILAND


Abstract. In this paper, we study the nonlinear equation of the form

\begin{displaymath}\frac{\partial^{2}}{\partial t^{2}}u(x,t)-c^2\triangle u(x,t)=f(x,t,u(x,t))\end{displaymath}

with the initial conditions

\begin{displaymath}u(x,0)~~and~~ \frac{\partial}{\partial t}u(x,0)=\delta(x),\end{displaymath}

where $u(x,t)$ is an unknown for $(x,t)=(x_1,x_2,\dots,x_n,t)\in\mathbb{R}^n\times(0,\infty)$,$\mathbb{R}^n$ is the dimension of the Euclidean space, $\triangle$ is the Laplacian operator defined by ([*]) and $c$ is a positive constant, $\delta(x)$ is the Dirac delta distribution, $f$ is the given function in nonlinear form depending on $x, t$ and $u(x,t)$. By method of convolution in the distribution theory we obtain the solution of the nonlinear wave equation.

Received: June 6, 2012

AMS Subject Classification:

Key Words and Phrases: ultra-hyperbolic, tempered distribution, Fourier transform

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 81
Issue: 4