IJPAM: Volume 81, No. 4 (2012)


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