On an initial boundary value problem involving the generalized EPD equation

Abstract

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In the present paper, a solution is given to the following singular initial boundary value problem u(0,t)=u(a,t) = 0, u(x,0) = l, $\frac{u}{t}(x,0)=0,\frac{\partial^p u}{\partial t^p}(x,0)=\frac{1}{p!}(p=2,3,...,m-1)$, for the homogeneous generalized Euler Poisson Darboux equation $\Delta u-\frac{\partial^m u}{\partial t^m}-\sum\limits_{p=1}^{m-1}\frac{k_p}{t^p}\frac{\partial^{m-p} u}{\partial t^{m-p}}-k_m u=0$, where x, a $\in\Bbb{R}^n, k_1,k_2,...,k_m $ are real parameters, t is the time variable and $\Delta$ is the n-dimensional Laplace operator, The solution is obtained using the finite integral transformation method and is given in terms of absolutely and uniformly convergent power series.