On the Equation of Euler–Poisson–Darboux

Weak solutions of the initial value problem for the EPD equation are constructed using distributional methods. After taking the Fourier transform with respect to the space variables we obtain an equation related to the Bessel differential equation which can easily be solved. The inverse transforms are then found using some results obtained earlier by the author. It is shown that for values of the parameter $\lambda $ which are greater than $n - 1$ (n being the space dimension) the solution is the same as the one obtained by Weinstein [16]. However, the method of this paper can be used for all values of the parameter. Also the exceptional values $\lambda = - 1, - 3, - 5, \cdots $ fit in quite naturally. Conditions for the regularity of the solutions are given for all values of $\lambda $.

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