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On the distribution related to the ultra-hyperbolic equations. (English) Zbl 0907.46034

Summary: We consider the distribution \(e^{\alpha t}\square^k\delta\), where \(\alpha\) is constant and \(\alpha= (\alpha_1,\alpha_2,\dots, \alpha_n)\in \mathbb{R}^n\) the \(n\)-dimensional Euclidean space and the variable \(t= (t_1,t_2,\dots, t_n)\in \mathbb{R}^n\) and \(\square^k\) is the \(n\)-dimensional ultra-hyperbolic operator iterated \(k\)-times, \(\delta\) is the Dirac-delta distribution with \(\square^0\delta= \delta\) and \(\square^1\delta= \square\delta\).
At first, all properties of \(e^{\alpha t}\square^k\delta\) are studied and after that we study the application of \(e^{\alpha t}\square^k\delta\) for solving the elementary solution of the equation of the ultra-hyperbolic type by using the convolution method.

MSC:

46F10 Operations with distributions and generalized functions
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References:

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