Kananthai, Amnuay On the distribution related to the ultra-hyperbolic equations. (English) Zbl 0907.46034 J. Comput. Appl. Math. 84, No. 1, 101-106 (1997). 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. Cited in 7 Documents MSC: 46F10 Operations with distributions and generalized functions Keywords:Schwartz space; ultra-hyperbolic operator; Dirac-delta distribution; elementary solution; convolution method PDFBibTeX XMLCite \textit{A. Kananthai}, J. Comput. Appl. Math. 84, No. 1, 101--106 (1997; Zbl 0907.46034) Full Text: DOI References: [1] Donoghue, W. F., Distributions and Fourier Transforms (1969), Academic Press: Academic Press New York · Zbl 0188.18102 [2] Gelfand, I. M.; Shilov, G. E., Generalized Function (1964), Academic Press: Academic Press New York [3] Kananthai, A., On the distribution \(e^{ αt }δ^{(k)}\) and its applications, (Proc. annual conf. in Mathematics (1996), King Mongkut’s Institute of Technology: King Mongkut’s Institute of Technology Ladkrabang, Thailand), to appear [4] A. Kananthai, On the solutions of the \(n\); A. Kananthai, On the solutions of the \(n\) · Zbl 0922.47042 [5] Nozaki, Y., On Riemann-Liouville integral of ultra-hyperbolic type, Kodai Math. Seminar Rep., 6, 2, 69-87 (1964) · Zbl 0168.37201 [6] Schwartz, L., (Theories des distributions, vols. 1 and 2, Actualités Scientifiques et Industriel (1957, 1959), Hermann & Cie: Hermann & Cie Paris) [7] Tellez, M. A., The distributional Hankel transform of Marcel Riesz’s ultra-hyperbolic Kernel, (Studies in Applied Mathematics, Vol. 93 (1994), Massachusetts Inst. of Technology, Elsevier: Massachusetts Inst. of Technology, Elsevier Amsterdam), 133-162 · Zbl 0820.46037 [8] Trione, S. E., (On Marcel Riesz’s ultra-hyperbolic Kernel, Studies in applied Mathematics, Vol. 79 (1988), Massachusetts Institute of Technology: Massachusetts Institute of Technology Cambridge, Massachusetts, USA), 185-191 · Zbl 0678.46030 [9] Zemanian, A. H., Distribution Theory and Transform Analysis (1965), McGraw-Hill: McGraw-Hill New York · Zbl 0127.07201 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.