Solutions to a Simplified Initial Boundary Value Problem for 1D Hyperbolic Equation with Interior Degeneracy

Vladimir L. Borsch, Peter I. Kogut

Анотація


A 1-parameter initial boundary value problem (IBVP) for a linear homogeneous
degenerate wave equation (JODEA, 28(1), 1 â“ 42) in a space-time rectangle is considered. The origin of degeneracy is the power law coefficient function with respect to the spatial distance to the symmetry line of the rectangle, the exponent being the only parameter of the problem, ranging in (0,1) and (1,2) and producing the weak and strong degeneracy respectively. In the case of weak degeneracy separation of variables is used in the rectangle to obtain the unique bounded continuous solution to the IBVP, having the continuous flux. In the case of strong degeneracy the IBVP splits into the two derived IBVPs posed respectively in left and right half-rectangles and solved separately using separation of variables. Continuous matching of the obtained left and right families of bounded solutions to the IBVPs results in a linear integro-differential equation of convolution type. The Laplace transformation is used to solve the equation and obtain a family of bounded solutions to the IBVP, having the continuous flux and depending on one undetermined function.


Ключові слова


degenerate wave equation; separation of variables; linear integro-differential equation of convolution type; Laplace transformation

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Посилання


V. L. Borsch, On initial boundary value problems for the degenerate 1D wave equation, Journal of Optimization, Differential Equations, and their Applications (JODEA), 27(2) (2019), 27–44.

V. L. Borsch, P. I. Kogut, G. Leugering, On an initial boundary-value problem for 1D hyperbolic equation with interior degeneracy: series solutions with the continuously differentiable fluxes, Journal of Optimization, Differential Equations, and their Applications (JODEA), 28(1) (2020), 1 – 42.

G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer, NY, 1974.

A. M. Kohen, Numerical Methods for Laplace Transform Inversion, Springer Science+ Business Media, LLC, NY, 2007.

Yu. V. Sidorov, M. V. Fedoryuk, M. I. Shabunin, Lectures on the Theory of Functions of a Complex Variable, Mir Publishers, Moscow, 1985.

G. P. Tolstov, Fourier Series, Dover, NY, 1962.

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1922.




DOI: http://dx.doi.org/10.15421/142101

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Адреса редколегії: 49050, Україна, Дніпровський національний університет імені Олеся Гончара, вул. Козакова 18, корп. 14, механіко-математичний факультет, д-р фіз.-мат. наук, проф. Когут П.І. 

email: p.kogut@i.ua

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