Geodesic
Goursat problem for loaded degenerate second order hyperbolic equation with Gellerstedt operator in principal part
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 1, pp. 7-21

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In the paper we study a loaded degenerate hyperbolic equation of the second order with variable coefficients. The principal part of the equation is the Gellerstedt operator. The loaded term is given in the form of the trace of desired solution on the degenerate line. The latter is located in the inner part of the domain. We investigate a boundary value problem. The boundary conditions are given on a characteristics line of the equation under study. For the model equation (when all subordinated coefficients are zero) we construct an explicit representation for solution of the Goursat problem. In the general case, we reduce the problem to an integral Volterra equation of the second kind. We apply the method realized by Sven Gellerstedt solving the second Darboux problem. In both cases, model and general, we use widely properties of the Green–Hadamard function.
Mots-clés : Goursat problem
Keywords: loaded equation, hyperbolic equation, degenerate equation, Gellerstedt operator, the Green–Hadamard's function method. 
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     author = {A. H. Attaev},
     title = {Goursat problem for loaded degenerate second order hyperbolic equation with {Gellerstedt} operator in principal part},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {7--21},
     publisher = {mathdoc},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2016_20_1_a0/}
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A. H. Attaev. Goursat problem for loaded degenerate second order hyperbolic equation with Gellerstedt operator in principal part. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 1, pp. 7-21. http://geodesic.mathdoc.fr/item/VSGTU_2016_20_1_a0/