Geodesic
The problem with Saigo operators for a hyperbolic equation that degenerates inside the domain
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 21 (2017) no. 3, pp. 473-480

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A nonlocal problem is investigated for a degenerate hyperbolic equation $$ |y|^{m} u_{xx}-u_{yy}+a |y|^{\frac{m}{2}-1} u_{x}=0 $$ in a domain bounded by the characteristics of this equation. The boundary condition for this problem contains a linear combination of generalized fractional integro-differentiation operators with a hypergeometric Gauss function in the kernel. The uniqueness of the solution is proved using the Tricomi method. The existence of a solution is equivalent to the solvability of a singular integral equation with a Cauchy kernel.
Keywords: boundary value problem, fractional integro-differentiation operators, Gauss function, singular integral equation. 
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     author = {O. A. Repin},
     title = {The problem with {Saigo} operators for a hyperbolic equation that degenerates inside the domain},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {473--480},
     publisher = {mathdoc},
     volume = {21},
     number = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2017_21_3_a4/}
}
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O. A. Repin. The problem with Saigo operators for a hyperbolic equation that degenerates inside the domain. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 21 (2017) no. 3, pp. 473-480. http://geodesic.mathdoc.fr/item/VSGTU_2017_21_3_a4/