Geodesic
Solving one boundary-value problem for mixed type equation with two singular lines with the use of spectral method
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2016), pp. 3-9

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For mixed type equation with two perpendicular singularity lines, we consider one boundary problem in the domain whose elliptic and hyperbolic part is rectangle and vertical half-strip, respectively. This problem differs from the Dirichlet problem by the fact that at the left boundary of the rectangle and of half-strip we specify not the unknown function, but the order of zero. We prove uniqueness of boundary problem solution by a spectral method with the use of Fourier–Bessel series. We give substantiation of uniform convergence of corresponding series with some restrictions upon the conditions of the problem.
Keywords: mixed type equations, equation with singular coefficients, spectral method, Fourier–Bessel series, Bessel functions. 
@article{IVM_2016_2_a0,
     author = {A. A. Abashkin},
     title = {Solving one boundary-value problem for mixed type equation with two singular lines with the use of spectral method},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {3--9},
     publisher = {mathdoc},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2016_2_a0/}
}
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A. A. Abashkin. Solving one boundary-value problem for mixed type equation with two singular lines with the use of spectral method. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2016), pp. 3-9. http://geodesic.mathdoc.fr/item/IVM_2016_2_a0/