Geodesic
An internal boundary value problem with~the Riemann--Liouville operator for the mixed type equation of the third order
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 1, pp. 43-53

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The unique solvability of the internal boundary value problem is investigated for the mixed type equation of the third order with Riemann–Liouville operators in boundary condition. The uniqueness theorem is proved for the different orders of operators of fractional integro-differentiation when the inequality constraints on the known functions exist. The existence of solution is verified by the method of reduction to Fredholm equations of the second kind, which unconditional solvability follows from the uniqueness of the solution of the problem.
Keywords: mixed type equation, Fredholm equation, Cauchy problem, fractional operators in the sense of Riemann–Liouville integro-differentiation. 
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     author = {O. A. Repin and S. K. Kumykova},
     title = {An internal boundary value problem with~the {Riemann--Liouville} operator for the mixed type equation of the third order},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {43--53},
     publisher = {mathdoc},
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     url = {http://geodesic.mathdoc.fr/item/VSGTU_2016_20_1_a3/}
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O. A. Repin; S. K. Kumykova. An internal boundary value problem with~the Riemann--Liouville operator for the mixed type equation of the third order. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 1, pp. 43-53. http://geodesic.mathdoc.fr/item/VSGTU_2016_20_1_a3/