Geodesic
Nonlocal boundary value problems for~Sobolev-type fractional equations and grid methods for~solving them
Matematičeskie trudy, Tome 21 (2018) no. 2, pp. 72-101

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We consider nonlocal boundary value problems for a Sobolev-type equation with variable coefficients with fractional Gerasimov–Caputo derivative. The main result of the article consists in proving a priori estimates for solutions to nonlocal boundary value problems both in differential and difference form obtained under the assumption of the existence of a solution $u(x,t)$ in a class of sufficiently smooth functions. These inequalities imply the uniqueness and stability of a solution with respect to the initial data and right-hand side and also the convergence of the solution to the difference problem to the solution to the differential problem. 
@article{MT_2018_21_2_a2,
     author = {M. Kh. Beshtokov},
     title = {Nonlocal boundary value problems {for~Sobolev-type} fractional equations and grid methods for~solving them},
     journal = {Matemati\v{c}eskie trudy},
     pages = {72--101},
     publisher = {mathdoc},
     volume = {21},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2018_21_2_a2/}
}
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M. Kh. Beshtokov. Nonlocal boundary value problems for~Sobolev-type fractional equations and grid methods for~solving them. Matematičeskie trudy, Tome 21 (2018) no. 2, pp. 72-101. http://geodesic.mathdoc.fr/item/MT_2018_21_2_a2/