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.

中文翻译：

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Sobolev型分数阶方程的非局部边值问题和求解它们的网格方法

我们考虑具有分数系数Gerasimov–Caputo导数的变系数Sobolev型方程的非局部边值问题。本文的主要结果在于，在一类足够光滑的函数中，假设存在解u（x，t）的情况下，以微分和差分形式证明非局部边值问题的解的先验估计。这些不等式意味着解决方案相对于初始数据和右手边的唯一性和稳定性，并且还意味着对差分问题的解决方案与对差分问题的解决方案的收敛。