Substantiation of a Quadrature-Difference Method for Solving Integro-Differential Equations with Derivatives of Variable Order

A new definition of the fractional derivative based on the interpolation of natural-order derivatives is given. The main advantage of the new definition is the locality of such derivatives. In other words, the value of the derivative at a point does not depend on the domain of the function, in contrast to the cases of Riemann-Liouville and Caputo derivatives. This enables one to construct and justify simple computational methods for solving equations containing such derivatives. Moreover, this definition allows one to generalize the concept of a derivative to the case of differentiation of variable order. The paper consideres a class of equations containing the introduced derivatives. The unique solvability of the initial equations is proved, and the quadrature-difference method for solving them is substantiated. Effective error estimates for approximate solutions are obtained. Theoretical conclusions are confirmed by a numerical solution of a model problem.