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.