Well-posedness of boundary-value problems for conditionally well-posed integro-differential equations and polynomial approximations of their solutions

The this paper, we introduce a pair of Sobolev spaces with special Jacobi–Gegenbauer weights, in which the general boundary-value problem for a class of ordinary integro-differential equations characterized by the positivity of the difference of orders of the inner and outer differential operators is well-posed in the Hadamard sense. Based on this result, a justification of the general polynomial projection method for solving the corresponding problem is performed. An application of general results to the proof of the convergence of the polynomial Galerkin method for solving the Cauchy problem in the Sobolev weighted space is given. The convergence rate of the method is characterized in terms of the best polynomial approximations of an exact solution, which automatically responds to the smoothness properties of the coefficients of the equation.