Hulianytskyi A. Qualitative analysis and numerical simulation of hereditary systems

The dissertation deals with the solvability and numerical approximation for partial integro-differential and fractional differential equations. Weak statements of the initial-boundary value problems are studied. By modifying the abc-method, we prove the a priori estimates for parabolic, pseudoparabolic, and pseudohyperbolic integro-differential operators. We also prove the convergence of semidicrete Galerkin approximations for inter-differential as well as constant-order fractional differental equations. For the constant-order subdiffusion equation, we additionally prove the continuity of the solution. For the variable-order subdiffusion equation, we establish the existence and uniqueness of the weak solution. Also, a fully discrete Galerkin method is suggested, and the approximate solutions are computed.