ABSTRACT
Along with studying the basic boundary-value problems for such equations, since the 70s, much attention has been paid to the formulation and study of non-local boundary-value problems. The main goal of this chapter is to formulate and investigate the unique solvability of boundary-value problems for a third-order loaded partial differential equation of mixed type, with the Caputo operators. The construction of the theory of unique solvability of problems for a loaded third-order differential equations of parabolic-hyperbolic and elliptic-hyperbolic operators with real and variable fractional order coefficients are required both for the internal completeness of the fractional integro-differentiation theory and numerous applications. The boundary-value problems for loaded integro-differential equations connect to the non-local boundary value problems, wherein boundary-value problems are reduced to the integral equation of Volterra type with the shift, and using the successive approximations method was proven an existence of unique solutions of equations.
