In this paper, we consider a boundary value problems for a systems of loaded integro-differential equations with an involutory transformation. The parameterization method is applied to the boundary value problem for a system with continuous kernel. By using the properties of involutory transformation, the problem is transformed to a boundary value problem for systems of loaded integro-differential equations. The latter problem, in turn, is reduced to solving a special Cauchy problem and a system of algebraic equations in parameters introduced. An algorithm for solving the boundary value problem for systems of loaded integro-differential equations is proposed. On the basis of this algorithm, necessary conditions for the unique solvability of the original problem are established. We also consider a boundary value problem for a systems of loaded integro-differential equations with involution in the case of degenerate kernels. By applying the parametrization method and the theory of integral equations, the problem is reduced to solving a system of algebraic equations. Based on the invertibility of the matrix of that system, necessary and sufficient conditions for the unique solvability of the problem under study are established.

In this paper, a multipoint boundary value problem for systems of integro-differential equations with involution has been studied. To solve the studied problem, the parameterization method is used. Based on the parametrization method, the studied problem is decomposed into two parts, i.e., into the Cauchy problem and a system of linear equations. Necessary and sufficient conditions for the unique solvability of the studied problem are determined.

In this paper, solvability of some inverse problems for a nonlocal analog of a pseudoparabolic equation is studied. The nonlocal analog of a pseudoparabolic equation is formed using transformations that have the involution property. Two types of inverse problems are considered. In the first problem, in addition to the solution, a function in the right-hand side of the equation depending on the spatial variable is determined. In the second problem, a function depending on the time variable is found. The first problem is solved using the Fourier method, and the second problem is solved by reducing to the integral Volterra equation.

In this paper, in the class of smooth functions, integration and differentiation operators connected with fractional conformable derivatives are introduced. The mutual reversibility of these operators is proved, and the properties of these operators in the class of smooth functions are studied. Using transformations generalizing involutive transformations, a nonlocal analogue of the Laplace operator is introduced. For the corresponding nonlocal analogue of the Poisson equation, the solvability of some boundary value problems with fractional conformable derivatives is studied. For the problems under consideration, theorems on the existence and uniqueness of solutions are proved. Necessary and sufficient conditions for solvability of the studied problems are obtained, and integral representations of solutions are given.

In the paper we study properties of some integro - differential operators of fractional order. As an application of the properties of these operators for Poisson equation we examine questions on solvability of a fractional analogue of the Neumann problem and analogues of periodic boundary value problems for circular domains. The exact conditions for solvability of these problems are found.