On a nonlocal problem for a fourth-order mixed-type equation with the Hilfer operator

The present work is devoted to the study of the solvability questions for a nonlocal problem with an integrodifferential conjugation condition for a fourth-order mixed-type equation with a generalized RiemannLiouville operator. Under certain conditions on the given parameters and functions, we prove the theorems of uniqueness and existence of the solution to the problem. In the paper, given example indicates that if these conditions are violated, the formulated problem will have a nontrivial solution. To prove uniqueness and existence theorems of a solution to the problem, the method of separation of variables is used. The solution to the problem is constructed as a sum of an absolutely and uniformly converging series in eigenfunctions of the corresponding one-dimensional spectral problem. The Cauchy problem for a fractional equation with a generalized integro-differentiation operator is studied. A simple method is illustrated for finding a solution to this problem by reducing it to an integral equation equivalent in the sense of solvability. The authors of the article also establish the stability of the solution to the considered problem with respect to the nonlocal condition

On a nonlocal problem for a fourth-order mixed-type equation with the Hilfer operator

The present work is devoted to the study of the solvability questions for a nonlocal problem with an integrodifferential conjugation condition for a fourth-order mixed-type equation with a generalized RiemannLiouville operator. Under certain conditions on the given parameters and functions, we prove the theorems of uniqueness and existence of the solution to the problem. In the paper, given example indicates that if these conditions are violated, the formulated problem will have a nontrivial solution. To prove uniqueness and existence theorems of a solution to the problem, the method of separation of variables is used. The solution to the problem is constructed as a sum of an absolutely and uniformly converging series in eigenfunctions of the corresponding one-dimensional spectral problem. The Cauchy problem for a fractional equation with a generalized integro-differentiation operator is studied. A simple method is illustrated for finding a solution to this problem by reducing it to an integral equation equivalent in the sense of solvability. The authors of the article also establish the stability of the solution to the considered problem with respect to the nonlocal condition

On a mixed problem for Hilfer type differential equation of higher order

The study considers the solvability of a mixed problem for a Hilfer type partial differential equation of the even order with initial value conditions and small positive parameters in mixed derivatives in threedimensional domain. It studies the solution to this fractional differential equation of higher order in the class of regular functions. The case, when the order of fractional operator is 1 <α< 2, is examined. During this study the authors use the Fourier series method and obtain a countable system of ordinary differential equations. The initial value problem is integrated as an ordinary differential equation and the integrated constants find by the aid of given initial value conditions. Using the Cauchy-Schwarz inequality and the Bessel inequality, it is proved the absolute and uniform convergence of the obtained Fourier series. The stability of the solution to the mixed problem on the given functions is studied

The Samarskii–Ionkin type problem for the fourth order parabolic equation with fractional differential operator

AbstractIn the present work the Samarskii–Ionkin type non-local problem with Caputo fractional order differential operator is studied. The considered problem generalizes some previous known problems formulated for some fourth order parabolic equations. We prove the existence and uniqueness of a regular solution of the formulated problem applying the method of separation of variables

An inverse problem for Hilfer type differential equation of higher order

In three-dimensional domain, an identification problem of the source function for Hilfer type partial differential equation of the even order with a condition in an integral form and with a small positive parameter in the mixed derivative is considered. The solution of this fractional differential equation of a higher order is studied in the class of regular functions. The case, when the order of fractional operator is 0 <α< 1, is studied. The Fourier series method is used and a countable system of ordinary differential equations is obtained. The nonlocal boundary value problem is integrated as an ordinary differential equation. By the aid of given additional condition, we obtained the representation for redefinition (source) function. Using the Cauchy-Schwarz inequality and the Bessel inequality, we proved the absolute and uniform convergence of the obtained Fourier series

Solvability of a Nonlocal Boundary Value Problem Involving Fractional Derivative Operators

In the present work solvability questions of a nonlocal boundary value problem involving fractional operator of Riemann-Liouville type have been studied. Theorem on a solvability of considered problem is proved

The Samarskii–Ionkin type problem for the fourth order parabolic equation with fractional differential operator

AbstractIn the present work the Samarskii–Ionkin type non-local problem with Caputo fractional order differential operator is studied. The considered problem generalizes some previous known problems formulated for some fourth order parabolic equations. We prove the existence and uniqueness of a regular solution of the formulated problem applying the method of separation of variables

On a Mixed Problem for Hilfer Type Fractional Differential Equation with Degeneration

Abstract: In three-dimensional domain the single-value solvability of a mixed problem for a Hilfer type nonlinear partial differential equation of the even order with small positive parameters in mixed derivatives is considered. The regular solution of this fractional differential equation is studied in the case 0< 1. The Fourier series method is used and a countable system of nonlinear ordinary differential equations is obtained. The ordinary differential equation is integrated and the obtained constants are found by the aid of given initial value conditions. The countable system of nonlinear functional-integral equations is solved by the method of successive approximations. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proved. © 2022, Pleiades Publishing, Ltd

Solvability of a Nonlocal Boundary Value Problem Involving Fractional Derivative Operators

In the present work solvability questions of a nonlocal boundary value problem involving fractional operator of Riemann-Liouville type have been studied. Theorem on a solvability of considered problem is proved