E. T. Karimov

On a Differential Equation with Caputo-Fabrizio Fractional Derivative of Order 1 <β ≤ 2 and Application to Mass-Spring-Damper System

In this work, we investigate a linear differential equation involving Caputo-Fabrizio fractional derivative of order

On the Volterra property of a boundary problem with integral gluing condition for a mixed parabolic-hyperbolic equation

1<\beta\leq 2

On Boundary-Value Problems for a Partial Differential Equation with Caputo and Bessel Operators

. Under some assumptions the considered equation is reduced to an integer order differential equation and solutions for different cases are obtained in explicit forms. We also prove a uniqueness of a solution of an initial value problem with a nonlinear differential equation containing the Caputo-Fabrizio derivative. Application of our result to the mass-spring-damper motion is also presented.

Solvability of a non-local problem with integral transmitting condition for mixed type equation with Caputo fractional derivative

In the present work, we consider a boundary value problem with gluing conditions of an integral form for the parabolic-hyperbolic type equation. We prove that the considered problem has the Volterra property. The main tools used in the work are related to the method of the integral equations and functional analysis.

On a boundary-value problem for the parabolic-hyperbolic equation with the fractional derivative and the sewing condition of the integral form

In this work, we investigate a unique solvability of a direct and inverse source problem for a time-fractional partial differential equation with the Caputo and Bessel operators. Using spectral expansion method, we give explicit forms of solutions to formulated problems in terms of multinomial Mittag-Leffler and first kind Bessel functions.

Certain image formulas of generalized hypergeometric functions

In the present paper, we discuss solvability questions of a non-local problem with integral form transmitting conditions for a mixed parabolic–hyperbolic type equation with the Caputo fractional derivative in a domain bounded by smooth curves. A uniqueness of the solution for a formulated problem we prove using energy integral method with some modifications. The existence of solution will be proved by equivalent reduction of the studied problem into a system of second kind Fredholm integral equations.

Initial boundary value problems for a fractional differential equation with hyper-Bessel operator

The main aim of the present work is an investigation of the analogue of the Tricomi problem with the integral sewing condition for parabolic-hyperbolic equation with the fractional derivative. The uniqueness of the solution for considered problem we prove by the method of energy integrals. The existence of the solution have been proved by reducing the considered problem to the Fredholm integral equation. We represent solution in an explicit form using Green’s function.

Inverse source problem for multi-term fractional mixed type equation

We aim at establishing certain new image formulas of generalized hypergeometric functions by applying the operators of fractional derivative involving Appells function F3(.) due to Saigo-Maeda. Furthermore, by employing some integral transforms on the resulting formulas, we presented some more image formulas. All the results derived here are of general character and can yield a number of (known and new) results in the theory of special functions.

Spatial boundary problem with the Dirichlet–Neumann condition for a singular elliptic equation

Abstract Direct and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart of a hyper-Bessel differential operator are considered. Solutions to these problems are constructed based on appropriate eigenfunction expansions and results on existence and uniqueness are established. To solve the resultant equations, a solution to such kind of non-homogeneous fractional differential equation is also presented.

Numerical Computation of a Fractional Derivative with Non-Local and Non-Singular Kernel

In this work, we investigate an inverse source problem for multi-term fractional mixed type equation in a rectangular domain. We seek solutions in a form of series expansions using orthogonal basis obtained by using the method of a separation of variables. The obtained solutions involve multi-variable Mittag-Leffler functions, and hence, certain properties of the multi-variable Mittag-Leffler function needed for our calculations were established and proved. Imposing certain conditions to the given data, the convergence of the infinite series solutions was proved as well.