On a boundary value problem for an equation of mixed type with a Riemann–Liouville fractional partial derivative

For an equation of mixed type with a Riemann–Liouville fractional partial derivative, we prove the uniqueness and existence of a solution of a nonlocal problem whose boundary condition contains a linear combination of generalized fractional integro-differentiation operators with the Gauss hypergeometric function in the kernel. A closed-form solution of the problem is presented.     

A problem with generalized fractional integro-differentiation operators of arbitrary order

For a degenerate hyperbolic equation we study a problem with fractional integro-differentiation operators in the boundary condition on the characteristic part of the boundary. We determine intervals for parameters of generalized operators of arbitrary order with a Gauss hypergeometric function such that the problem either is uniquely solvable or has more than one solution.     

A nonlocal problem for degenerate hyperbolic equation

We consider a nonlocal problem for a degenerate equation in a domain bounded by characteristics of this equation. The boundary-value conditions of the problem include linear combination of operators of fractional integro-differentiation in the Riemann–Liouville sense. The uniqueness of solution of the problem under consideration is proved by means of the modified Tricomi method, and existence is reduced to solvability of either singular integral equation with the Cauchy kernel or Fredholm integral equation of second kind.     

Boundary-value problem with saigo operators for mixed type equation of the third order with multiple characteristics

We study a boundary-value problem with shift formixed-type equation of the third order. In the hyperbolic field boundary condition contains a linear combination of generalized operators of fractional integro-differentiation. We prove a unique solvability of the problem.     

On Solvability of Nonlocal Problem for Loaded Parabolic-Hyperbolic Equation

We study unique solvability of a nonlocal problem for equations of mixed type in a finite domain. This equation contains the partial fractional Riemann–Liouville derivative. The boundary condition of the problem contains a linear combination of operators of fractional differentiation in the sense of Riemann–Liouville of values of function derivative on the degeneration line and generalized operators of fractional integro-differentiation in the sense of M. Saigo. The uniqueness theorem of the problem is proved by a modified Tricomi method. The existence of solutions is equivalently reduced to the solvability of Fredholm integral equation of the second kind.     

On a problem with shift for mixed type equation with two degeneration lines

For mixed type equation with two perpendicular lines of degeneracy we consider the boundary-value problem with nonlocal condition, connecting with the help of generalized operators of fractional integro-differentiation the trace of the normal derivative of the unknown function on the transition line and its own trace on the control characteristics and the line of degeneracy. The author proves the unique solvability of the problem.     

Solvability of a nonlocal problem for a loaded parabolic-hyperbolic equation

For a mixed-type equation we study a problem with generalized fractional integrodifferentiation operators in the boundary condition. We prove its unique solvability under inequality-type conditions imposed on the known functions for various orders of fractional integrodifferentiation operators. We prove the existence of a solution to the problem by reducing the latter to a fractional differential equation.     

On a Problem for a Mixed-Type Equation With Fractional Derivative

For a mixed-type equation with the Riemann–Liouville partial fractional derivative we study a problem where the boundary condition contains a linear combination of generalized fractional operators with the Gauss hypergeometric function. We find a solution to the considered problem explicitly by solving an equation with fractional derivatives of various orders and prove the uniqueness of the solution for various values of parameters of the mentioned operators.     

On a generalization of the Neumann problem for the Laplace equation

We investigate solvability of a fractional analogue of the Neumann problem for the Laplace equation. As a boundary operator we consider operators of fractional differentiation in the Hadamard sense. The problem is solved by reduction to an integral Fredholm equation. A theorem on existence and uniqueness of the problem solution is proved.     

Boundary value problem for a multidimensional fractional partial differential equation

We construct a fundamental solution of a linear fractional partial differential equation. For an equation with Dzhrbashyan-Nersesyan fractional differentiation operators, we solve a boundary value problem and find a closed-form representation for its solution. The corresponding results for equations with Riemann-Liouville and Caputo derivatives are special cases of the assertions proved here.     

Nonlocal problem for an equation of mixed type in a domain whose elliptic part is a half-strip

For the Gellerstedt equation, we study a problem with shift in a domain whose elliptic part is an infinite half-strip. By using generalized fractional differentiation operators, we specify a linear combination that relates the value of the solution on characteristics of the equation with the value of the solution and its derivative on the parabolic degeneration line. We prove the unique solvability of this problem.     

Boundary value problem for a first-order partial differential equation with a fractional discretely distributed differentiation operator

We solve a boundary value problem for a first-order partial differential equation in a rectangular domain with a fractional discretely distributed differentiation operator. The fractional differentiation is given by Dzhrbashyan–Nersesyan operators. We construct a representation of the solution and prove existence and uniqueness theorems. The results remain valid for the corresponding equations with Riemann–Liouville and Caputo derivatives. In terms of parameters defining the fractional differential operator, we derive necessary and sufficient conditions for the solvability of the problem.     

On a boundary value problem with shift for an equation of mixed type in an unbounded domain

We study the unique solvability of a problem with shift for an equation of mixed type in an unbounded domain. We prove the uniqueness theorem under inequality-type constraints for known functions for various orders of the fractional differentiation operators in the boundary condition. The existence of a solution is proved by reduction to a Fredholm equation of the second kind, whose unconditional solvability follows from the uniqueness of the solution of the problem.     

A problem of cauchy type for a fractal string

We discuss the characteristics of the statement and solution of a problem of Cauchy type using the example of the plastic subsystem of a fractal string. To solve the basic dynamic equation in fractional derivatives we propose two approaches: reduction to a system of equations and the use of composition formulas for fractional derivative operators. The results obtained are generalized to the solution of the Cauchy problem in matrix form. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 137–145.     

Boundary-Value Problem With Saigo Operators for Mixed Type Equation With Fractional Derivative

We set up and solve a non-local problem for a differential equation, which contains the diffusion equation of fractional order. The boundary condition contains a linear combination of generalized operators with the Gauss hypergeometric function in the kernel. For various values of parameters of these operators we write a solution in explicit form.     

Analytic investigation of features of stresses in plane nonstationary contact problems with moving boundaries

We propose a technique for the analytic investigation of features of contact stresses in the vicinity of the nonstationary moving boundary of a contact region in plane nonstationary contact problems with moving boundaries, which is based on the reduction of a boundary two-dimensional singular integral equation resolving the problem to a system of two one-dimensional singular equations. As tools of research, a method for the reduction of singular integral equations to an equivalent Riemann type problem for piecewise analytic functions and a technique of fractional integro-differentiation are used. It is shown that, on the moving boundary of the contact region, a power singularity, the order of which depends on the velocity of the boundary, takes place.     

Inversion of the Kipriyanov–Radon transform via fractional derivatives in a one-dimensional parameter

This paper considers the Kipriyanov–Radon transform constructed as a special Radon transform adopted for dealing with singular Bessel differential operators of the corresponding indices acting on a part of the variables. The authors obtain inversion formulas generalizing the classical formulas for the Radon transform of axially-symmetric functions and relating to the integro-differentiation of fractional order in a one-dimensional parameter. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 54, Suzdal Conference–2006, Part 2, 2008.     

Problem with generalized fractional differentiation operators for a hyperbolic equation degenerating in the interior of the domain

We study the unique solvability of a nonlocal problem with generalized fractional differentiation operators in the boundary condition for a hyperbolic equation degenerating in the interior of the domain.     

On a class of fractional non-selfadjoint operators associated with differential equations

We present a research method for non-selfadjoint integral operators associated with fractional differential equations. With the help of this method we, in particular, estimate eigen-functions and eigenvalues of the boundary-value problem for a fractional oscillatory equation.     

On a non-local boundary problem for a parabolic-hyperbolic equation involving a Riemann-Liouville fractional differential operator

In the present work, a non-local boundary value problem with special gluing conditions for a mixed parabolic-hyperbolic equation with parameter is considered. The parabolic part of this equation is a fractional analogue of heat equation and the hyperbolic part is the telegraph equation. The considered problem is reduced, for positive values of the parameter, to an equivalent system of the second kind Volterra integral equations. Due to the influence of the fractional diffusion equation, the looked for solution belongs to a specific class of functions. The method of the Green functions and the properties of integro-differential operators are on the basis of the investigation.