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.     

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 the solvability of a nonlocal problem for a hyperbolic equation of the second kind

In the characteristic triangle for a hyperbolic equation of the second kind we study a nonlocal problem, where the boundary value condition contains a linear combination of Riemann–Liouville fractional integro-differentiation operators. We establish variation intervals of orders of fractional integro-differentiation operators, taking into account parameters of the considered equation with which the mentioned problem has either a unique solution or more than one solution.     

On fractional powers of a pair of matrices and a plate-beam problem

In this paper we consider the transversal deflections of a dynamically-coupled Von Kármán system consisting of a plate which has a beam attached to its one edge. The problem is considered in the form of a non-linear evolution problem in a product space. We show the existence of a unique local solution by following a fractional powers approach to first construct a “weak” solution in a larger space. Regularity properties for this solution yield a unique local strong solution for the original boundary-value problem. This approach entails the introduction of fractional powers of a pair of matrices.     

Solving a final value fractional diffusion problem by boundary condition regularization

The evolution process of fractional order describes some phenomenon of anomalous diffusion and transport dynamics in complex system. The equation containing fractional derivatives provides a suitable mathematical model for describing such a process. The initial boundary value problem is hard to solve due to the nonlocal property of the fractional order derivative. We consider a final value problem in a bounded domain for fractional evolution process with respect to time, which means to recover the initial state for some slow diffusion process from its present status. For this ill-posed problem, we construct a regularizing solution using quasi-reversible method. The well-posedness of the regularizing solution as well as the convergence property is rigorously analyzed. The advantage of the proposed scheme is that the regularizing solution is of the explicit analytic solution and therefore is easy to be implemented. Numerical examples are presented to show the validity of the proposed scheme.     

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.     

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.     

一类分数阶微分方程四点边值问题解的存在性

研究了一类分数阶微分方程四点边值问题解的存在性,利用Schauder不动点定理,得到了边值问题至少存在一个解的充分条件.     

Vibrations of a fractal elastic string

We construct the solution of the fractional space-time equations that describe the vibrations of a quasi-one-dimensional fractal elastic string. We give the solution of the Cauchy problem for fractional differential equations with initial conditions. We carry out a numerical analysis and construct the graphic variation of the displacement function of a fractal elastic string. Three figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 142–147     

An inverse problem for a family of two parameters time fractional diffusion equations with nonlocal boundary conditions

The determination of a space‐dependent source term along with the solution for a 1‐dimensional time fractional diffusion equation with nonlocal boundary conditions involving a parameter β>0 is considered. The fractional derivative is generalization of the Riemann‐Liouville and Caputo fractional derivatives usually known as Hilfer fractional derivative. We proved existence and uniqueness results for the solution of the inverse problem while over‐specified datum at 2 different time is given. The over‐specified datum at 2 time allows us to avoid initial condition in terms of fractional integral associated with Hilfer fractional derivative.     

Dirichlet problem for a nonlocal wave equation

We study the Dirichlet problem for a nonlocal wave equation in a rectangular domain. We prove the existence and uniqueness of a solution of the problem and show that determining whether the solution is unique can be reduced to determining whether a function of Mittag-Leffler type has real zeros. The obtained uniqueness condition turns into the uniqueness condition for the solution of the Dirichlet problem for the wave equation as the order of the fractional derivative in the equation tends to 2.     

On a singular nonlocal time fractional order mixed problem with a memory term

We use the priori estimate method to prove the existence and uniqueness of a solution as well as its dependence on the given data of a singular time fractional mixed problem having a memory term. The considered fractional equation is associated with a nonlocal condition of integral type and a Neuman condition. Our results develop and show the efficiency and effectiveness of the energy inequalities method for the time fractional order differential equations with a nonlocal condition.     

Solution of a three-dimensional boundary-value problem for a fractional differential heat conduction equation

We consider a three-dimensional axisymmetric problem for a differential heat conduction equation with fractional time derivatives. Using the method of homogeneous solutions and integral transformations, we obtain an asymptotic and a numerical solution of the problem. The results of calculations are presented.     

On Linear Fractional Programming Problem and its Computation Using a Neural Network Model

In this paper we consider linear fractional programming problem and look at its linear complementarity formulation. In the literature, uniqueness of solution of a linear fractional programming problem is characterized through strong quasiconvexity. We present another characterization of uniqueness through complementarity approach and show that the solution set of a fractional programming problem is convex. Finally we formulate the complementarity condition as a set of dynamical equations and prove certain results involving the neural network model. A computational experience is also reported.      

A Generalization of Ritz-Variational Method for Solving a Class of Fractional Optimization Problems

This paper presents an approximate method for solving a class of fractional optimization problems with multiple dependent variables with multi-order fractional derivatives and a group of boundary conditions. The fractional derivatives are in the Caputo sense. In the presented method, first, the given optimization problem is transformed into an equivalent variational equality; then, by applying a special form of polynomial basis functions and approximations, the variational equality is reduced to a simple linear system of algebraic equations. It is demonstrated that the derived linear system has a unique solution. We get an approximate solution for the initial optimization problem by solving the final linear system of equations. The choice of polynomial basis functions provides a method with such flexibility that all initial and boundary conditions of the problem can be easily imposed. We extensively discuss the convergence of the method and, finally, present illustrative test examples to demonstrate the validity and applicability of the new technique.     

On the solvability of a boundary value problem for a fractional differential equation

We obtain sufficient conditions for the existence and uniqueness of a solution of a boundary value problem for a differential equation that contains a mixed fractional derivative.     

Inverse problem with a time‐integral condition for a fractional diffusion equation

We find the conditions for the unique solvability of the inverse problem for a time‐fractional diffusion equation with Schwarz‐type distributions in the right‐hand sides. This problem is to find a generalized solution of the Cauchy problem and an unknown space‐dependent part of an equation's right‐hand side under a time‐integral overdetermination condition.     

On a nonclassical fractional boundary-value problem for the Laplace operator

In this paper we consider a boundary-value problem for the Poisson equation with a boundary condition comprising the fractional derivative in time and the right-hand sides dependent on time. We prove the one-valued solvability of this problem, and provide the coercive estimates of the solution.     

Parametric computation of a fuzzy set solution to a class of fuzzy linear fractional optimization problems

The class of fuzzy linear fractional optimization problems with fuzzy coefficients in the objective function is considered in this paper. We propose a parametric method for computing the membership values of the extreme points in the fuzzy set solution to such problems. We replace the exhaustive computation of the membership values—found in the literature for solving the same class of problems—by a parametric analysis of the efficiency of the feasible basic solutions to the bi-objective linear fractional programming problem through the optimality test in a related linear programming problem, thus simplifying the computation. An illustrative example from the field of production planning is included in the paper to complete the theoretical presentation of the solving approach, but also to emphasize how many real life problems may be modelled mathematically using fuzzy linear fractional optimization.