Solvability of a coefficient recovery problem for a time-fractional diffusion equation with periodic boundary and overdetermination conditions

In this paper, we discuss a boundary value problem for an impulsive fractional differential equation. By transforming the boundary value problem into an equivalent integral equation, and employing the Banach fixed point theorem and the Schauder fixed point theorem, existence results for the solutions are obtained. For application, we provide some examples to illustrate our main results.

This article studies the inverse problem of determining the pressure and convolution kernel in an integro-differential Moore-Gibson-Thompson equation with initial, periodic boundary and integral overdetermination conditions on the rectangular domain. By Fourier method this problem is reduced to an equivalent integral equation and on based of Banach’s f ixed point argument in a suitably chosen function space, the local solvability of the problem is proven. Then, the found solutions are continued throughout the entire domain of definition of the unknowns.

We investigate the solution of the inverse problem for a linear two-dimensional parabolic equation with periodic boundary and integral overdetermination conditions. Under certain natural regularity and consistency conditions imposed on the input data, we establish the existence, uniqueness of the solution, and its continuous dependence on the data by using the generalized Fourier method. In addition, an iterative algorithm is constructed for the numerical solution of the problem.

In this paper, we consider the problem of determining the time-dependent thermal diffusivity and the temperature distribution in one-dimensional quasilinear heat equation with periodic boundary and integral overdetermination conditions. We establish conditions for the existence, uniqueness and continuous dependent of a classical solution of the problem under considerations. We present some results on the numerical solution with two examples.

Fractional differential equations and systems are gradually becoming an essential approach to real world applications in science and engineering technology. The boundary value problems of the fractional differential equations and systems were investigated by using several different methods. Recently, much attention has been paid to investigate fractional differential equations with nonlocal conditions by using the fixed point method. In this paper, by using the Banach fixed point theorem and Krasnoselkii fixed point theorem, the existence and uniqueness of the solutions to the initial value problem of the nonlinear fractional differential equations with nonlocal conditions are investigated, and some sufficient conditions are obtained. We extend some results that already exist. Finally, an example is given to show the usefulness of the theoretical results.

This paper investigates the inverse problem of finding a time-dependent diffusion coefficient in a parabolic equation with the periodic boundary and integral overdetermination conditions. Under some assumption on the data, the existence, uniqueness, and continuous dependence on the data of the solution are shown by using the generalized Fourier method. The accuracy and computational efficiency of the proposed method are verified with the help of the numerical examples.

This paper concerns the investigation of an eigenvalue problem for a nonlinear fractional differential equation. Using the properties of the Green function, Banach contraction principle, Leray-Schauder nonlinear alternative and Guo-Krasnosel'skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation are considered. Some sufficient conditions for the existence of at least one positive solution is established. Some examples are presented to illustrate the main results.

In this work we investigate the existence and uniqueness of solutions of boundary value problems for fractional differential equations involving the Caputo fractional derivative with integral conditions and the nonlinear term depends on the fractional derivative of an unknown function. Our existence results are based on Banach contraction principle and Schauder fixed point theorem. Two examples are provided to illustrate our results.

In this paper, the existence and uniqueness results of solution for an initial value problem of fractional differential equations with a new concept of fractional derivative in the case of the order $\alpha\in(1,2)$ have been investigated. This concept is based on the Caputo fractional derivative and Caputo-Hadamard fractional derivative into a single form Banach ' s contraction mapping principle is used to obtain the existence and uniqueness results.

In this paper the inverse problem of finding the time-dependent coefficient of heat capacity together with the solution of a heat equation with periodic boundary and integral overdetermination conditions is considered. The conditions for the existence and uniqueness of a classical solution of the problem under consideration are established. A numerical example using the Crank-Nicolson finite-difference scheme combined with the iteration method is presented.

In this paper, we study a new approach of investigation of existence, uniqueness and stability of the periodic solution of the nonlinear fractional integro-differential equation of type Caputo-Fabrizio fractional derivative with the initial condition, periodic boundary conditions, and integral boundary conditions by using successive approximations method and Banach fixed point theorem. Finally, some examples are present to illustrate the theorems.

In this work, two-dimensional inverse quasi-linear parabolic problem with periodic boundary and integral overdetermination conditions is investigated. The formal solution is obtained by the Fourier approximation. Under some natural regularity and consistency conditions on the input data,the existence, uniqueness and continuously dependence upon the data of the solution are proved by iteration method. The inverse problem is first examined by linearization and then used implicit finite difference scheme for the numerical solution. Also predictor corrector method is considered in the numerical approach. Some results on the numerical solution with two examples are presented with figures and tables. The sensitivity of the scheme with respect to noisy overdetermination data is illustrated.

In this paper, we study the inverse problem of determining the time-dependent coefficient in a one-dimensional fractional order equation with initial-boundary conditions and over-determination conditions. Using the Fourier method, this problem is reduced to equivalent integral equations. Then, using estimates of the Mittag--Leffler function and the method of successive approximations, an estimate of the solution to the direct problem is obtained through the norm of the unknown coefficient, which will be used in the study of the inverse problem. The inverse problem is reduced to an equivalent integral equation. To solve this equation, the contraction mapping principle is used. The results of local existence and uniqueness are proved.обратная задача; адвекция-дисперсия; дробная производная; функция Мит\-таг-Леффлера; $H$-функция.

The initial value problem of fractional differential equations and its solving method are studied in this paper. Firstly, for easy understanding, a different version of the initialized operator theory is presented for Riemann-Liouville’s fractional-order derivative, addressing the initial history in a straightforward form. Then, the initial value problem of a single-term fractional differential equation is converted to an equivalent integral equation, a form that is easy for both theoretical and numerical analysis, and two illustrative examples are given for checking the correctness of the integral equation. Finally, the counter-example proposed in a recent paper, which claims that the initialized operator theory results in wrong solution of a fractional differential equation, is checked again carefully. It is found that solving the equivalent integral equation gives the exact solution, and the reason behind the result of the counter-example is that the calculation therein is based on the conventional Laplace transform for fractional-order derivative, not on the initialized operator theory. The counter-example can be served as a physical model of creep phenomena for some viscoelastic materials, and it is found that it fits experimental curves well.

This paper deals with a class of elliptic differential eigenvalue problems (EVPs) of second order on a rectangular domain Ω⊂ℝ2, with periodic or semi-periodic boundary conditions (BCs) on two adjacent sides of Ω. On the remaining sides, classical Dirichlet or Robin type BCs are imposed. First, we pass to a proper variational formulation, which is shown to fit into the framework of abstract EVPs for strongly coercive, bounded and symmetric bilinear forms in Hilbert spaces. Next, the variational EVP serves as the starting point for finite element approximations. We consider finite element methods (FEMs) without and with numerical quadrature, both with triangular and with rectangular meshes. The aim of the paper is to show that well-known error estimates, established for finite element approximations of elliptic EVPs with classical BCs, remain valid for the present type of EVPs, including the case of multiple exact eigenvalues. Finally, the analysis is illustrated by a non-trivial numerical example, the exact eigenpairs of which can be determined.