Inverse problems for a perturbed time fractional diffusion equation with final overdetermination

Inverse problems to recover a space‐dependent factor of a source term and an initial condition in a perturbed time fractional diffusion equation containing an additional convolution term from final data are considered. Existence, uniqueness, and stability of solutions to these problems are proved.     

Finite difference scheme for time-space fractional diffusion equation with fractional boundary conditions

In this paper, the finite difference scheme is developed for the time-space fractional diffusion equation with Dirichlet and fractional boundary conditions. The time and space fractional derivatives are considered in the senses of Caputo and Riemann-Liouville, respectively. The stability and convergence of the proposed numerical scheme are strictly proved, and the convergence order is O(τ^2−α+h²). Numerical experiments are performed to confirm the accuracy and efficiency of our scheme.     

An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions

We consider the inverse source problem for a time fractional diffusion equation. The unknown source term is independent of the time variable, and the problem is considered in two dimensions. A biorthogonal system of functions consisting of two Riesz bases of the space L²[(0,1) × (0,1)], obtained from eigenfunctions and associated functions of the spectral problem and its adjoint problem, is used to represent the solution of the inverse problem. Using the properties of the biorthogonal system of functions, we show the existence and uniqueness of the solution of the inverse problem and its continuous dependence on the data. Copyright © 2012 John Wiley & Sons, Ltd.     

Nonlocal boundary value problem for generalized Hilfer implicit fractional differential equations

In this paper, we derive the equivalent fractional integral equation to the nonlinear implicit fractional differential equations involving Ψ-Hilfer fractional derivative subject to nonlocal fractional integral boundary conditions. The existence of a solution, Ulam–Hyers, and Ulam–Hyers–Rassias stability have been acquired by means of an equivalent fractional integral equation. Our investigations depend on the fixed-point theorem due to Krasnoselskii and the Gronwall inequality involving Ψ-Riemann–Liouville fractional integral. Finally, examples are provided to show the utilization of primary outcomes.     

Nonlocal fractional elliptic equations and applications

The L_p-coercive properties of a nonlocal fractional elliptic equation is studied. Particularly, it is proved that the fractional elliptic operator generated by this equation is sectorial in L_p space and also is a generator of an analytic semigroup. Moreover, by using the L_p-separability properties of the given elliptic operator the maximal regularity of the corresponding nonlocal fractional parabolic equation is established.     

L∞‐solutions for the ultra‐slow reaction‐diffusion equation

The existence and uniqueness of solutions for a reaction‐diffusion ultra‐slow equation are proved. We also show that they can be extended up a maximal time and are stable as long as they exist. Symmetric and positive solutions are also proved to exist.     

On positive solutions to fractional sum boundary value problems for nonlinear fractional difference equations

In this paper, we study a new class of 3‐point boundary value problems of nonlinear fractional difference equations. Our problems contain difference and fractional sum boundary conditions. Existence and uniqueness of solutions are proved by using the Banach fixed‐point theorem, and existence of the positive solutions is proved by using the Krasnoselskii's fixed‐point theorem. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence and regularity results of a backward problem for fractional diffusion equations

In this paper, we study a backward problem for an inhomogeneous fractional diffusion equation in a bounded domain. By applying the properties of Mittag‐Leffler functions and the method of eigenvalue expansion, we establish some results about the existence, uniqueness, and regularity of the mild solutions as well as the classical solutions of the proposed problem in a weighted Hölder continuous function space.     

On a nonlocal problem for the Laplace equation in the unit ball with fractional boundary conditions

In this paper, we investigate the correct solvability for the Laplace equation with a nonlocal boundary condition in the unit ball. The considered boundary operator is of fractional order. This problem is a generalization of the well‐known Bitsadze–Samarskii problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of positive solutions for fractional boundary value problems

In this paper, by introducing a new operator, improving and generating a p-Laplace operator for some $p > 1$, we discuss the existence and multiplicity of positive solutions to the four point boundary value problems of nonlinear fractional differential equations. Our results extend some recent works in the literature.     

Positive solutions for a singular fractional boundary value problem

We investigate the existence of positive solutions for a system of Riemann-Liouville fractional differential equations, supplemented with uncoupled nonlocal boundary conditions which contain various fractional derivatives and Riemann-Stieltjes integrals, and the nonlinearities of the system are nonnegative functions and they may be singular at the time variable. In the proof of our main theorems, we use the Guo-Krasnosel'skii fixed point theorem.     

On the solution of a fractional diffusion integrodifferential equation with Rothe time discretization

This article deals with a fractional diffusion equation of the second-order differential Volterra operator and fractional integral condition. Existence and uniqueness of a weak solution in an appropriate sense as well as some regularity results are obtained by the use of Rothe’s method. Finally, an example is given to demonstrate the effectiveness of the proposed approach.     

Fourth-order problems with fully nonlinear boundary conditions

Differential inequality method, bounding function method and topological degree are applied to obtain the existence criterions of at least one solution for the general fourth-order differential equations under nonlinear boundary conditions, and many existing results are complemented.     

A second‐order finite difference method for fractional diffusion equation with Dirichlet and fractional boundary conditions

In this paper, we develop a practical numerical method to approximate a fractional diffusion equation with Dirichlet and fractional boundary conditions. An approach based on the classical Crank–Nicolson method combined with spatial extrapolation is used to obtain temporally and spatially second‐order accurate numerical estimates. The solvability, stability, and convergence of the proposed numerical scheme are proved via the Gershgorin theorem. Numerical experiments are performed to confirm the accuracy and efficiency of our scheme.     

Second-order boundary value problems with nonhomogeneous boundary conditions (II)

Sufficient conditions are given for the existence of solutions of the following nonlinear boundary value problem with nonhomogeneous multi-point boundary condition

We prove that the whole plane is divided by a “continuous decreasing curve” Γ into two disjoint connected regions Λ^E and Λ^N such that the above problem has at least one solution for (λ₁,λ₂)Γ, has at least two solutions for (λ₁,λ₂)Λ^EΓ, and has no solution for (λ₁,λ₂)Λ^N. We also find explicit subregions of Λ^E where the above problem has at least two solutions and two positive solutions, respectively.     

Difference methods for solving boundary value problems for fractional differential equations

Difference schemes for second-order ordinary and partial differential equations with a fractional time derivative are considered. Stationary and nonstationary problems for the diffusion equation in one-and multidimensional domains are examined separately. The stability and convergence of the difference schemes for these equations are proved.     

An inverse diffusion problem with nonlocal boundary conditions

This article considers the inverse problem of identification of a time‐dependent thermal diffusivity together with the temperature in an one‐dimensional heat equation with nonlocal boundary and integral overdetermination conditions when a heat exchange takes place across boundary of the material. The well‐posedness of the problem is studied under some regularity, and consistency conditions on the data of the problem together with the nonnegativity condition on the Fourier coefficients of the initial data and source term. The inverse problem is also studied numerically by using the Crank–Nicolson finite difference scheme combined with predictor‐corrector technique. The numerical examples are presented and discussed. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 564–590, 2016