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.     

Identification of a time-dependent source term for a time fractional diffusion problem

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.     

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.     

A fractional nonlocal time-space viscoelasticity theory and its applications in structural dynamics

To overcome the long wavelength and time limits of classical elastic theory, this paper presents a fractional nonlocal time-space viscoelasticity theory to incorporate the non-locality of both time and spatial location. The stress (strain) at a reference point and a specified time is assumed to depend on the past time history and the stress (strain) of all the points in the reference domain through nonlocal kernel operators. Based on an assumption of weak non-locality, the fractional Taylor expansion series is used to derive a fractional nonlocal time-space model. A fractional nonlocal Kevin–Voigt model is considered as the simplest fractional nonlocal time-space model and chosen to be applied for structural dynamics. The correlation between the intrinsic length and time parameters is discussed. The effective viscoelastic modulus is derived and, based on which, the tension and vibration of rods and the bending, buckling and vibration of beams are studied. Furthermore, in the context of Hamilton’s principle, the governing equation and the boundary condition are derived for longitudinal dynamics of the rod in a more rigorous manner. It is found that when the external excitation frequency and the wavenumber interact with the intrinsic microstructures of materials and the intrinsic time parameter, the nonlocal space-time effect will become substantial, and therefore the viscoelastic structures are sensitive to both microstructures and time.     

Approximation Technique for Fractional Evolution Equations with Nonlocal Integral Conditions

We consider a class of fractional evolution equations with nonlocal integral conditions in Banach spaces. New existence of mild solutions to such a problem are established using Schauder fixed-point theorem, diagonal argument and approximation techniques under the hypotheses that the nonlinear term is Carathéodory continuous and satisfies some weak growth condition, the nonlocal term depends on all the value of independent variable on the whole interval and satisfies some weak growth condition. This work may be viewed as an attempt to develop a general existence theory for fractional evolution equations with general nonlocal integral conditions. Finally, as a sample of application, the results are applied to a fractional parabolic partial differential equation with nonlocal integral condition. The results obtained in this paper essentially extend some existing results in this area.     

Fourier regularization for a final value time-fractional diffusion problem

The evolution process of fractional order describes some phenomenon of anomalous diffusion and transport dynamics in complex system. The equation containing time-fractional derivative provides a suitable mathematical model for describing such a process. The backward problem for this system, which means to recover the initial state for some slow diffusion process from its present status, is very hard to solve due to the nonlocal property of fractional derivative and the irreversibility of time. For this ill-posed problem, we construct a regularizing solution using the Fourier transform method. Both the a-priori choice strategy and the a-posteriori choice strategy for the regularizing parameter are given, with the convergence analysis on the regularizing solution. Numerical implementations are presented to show the validity of the proposed scheme.     

Solvability of a boundary value problem for a mixed-type equation with a partial Riemann-Liouville fractional derivative

We prove existence and uniqueness theorems for the solution of a nonlocal problem for a partial differential equation with a Riemann-Liouville fractional derivative in which the boundary condition contains a generalized fractional differentiation operator with the Gauss hypergeometric function in the kernel. We present a closed-form solution of the problem.     

A class of time‐fractional reaction‐diffusion equation with nonlocal boundary condition

The purpose of the study is to analyze the time‐fractional reaction‐diffusion equation with nonlocal boundary condition. The proposed model is used to predict the invasion of tumor and its growth. Further, we establish the existence and uniqueness of a weak solution of the proposed model using the Faedo‐Galerkin method and compactness arguments.     

Finite Element Approximation of a Contact Vector Eigenvalue Problem

We consider a nonstandard elliptic eigenvalue problem of second order on a two-component domain consisting of two intervals with a contact point. The interaction between the two domains is expressed through a coupling condition of nonlocal type, more specifically, in integral form. The problem under consideration is first stated in its variational form and next interpreted as a second-order differential eigenvalue problem. The aim is to set up a finite element method for this problem. The error analysis involved is shown to be affected by the nonlocal condition, which requires a suitable modification of the vector Lagrange interpolant on the overall finite element mesh. Nevertheless, we arrive at optimal error estimates. In the last section, an illustrative numerical example is given, which confirms the theoretical results.     

Controllability of fractional impulsive neutral integrodifferential systems with a nonlocal Cauchy condition in Banach spaces

In this work the controllability of fractional impulsive neutral functional integrodifferential systems with a nonlocal Cauchy condition in a Banach space has been addressed. Sufficient conditions for the controllability are established using fractional powers of operators and the Banach contraction mapping theorem.     

On the nonlocal Cauchy problem for semilinear fractional order evolution equations

In this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first result relies on a growth condition on the whole time interval via Schaefer fixed point theorem. The second result relies on a growth condition splitted into two parts, one for the subinterval containing the points associated with the nonlocal conditions, and the other for the rest of the interval via O’Regan fixed point theorem.     

On a first-order semipositone discrete fractional boundary value problem

In this paper we demonstrate the existence of at least one positive solution to a discrete fractional semipositone boundary value problem. Our results extend existing results not only due to the fact that the problem we treat here is of fractional order but also due to the fact that the boundary condition we study is potentially very general being as it can be both nonlocal and nonlinear.     

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 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.     

A note on the fractional Cauchy problems with nonlocal initial conditions

Of concern is the Cauchy problems for fractional integro-differential equations with nonlocal initial conditions. Using a new strategy in terms of the compactness of the semigroup generated by the operator in the linear part and approximating technique, a new existence theorem for mild solutions is established. An application to a fractional partial integro-differential equation with a nonlocal initial condition is also considered.     

An inverse problem of identifying the time-dependent potential in a fourth-order pseudo-parabolic equation from additional condition

The aim of this work is to identify numerically, for the first time, the time-dependent potential coefficient in a fourth-order pseudo-parabolic equation with nonlocal initial data, nonlocal boundary conditions, and the boundary data as overdetermination condition. This problem emerges significantly in the modeling of various phenomena in physics and engineering. From literature we already know that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data. For the numerical realization, we apply the quintic B-spline (QB-spline) collocation method for discretizing the pseudo-parabolic problem and the Tikhonov regularization for finding a stable and accurate solution. The resulting nonlinear minimization problem is solved using the MATLAB subroutine lsqnonlin. Moreover, the von Neumann stability analysis is also discussed.     

Approximate Controllability of Fractional Neutral Stochastic Evolution Equations with Nonlocal Conditions

In this paper, the approximate controllability of neutral stochastic fractional differential equations involving nonlocal initial conditions is studied. By using Sadovskii’s fixed point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of semilinear fractional stochastic differential equations with nonlocal conditions under the assumption that the corresponding linear system is approximately controllable. Finally, an application to a fractional partial stochastic differential equation with nonlocal initial condition is provided to illustrate the obtained theory.     

HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY

In this paper, efficient numerical schemes are proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water wave. By using the Caputo fractional derivative definition to approximate the nonlocal fractional operator, finite difference method in time and spectral method in space are constructed for the considered model. The proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear term. The analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time, and spectral accuracy in space. Several numerical results are provided to verify the efficiency and accuracy of our theoretical claims. Finally, the decay rate of solutions is investigated.     

Finite Element Approximation of Space Fractional Optimal Control Problem with Integral State Constraint

In this paper finite element approximation of space fractional optimal control problem with integral state constraint is investigated. First order optimal condition and regularity of the control problem are discussed. A priori error estimates for control, state, adjoint state and lagrange multiplier are derived. The nonlocal property of the fractional derivative results in a dense coefficient matrix of the discrete state and adjoint state equation. To reduce the computational cost a fast projection gradient algorithm is developed based on the Toeplitz structure of the coefficient matrix. Numerical experiments are carried out to illustrate the theoretical findings.     

Signed and sign-changing solutions for a Kirchhoff-type problem involving the fractional p-Laplacian with critical Hardy nonlinearity

In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p-Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on mountain pass theorem and constrained minimization in Nehari sets.