Existence and multiplicity of solutions for nonlocal systems involving fractional Laplacian with non-differentiable terms

In this paper, we devote to the study of the existence and multiplicity of solutions of nonlocal systems involving fractional Laplacian with non-differentiable terms using some extended critical point theorems for locally Lipschitz function on product spaces.     

Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces

We consider a transmission problem consisting of two semilinear parabolic equations involving fractional diffusion operators of different orders in a general non-smooth setting with emphasis on Lipschitz interfaces and transmission conditions along the interface. We give a unified framework for the existence and uniqueness of strong and mild solutions, and their global regularity properties.     

Continuity in law with respect to the Hurst index of some additive functionals of sub-fractional Brownian motion

In this article, first, we prove some properties of the sub-fractional Brownian motion introduced by Bojdecki et al. [Statist. Probab. Lett. 69(2004):405–419]. Second, we prove the continuity in law, with respect to small perturbations of the Hurst index, in some anisotropic Besov spaces, of some continuous additive functionals of the sub-fractional Brownian motion. We prove that our result can be obtained easily, by using the decomposition in law of the sub-fractional Brownian motion given by Bardina and Bascompte [Collect. Math. 61(2010):191–204] and Ruiz de Chavez and Tudor [Math. Rep. 11(2009):67–74], without using the result of Wu and Xiao [Stoch. Proc. Appl. 119(2009):1823–1844] by connecting the sub-fractional Brownian motion to its stationary Gaussian process through Lamperti’s transform. This decomposition in law leads to a better understanding and simple proof of our result.     

A fast discontinuous finite element discretization for the space‐time fractional diffusion‐wave equation

In this article, we study fast discontinuous Galerkin finite element methods to solve a space‐time fractional diffusion‐wave equation. We introduce a piecewise‐constant discontinuous finite element method for solving this problem and derive optimal error estimates. Importantly, a fast solution technique to accelerate Toeplitz matrix‐vector multiplications which arise from discontinuous Galerkin finite element discretization is developed. This fast solution technique is based on fast Fourier transform and it depends on the special structure of coefficient matrices. In each temporal step, it helps to reduce the computational work from required by the traditional methods to log , where is the size of the coefficient matrices (number of spatial grid points). Moreover, the applicability and accuracy of the method are verified by numerical experiments including both continuous and discontinuous examples to support our theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2043–2061, 2017     

High‐order compact schemes for fractional differential equations with mixed derivatives

In this article, we consider two‐dimensional fractional subdiffusion equations with mixed derivatives. A high‐order compact scheme is proposed to solve the problem. We establish a sufficient condition and show that the scheme converges with fourth order in space and second order in time under this condition.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2141–2158, 2017     

Error and stability analysis of numerical solution for the time fractional nonlinear Schrödinger equation on scattered data of general‐shaped domains

In present work, a kind of spectral meshless radial point interpolation (SMRPI) technique is applied to the time fractional nonlinear Schrödinger equation in regular and irregular domains. The applied approach is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. It is proved the scheme is unconditionally stable with respect to the time variable in and also convergent by the order of convergence , . In the current work, the thin plate spline are used as the basis functions and to eliminate the nonlinearity, a simple predictor‐corrector (P‐C) scheme is performed. It is shown that the SMRPI solution, as a complex function, is suitable one for the time fractional nonlinear Schrödinger equation. The results of numerical experiments are compared to analytical solutions to confirm the reliable treatment of these stable solutions. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1043–1069, 2017     

Some inverse problems for the nonlocal heat equation with Caputo fractional derivative

A class of inverse problems for restoring the right‐hand side of a fractional heat equation with involution is considered. The results on existence and uniqueness of solutions of these problems are presented.     

A self‐adaptive intelligence gray prediction model with the optimal fractional order accumulating operator and its application

The self‐adaptive intelligence gray predictive model (SAIGM) has an alterable‐flexible model structure, and it can build a dynamic structure to fit different external environments by adjusting the parameter values of SAIGM. However, the order number of the raw SAIGM model is not optimal, which is an integer. For this, a new SAIGM model with the fractional order accumulating operator (SAIGM_FO) was proposed in this paper. Specifically, the final restored expression of SAIGM_FO was deduced in detail, and the parameter estimation method of SAIGM_FO was studied. After that, the Particle Swarm Optimization algorithm was used to optimize the order number of SAIGM_FO, and some steps were provided. Finally, the SAIGM_FO model was applied to simulate China's electricity consumption from 2001 to 2008 and forecast it during 2009 to 2015, and the mean relative simulation and prediction percentage errors of the new model were only 0.860% and 2.661%, in comparison with the ones obtained from the raw SAIGM model, the GM(1, 1) model with the optimal fractional order accumulating operator and the GM(1, 1) model, which were (1.201%, 5.321%), (1.356%, 3.324%), and (2.013%, 23.944%), respectively. The findings showed both the simulation and the prediction performance of the proposed SAIGM_FO model were the best among the 4 models.     

Generalized Bessel functions: Theory and their applications

This paper presents 2 new classes of the Bessel functions on a compact domain [0,T] as generalized‐tempered Bessel functions of the first‐ and second‐kind which are denoted by GTBFs‐1 and GTBFs‐2. Two special cases corresponding to the GTBFs‐1 and GTBFs‐2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self‐adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders.     

Multi‐switching combination–combination synchronization of non‐identical fractional‐order chaotic systems

In this paper, multi‐switching combination–combination synchronization scheme has been investigated between a class of four non‐identical fractional‐order chaotic systems. The fractional‐order Lorenz and Chen's systems are taken as drive systems. The combination–combination of multi drive systems is then synchronized with the combination of fractional‐order Lü and Rössler chaotic systems. In multi‐switching combination–combination synchronization, the state variables of two drive systems synchronize with different state variables of two response systems simultaneously. Based on the stability of fractional‐order chaotic systems, the multi‐switching combination–combination synchronization of four fractional‐order non‐identical systems has been investigated. For the synchronization of four non‐identical fractional‐order chaotic systems, suitable controllers have been designed. Theoretical analysis and numerical results are presented to demonstrate the validity and feasibility of the applied method. Copyright © 2017 John Wiley & Sons, Ltd.