On the absence of global solutions for quantum versions of Schrödinger equations and systems

We, first, consider the quantum version of the nonlinear Schrödinger equation

$i^q{D}_{q|t}u(t,x)+\Delta u(qt,x)=\lambda |u(qt,x){|}^p,t>0,x\in {\mathbb{R}}^N,$

where $0<q<1$, $i^q$ is the principal value of $i^q$, $D_{q|t}$ is the $q$-derivative with respect to $t$, $\Delta$ is the Laplacian operator in ${\mathbb{R}}^N$, $\lambda \in ??\left\{0\right\}$, $p>1$, and $u(t,x)$ is a complex-valued function. Sufficient conditions for the nonexistence of global weak solution to the considered equation are obtained under suitable initial data. Next, we study the system of nonlinear coupled equations

$i^q{D}_{q|t}u(t,x)+\Delta u(qt,x)=\lambda |v(qt,x){|}^p,t>0,x\in {\mathbb{R}}^N,$

$i^q{D}_{q|t}v(t,x)+\Delta v(qt,x)=\lambda |u(qt,x){|}^m,t>0,x\in {\mathbb{R}}^N,$     

Application of Legendre wavelets for solving fractional differential equations

In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelet approximations. The properties of Legendre wavelets are first presented. These properties are then utilized to reduce the fractional ordinary differential equations (FODEs) to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution.     

L1 Fourier spectral methods for a class of generalized two-dimensional time fractional nonlinear anomalous diffusion equations

In this paper, L1 Fourier spectral methods are derived to obtain the numerical solutions for a class of generalized two-dimensional time-fractional nonlinear anomalous diffusion equations involving Caputo fractional derivative. Firstly, we establish the L1 Fourier Galerkin full discrete and L1 Fourier collocation schemes with Fourier spectral discretization in spatial direction and L1 difference method in temporal direction. Secondly, stability and convergence for both Galerkin and collocation approximations are proved. It is shown that the proposed methods are convergent with spectral accuracy in space and $(2?\alpha )$ order accuracy in time. For implementation, the equivalence between pseudospectral method and collocation method is discussed. Furthermore, a numerical algorithm of Fourier pseudospectral method is developed based on two-dimensional fast Fourier transform (FFT2) technique. Finally, numerical examples are provided to test the theoretical claims. As is shown in the numerical experiments, Fourier spectral methods are powerful enough with excellent efficiency and accuracy.     

Numerical solution for a class of space-time fractional equation by the piecewise reproducing kernel method

ABSTRACT

Due to the non-locality of fractional derivative, the analytical solution and good approximate solution of fractional partial differential equations are usually difficult to get. Reproducing kernel space is a perfect space in studying this type of equations, however the numerical results of equations by using the traditional reproducing kernel method (RKM) isn't very good. Based on this problem, we present the piecewise technique in the reproducing kernel space to solve this type of equations. The focus of this paper is to verify the stability and high accuracy of the present method by comparing the absolute error with traditional RKM and study the effect on absolute error for different values of α. Furthermore, we can study the distribution of entire space at a particular time period. Three numerical experiments are provided to verify the efficiency and stability of the proposed method. Meanwhile, it is tested by experiments that the change of the value of α has little effect on its accuracy.     

Periodic solutions for a class of nonlinear hyperbolic evolution equations

The existence of periodic solutions is constructively substantiated for a class of nonlinear hyperbolic evolution equations. A priori estimates are obtained. Examples are given to illustrate the results.     

Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion

In this paper, we study the time-space fractional order (fractional for simplicity) nonlinear subdiffusion and superdiffusion equations, which can relate the matter flux vector to concentration gradient in the general sense, describing, for example, the phenomena of anomalous diffusion, fractional Brownian motion, and so on. The semi-discrete and fully discrete numerical approximations are both analyzed, where the Galerkin finite element method for the space Riemann-Liouville fractional derivative with order 1+β∈[1,2] and the finite difference scheme for the time Caputo derivative with order α∈(0,1) (for subdiffusion) and (1,2) (for superdiffusion) are analyzed, respectively. Results on the existence and uniqueness of the weak solutions, the numerical stability, and the error estimates are presented. Numerical examples are included to confirm the theoretical analysis. During our simulations, an interesting diffusion phenomenon of particles is observed, that is, on average, the diffusion velocity for 0<α<1 is slower than that for α=1, but the diffusion velocity for 1<α<2 is faster than that for α=1. For the spatial diffusion, we have a similar observation.     

Using reproducing kernel for solving a class of time-fractional telegraph equation with initial value conditions

Today, most of the real physical world problems can be best modelled with fractional telegraph equation. Besides modelling, the solution techniques and their reliability are the most important. Therefore, high accuracy solutions are always needed. As we all know, reproducing kernel method (RKM) has been successfully presented for solving various ordinary differential equations. However, the numerical results are not perfectly satisfactory when we directly use the traditional RKM for solving fractional partial differential equation. The aim of this paper is to fill this gap. In this paper, a new method is provided for solving fractional telegraph equation in the reproducing kernel space by piecewise technique, which can obtain more accurate solution than traditional method. Three experiments are given to demonstrate the effectiveness of the present method.     

Computational algorithms for computing the fractional derivatives of functions

In this paper, we propose algorithms to compute the fractional derivatives of a function by a weighted sum of function values at specified points. The fractional derivatives are considered in the Caputo sense. The error analysis of the algorithms and some illustrative examples are presented. The numerical results confirm that the new algorithms are accurate, efficient and readily implemented.     

一类耦合抽象非线性梁方程组的整体解

研究了一类抽象耦合非线性梁方程组在Hilbert空间中的初值问题.首先运用Galerkin方法对两个方程进行一定的处理,然后证明收敛性,最后证明了上述非线性梁方程组的整体弱解的存在性.     

分数阶非线性方程精确解的新方法

将分数阶复变换方法和[(G/G)]方法相结合得到了一种辅助方程方法,用来求解分数阶非线性微分方程。利用该方法并借助于软件Mathematica的符号计算功能求解了分数阶Calogero KDV方程,得到了该方程新的精确解。     

分数阶非线性Duffing振子方程的特性研究

将Caputo分数阶微分算子引入到非线性的Duffing振子方程中,运用同伦扰动变换法--一种同伦扰动法和Laplace变换相结合的方法来求解分数阶的非线性方程,借助Mathematica软件的符号计算功能得到了分数阶非线性Duffing振子方程的近似解,研究了振子运动过程与分数阶导数之间的关系。     

The global solution for a class of systems of fractional nonlinear Schrödinger equations with periodic boundary condition

In this paper, we consider a class of systems of fractional nonlinear Schrödinger equations. We prove the existence and uniqueness of the global solution to the periodic boundary value problem by using the Faedo-Galërkin method.     

Existence results for fractional integrodifferential equations with nonlocal condition via resolvent operators

In this paper, we prove the existence of solutions of fractional integrodifferential equations by using the resolvent operators and fixed point theorem. An example is given to illustrate the abstract results.     

一类具有记忆项的耦合非线性抽象方程组的整体解

运用Galerkin方法讨论了一类具有记忆项的耦合非线性抽象方程组的初值问题,根据方程组的特点,巧妙地对两个方程进行相加,并结合微积分的性质得到了所要的结果,然后研究收敛性,最后证明了方程组整体弱解的存在性.     

Well-posedness,optimal control and discretization for time-fractional parabolic equations with time-dependent coefficients on metric graphs

We study distributed optimal control problems governed by time-fractional parabolic equations with time dependent coefficients on metric graphs, where the fractional derivative is considered in the Caputo sense. Using the Galerkin method and compactness results, for the spatial part, and approximating the kernel of the time-fractional Caputo derivative by a sequence of more regular kernel functions, we first prove the well-posedness of the system. We then turn to the existence and uniqueness of solutions to the distributed optimal control problem. By means of the Lagrange multiplier method, we develop an adjoint calculus for the right Caputo derivative and derive the corresponding first order optimality system. Moreover, we propose a finite difference scheme to find the approximate solution of the state equation and the resulting optimality system on metric graphs. Finally, examples are provided on two different graphs to illustrate the performance of the proposed difference scheme.     

Operational method for solving fractional differential equations using cubic B-spline approximation

In the present paper we construct the cubic B-spline operational matrix of fractional derivative in the Caputo sense, and use it to solve fractional differential equation. The main characteristic of the approach is that it overcomes the computational difficulty induced by the memory effect. There is no need to save and call all historic information, which can save memory space and reduce computational complexity. Numerical results demonstrate the validity and applicability of the method to solve fractional differential equation. The results from this method are good in terms of accuracy.     

On global controllability for a class of polynomial affine nonlinear systems

In this paper,a necessary and sufficient condition of the global controllability for a class of low dimensional polynomial affine nonlinear systems with special structure is obtained.The condition is imposed on the coefficients of the system only and the methods are based on Green’s formula and the trajectory analysis of planar linear system.Furthermore, I point out that the global controllability does not hold for the corresponding high dimensional polynomial system.     

Regularity for solutions of nonlinear evolution equations with nonlinear perturbations

The approximate controllability for the nonlinear control system with nonlinear monotone hemicontinuous and the nonlinear perturbations is studied. The existence, uniqueness, and a variation of solutions of the system are also given.     

An efficient fractional-order wavelet method for fractional Volterra integro-differential equations

In this paper, an efficient and robust numerical technique is suggested to solve fractional Volterra integro-differential equations (FVIDEs). The proposed method is mainly based on the generalized fractional-order Legendre wavelets (GFLWs), their operational matrices and the Collocation method. The main advantage of the proposed method is that, by using the GFLWs basis, it can provide more efficient and accurate solution for FVIDEs in compare to integer-order wavelet basis. A comparison between the achieved results confirms accuracy and superiority of the proposed GFLWs method for solving FVIDEs. Error analysis and convergence of the GFLWs basis is provided.