A Split-Step Predictor-Corrector Method for Space-Fractional Reaction-Diffusion Equations with Nonhomogeneous Boundary Conditions

A split-step second-order predictor-corrector method for space-fractional reaction-diffusion equations with nonhomogeneous boundary conditions is presented and analyzed for the stability and convergence.The matrix transfer technique is used for spatial discretization of the problem.The method is shown to be unconditionally stable and second-order convergent.Numerical experiments are performed to confirm the stability and secondorder convergence of the method.The split-step predictor-corrector method is also compared with an IMEX predictor-corrector method which is found to incur oscillatory behavior for some time steps.Our method is seen to produce reliable and oscillatioresults for any time step when implemented on numerical examples with nonsmooth initial data.We also present a priori reliability constraint for the IMEX predictor-corrector method to avoid unwanted oscillations and show its validity numerically.

A fast finite difference/finite element method for the two-dimensional distributed-order time-space fractional reaction-diffusion equation

In this work,we develop a finite difference/finite element method for the two-dimensional distributed-order time-space fractional reaction-diffusion equation(2D-DOTSFRDE)with low regularity solution at the initial time.A fast evaluation of the distributedorder time fractional derivative based on graded time mesh is obtained by substituting the weak singular kernel for the sum-of-exponentials.The stability and convergence of the developed semi-discrete scheme to 2D-DOTSFRDE are discussed.For the spatial approximation,the finite element method is employed.The convergence of the corresponding fully discrete scheme is investigated.Finally,some numerical tests are given to verify the obtained theoretical results and to demonstrate the effectiveness of the method.

An Efficient Second-Order Finite Volume ADI Method for Nonlinear Three-Dimensional Space-Fractional Reaction-Diffusion Equations

In this paper,a three-dimensional time-dependent nonlinear Riesz spacefractional reaction-diffusion equation is considered.First,a linearized finite volume method,named BDF-FV,is developed and analyzed via the discrete energy method,in which the space-fractional derivative is discretized by the finite volume element method and the time derivative is treated by the backward differentiation formulae(BDF).The method is rigorously proved to be convergent with second-order accuracy both in time and space with respect to the discrete and continuous L2 norms.Next,by adding high-order perturbation terms in time to the BDF-FV scheme,an alternating direction implicit linear finite volume scheme,denoted as BDF-FV-ADI,is proposed.Convergence with second-order accuracy is also strictly proved under a rough temporal-spatial stepsize constraint.Besides,efficient implementation of the ADI method is briefly discussed,based on a fast conjugate gradient(FCG)solver for the resulting symmetric positive definite linear algebraic systems.Numerical experiments are presented to support the theoretical analysis and demonstrate the effectiveness and efficiency of the method for large-scale modeling and simulations.

Analytical solutions fractional order partial differential equations arising in fluid dynamics

This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high,and low-frequency waves.Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed.Graphical simulations of the numerical solutions are flaunted by MATLAB.

Memory effect in time fractional Schrödinger equation

A significant obstacle impeding the advancement of the time fractional Schrodinger equation lies in the challenge of determining its precise mathematical formulation.In order to address this,we undertake an exploration of the time fractional Schrodinger equation within the context of a non-Markovian environment.By leveraging a two-level atom as an illustrative case,we find that the choice to raise i to the order of the time derivative is inappropriate.In contrast to the conventional approach used to depict the dynamic evolution of quantum states in a non-Markovian environment,the time fractional Schrodinger equation,when devoid of fractional-order operations on the imaginary unit i,emerges as a more intuitively comprehensible framework in physics and offers greater simplicity in computational aspects.Meanwhile,we also prove that it is meaningless to study the memory of time fractional Schrodinger equation with time derivative 1<α≤2.It should be noted that we have not yet constructed an open system that can be fully described by the time fractional Schrodinger equation.This will be the focus of future research.Our study might provide a new perspective on the role of time fractional Schrodinger equation.

A STRONG POSITIVITY PROPERTY AND A RELATED INVERSE SOURCE PROBLEM FOR MULTI-TERM TIME-FRACTIONAL DIFFUSION EQUATIONS

In this article,we consider the diffusion equation with multi-term time-fractional derivatives.We first derive,by a subordination principle for the solution,that the solution is positive when the initial value is non-negative.As an application,we prove the uniqueness of solution to an inverse problem of determination of the temporally varying source term by integral type information in a subdomain.Finally,several numerical experiments are presented to show the accuracy and efficiency of the algorithm.

A Novel Accurate Method forMulti-Term Time-Fractional Nonlinear Diffusion Equations in Arbitrary Domains

Anovel accuratemethod is proposed to solve a broad variety of linear and nonlinear(1+1)-dimensional and(2+1)-dimensional multi-term time-fractional partial differential equations with spatial operators of anisotropic diffusivity.For(1+1)-dimensional problems,analytical solutions that satisfy the boundary requirements are derived.Such solutions are numerically calculated using the trigonometric basis approximation for(2+1)-dimensional problems.With the aid of these analytical or numerical approximations,the original problems can be converted into the fractional ordinary differential equations,and solutions to the fractional ordinary differential equations are approximated by modified radial basis functions with time-dependent coefficients.An efficient backward substitution strategy that was previously provided for a single fractional ordinary differential equation is then used to solve the corresponding systems.The straightforward quasilinearization technique is applied to handle nonlinear issues.Numerical experiments demonstrate the suggested algorithm’s superior accuracy and efficiency.

THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS

This study introduces a pre-orthogonal adaptive Fourier decomposition(POAFD)to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre(generalized Poisson equation).As a first step,the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence,and,as a second step,makes use of the semigroup or the reproducing kernel property of each of the expanding entries.Experiments show the effectiveness and efficiency of the proposed series solutions.

A Collocation Technique via Pell-Lucas Polynomials to Solve Fractional Differential EquationModel for HIV/AIDS with Treatment Compartment

In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.

Pseudo S-Asymptotically(ω,c)-Periodic Solutions to Fractional Differential Equations of Sobolev Type

In this paper,we firstly recall some basic results on pseudo S-asymptotically(ω,c)-periodic functions and Sobolev type fractional differential equation.We secondly investigate some existence of pseudo S-asymptotically(ω,c)-periodic solutions for a semilinear fractional differential equations of Sobolev type.We finally present a simple example.

Legendre-Weighted Residual Methods for System of Fractional Order Differential Equations

The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.

Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations

The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L2-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.

Fractional differential equations of motion in terms of combined Riemann-Liouville derivatives

In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A combined Riemann-Liouville fractional derivative operator is defined, and a fractional Hamilton principle under this definition is established. The fractional Lagrange equations and the fractional Hamilton canonical equations are derived from the fractional Hamilton principle. A number of special cases are given, showing the universality of our conclusions. At the end of the paper, an example is given to illustrate the application of the results.

Existence of Positive Solutions to Boundary Value Problem for a Nonlinear Fractional Differential Equation

In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D~α_(o^+)is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel'skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.

EXISTENCE OF SOLUTION FOR BOUNDARY VALUE PROBLEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

This paper is concerned with the boundary value problem of a nonlinear fractional differential equation. By means of Schauder fixed-point theorem, an existence result of solution is obtained.

A NOTE ON GLOBAL WELL-POSEDNESS OF SOLUTIONS TO BOUSSINESQ EQUATIONS WITH FRACTIONAL DISSIPATION

The goal of this paper is to consider the global well-posedness to n-dimensional （n 〉 3） Boussinesq equations with fractional dissipation. More precisely, it is proved that there exists a unique global regular solution to the Boussinesq equations provided the real parameter α satisfies α≥1/2 ＋n/4.

Lagrange equations of nonholonomic systems with fractional derivatives

This paper obtains Lagrange equations of nonholonomic systems with fractional derivatives. First, the exchanging relationships between the isochronous variation and the fractional derivatives are derived. Secondly, based on these exchanging relationships, the Hamilton＇s principle is presented for non-conservative systems with fractional derivatives. Thirdly, Lagrange equations of the systems are obtained. Furthermore, the d＇Alembert-Lagrange principle with fractional derivatives is presented, and the Lagrange equations of nonholonomic systems with fractional derivatives are studied. An example is designed to illustrate these results.

CONVERGENCE ANALYSIS OF THE JACOBI SPECTRAL-COLLOCATION METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L^2-norm. The numerical examples are given to illustrate the theoretical results.

Bright and dark soliton solutions for some nonlinear fractional differential equations

In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified Benjamin-Bona- Mahoney （mBBM） equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions： amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional deriva- tives are described in the modified Riemann-Liouville sense.

Exact solutions of nonlinear fractional differential equations by (G'/G)-expansion method

In this paper, we use the fractional complex transform and the （G＇/G）-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie＇s modified Riemann-Liouville derivative into its ordinary differential equation. It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations.