Difference Scheme For Fractional Equation Research Articles

A High Order Scheme for Fractional Differential Equations with the Caputo-Hadamard Derivative

A High Order Scheme for Fractional Differential Equations with the Caputo-Hadamard Derivative

PMNN: Physical model-driven neural network for solving time-fractional differential equations

In this paper, an innovative Physical Model-driven Neural Network (PMNN) method is proposed to solve time-fractional differential equations. It establishes a temporal iteration scheme based on physical model-driven neural networks which effectively combines deep neural networks (DNNs) with interpolation approximation of fractional derivatives. Specifically, once the fractional differential operator is discretized, DNNs are employed as a bridge to integrate interpolation approximation techniques with differential equations. On the basis of this integration, we construct a neural-based iteration scheme. Subsequently, by training DNNs to learn this temporal iteration scheme, approximate solutions to the differential equations can be obtained. The proposed method aims to preserve the intrinsic physical information within the equations as far as possible. It fully utilizes the powerful fitting capability of neural networks while maintaining the efficiency of the difference schemes for fractional differential equations. The experimental results show that the PMNN maintains precision comparable to traditional methods while exhibiting superior computational efficiency. This implies the potential of PMNN in addressing large-scale problems. Moreover, when considering both error and convergence rate, PMNN consistently outperforms fPINN. Additionally, the performance of PMNN on L2−1σ surpasses that on L1 in an overall comparison. The data and code can be found at https://github.com/DouMiao1226/PMNN.

Piecewise power law approximation of the interlayer relaxation curve for the long term viscoelastic fractional modeling of laminated glass

Laminated glass is composed of glass layers coupled by thin polymeric films which, being highly viscoelastic, influence the bending response over time. Here, an approximation of the polymer relaxation curve is proposed by means of continuously-connected power law branches. In the bi-logarithmic stress-time plane, this corresponds to a piecewise trilinear path, which well reproduces the experimental tests on most commercial polymers. A power law is naturally suited to be modeled via fractional calculus, as the Boltzmann convolution integral coincides with the definition of fractional derivative. However, the piecewise characterization through multiple power laws requires to properly extend the Grünwald–Letnikov integration scheme for fractional differential equations. For the paradigmatic case of a simply supported three-layer laminated beam, numerical results show an excellent correspondence with experiments under long duration loads. The comparison with the classic Prony series approach highlights, on the one hand, how calibration of material parameters is simplified; on the other hand, how the fractional approach is computationally more efficient and stable, even when the integration time step is wide enough to follow long term phenomena.

A novel numerical scheme for fractional differential equations using extreme learning machine

In this paper, we propose a neural network-based approach with an Extreme Learning Machine (ELM) for solving fractional differential equations. The solution procedure for the linear and nonlinear fractional differential equations has been derived. Also the convergence and stability of the proposed method is provided. Then we examine the numerical solution of several fractional-order ordinary and partial differential equations. As a last example the Burgers equation without an explicit exact solution. The effect of changing the number of neurons on the accuracy of the solution is obtained graphically.

Central part interpolation schemes for fractional differential equations

We consider a two-point boundary value problem for fractional integro-differential equations with weakly singular kernels. Using an integral equation reformulation of the underlying problem, a collocation method based on the central part interpolation by continuous piecewise polynomials on the uniform grid is constructed and analysed. Optimal convergence order of the proposed method is established and confirmed by numerical experiments.

A New Fifth-Order Finite Difference Compact Reconstruction Unequal-Sized WENO Scheme for Fractional Differential Equations

This paper designs a new finite difference compact reconstruction unequal-sized weighted essentially nonoscillatory scheme (CRUS-WENO) for solving fractional differential equations containing the fractional Laplacian operator. This new CRUS-WENO scheme uses stencils of different sizes to achieve fifth-order accuracy in smooth regions and maintain nonoscillatory properties near discontinuities. The fractional Laplacian operator of order β(0<β<1) is split into the integral part and the first derivative term. Using the Gauss–Jacobi quadrature method to solve the integral part of the fractional Laplacian operators, a new finite difference CRUS-WENO scheme is presented to discretize the first derivative term of the fractional equation. This new CRUS-WENO scheme has the advantages of a narrower large stencil and high spectral resolution. In addition, the linear weights of the new CRUS-WENO scheme can be any positive numbers whose sum is one, which greatly reduces the calculation cost. Some numerical examples are given to show the effectiveness and feasibility of this new CRUS-WENO scheme in solving fractional equations containing the fractional Laplacian operator.

Mittag-Leffler Euler [formula omitted]-differences for Caputo fractional-order systems

Exponential Euler differences have got rapid development recently for integer-order differential equations. But there are few papers focusing on this difference to fractional differential equations. This paper establishes the basic structure of Mittag-Leffler implicit Euler scheme for fractional differential equations by using the constant variation methods in fractional calculus. More critically, since the acquired difference forms belong to the scope of implicit Euler differences and then the fractional PECE algorithms are proposed to solve these implicit differences in numerical calculations effectively. Besides, a novel nonlocal difference operator is proposed and some relative properties are presented. In the end, the existence and boundedness of a unique globally exponentially stable solution of some nonlocal difference system with short-term memory are investigated. Some calculative examples and numerical simulations are given to illustrate the primary research findings.

On high order numerical schemes for fractional differential equations by block-by-block approach

The exact solutions to nonlinear fractional problems usually have initial singularity. Taking the singularity into account, the change of variable and the block-by-block approach are introduced to propose a novel high-order scheme. It is proved that our proposed scheme can be of order 3+α under the non-smooth solutions. Numerical examples are shown to validate our theoretical results.

Matlab Code for Lyapunov Exponents of Fractional-Order Systems, Part II: The Noncommensurate Case

In this paper, the Benettin–Wolf algorithm for determining all Lyapunov exponents of noncommensurate fractional-order systems modeled by Caputo’s derivative and the corresponding Matlab code are presented. The paper continues the work started in [ Danca & Kuznetsov, 2018 ], where the Matlab code of commensurate fractional-order systems is given. To integrate the extended systems, the Adams–Bashforth–Moulton scheme for fractional differential equations is utilized. Like the Matlab program for commensurate-order systems, the program presented in this paper prints and plots all Lyapunov exponents as function of time. The program can be simply adapted to plot the evolution of the Lyapunov exponents as a function of orders, or a function of a bifurcation parameter. Special attention is paid to the periodicity of fractional-order systems and its influences. The case of noncommensurate Lorenz system is demonstrated.

A New Sixth-Order Finite Difference WENO Scheme for Fractional Differential Equations

In this paper, we propose a new sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme for solving the fractional differential equations which may contain non-smooth solutions at a later time, even if the initial solution is smooth enough. After splitting the Caputo fractional derivative of order $$\alpha $$ $$(1<\alpha \le 2)$$ into a weakly singular integral and a classical second derivative, the classical Gauss–Jacobi quadrature is used to solve the weakly singular integral and a new spatial WENO-type reconstruction methodology is proposed to approximate the second derivative. There are two advantages of the new WENO scheme: the first is that the linear weights can be any positive numbers on condition that their summation equals one, and the second is its simplicity in the engineering applications. The new WENO reconstruction is a convex combination of a quartic polynomial with two linear polynomials defined on three unequal-sized spatial stencils in a traditional WENO fashion. This new sixth-order WENO scheme uses smaller number of cell average information than that of the same order classical WENO schemes and could eliminate non-physical oscillations near strong discontinuities when solving the fractional differential equations. Some benchmark examples are given to demonstrate the efficiency, robustness, and good performance of this new finite difference WENO scheme.

A self-singularity-capturing scheme for fractional differential equations

We develop a two-stage computational framework for robust and accurate time-integration of multi-term linear/nonlinear fractional differential equations. In the first stage, we formulate a self-singularity-capturing scheme, given available/observable data for diminutive time, experimentally obtained or sampled from an approximate numerical solution utilizing a fine grid nearby the initial time. The fractional differential equation provides the necessary knowledge/insight on how the hidden singularity can bridge between the initial and the subsequent short-time solution data. In the second stage, we utilize the multi-singular behaviour of solution in a variety of numerical methods, without resorting to making any ad-hoc/uneducated guesses for the solution singularities. Particularly, we employed an implicit finite-difference method, where the captured singularities, in the first stage, are taken into account through some Lubich-like correction terms, leading to an accuracy of order . We show that this novel framework can control the error even in the presence of strong multi-singularities.

A Gauss–Jacobi Kernel Compression Scheme for Fractional Differential Equations

A scheme for approximating the kernel w of the fractional $$\alpha $$ -integral by a linear combination of exponentials is proposed and studied. The scheme is based on the application of a composite Gauss–Jacobi quadrature rule to an integral representation of w. This results in an approximation of w in an interval $$[\delta ,T]$$ , with $$0<\delta $$ , which converges rapidly in the number J of quadrature nodes associated with each interval of the composite rule. Using error analysis for Gauss–Jacobi quadratures for analytic functions, an estimate of the relative pointwise error is obtained. The estimate shows that the number of terms required for the approximation to satisfy a prescribed error tolerance is bounded for all $$\alpha \in (0,1)$$ , and that J is bounded for $$\alpha \in (0,1)$$ , $$T>0$$ , and $$\delta \in (0,T)$$ .

Numerical discretization for fractional differential equations with feedback control

Abstract In this paper, we introduce a numerical scheme for fractional differential equations with feedback control. Due to the possibility of dealing with a feedback control as a functional delay, we construct a numerical method based on Euler method accompanied with piecewise constant interpolation. The method is based on the idea of separating the current state and the prehistory function. The convergence of the method is stated and proved. Numerical experiments are given to clarify the good agreement between numerical and theoretical results.

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

The Order of Convergence of Difference Schemes for Fractional Equations

ABSTRACTIn this paper, we continue our research on convergence of difference schemes for fractional differential equations. Using implicit difference scheme and explicit difference scheme, we have a deal with the full discretization of the solutions of fractional differential equations in time variables and get the order of convergence.

High-Order Accurate Adaptive Kernel Compression Time-Stepping Schemes for Fractional Differential Equations

High-order adaptive methods for fractional differential equations are proposed. The methods rely on a kernel compression scheme for the approximation and localization of the history term. To avoid complications typical to multistep methods, we focus our study on 1-step methods and approximate the local part of the fractional integral by integral deferred correction to enable high order accuracy. We study the local truncation error of integral deferred correction schemes for Volterra equations and present numerical results obtained with both implicit and the explicit methods applied to different problems.

A Kernel Compression Scheme for Fractional Differential Equations

The nonlocal nature of the fractional integral makes the numerical treatment of fractional dierential equations expensive in terms of computational eort and memory requirements. In this paper we propose a method to reduce these costs while controlling the accuracy of the scheme. This is achieved by splitting the fractional integral of a function f into a local term and a history term. Observing that the history term is a convolution of the history of f and a regular kernel, we derive a multipole approximation to the Laplace transform of the kernel. This enables the history term to be replaced by a linear combination of auxiliary variables dened as solutions to standard ordinary dierential equations. We derive a priori error estimates, uniform in f, and obtain estimates on the number of auxiliary variables required to satisfy an error tolerance. The resulting formulation is discretized to produce a time stepping method. The method is applied to some test cases to illustrate the performance of the scheme.

On the convergence of difference schemes for fractional differential equations with Robin boundary conditions

Locally one-dimensional difference schemes for partial differential equations with fractional order derivatives with respect to time and space in multidimensional domains are considered. Stability and convergence of locally one-dimensional schemes for this equation are proved.

Discrete and continuous fractional persistence problems – the positivity property and applications

In this article, we study the continuous and discrete fractional persistence problem which looks for the persistence of properties of a given classical (α=1) differential equation in the fractional case (here using fractional Caputo’s derivatives) and the numerical scheme which are associated (here with discrete Grünwald–Letnikov derivatives). Our main concerns are positivity, order preserving ,equilibrium points and stability of these points. We formulate explicit conditions under which a fractional system preserves positivity. We deduce also sufficient conditions to ensure order preserving. We deduce from these results a fractional persistence theorem which ensures that positivity, order preserving, equilibrium points and stability is preserved under a Caputo fractional embedding of a given differential equation. At the discrete level, the problem is more complicated. Following a strategy initiated by R. Mickens dealing with non local approximations, we define a non standard finite difference scheme for fractional differential equations based on discrete Grünwald–Letnikov derivatives, which preserves positivity unconditionally on the discretization increment. We deduce a discrete version of the fractional persistence theorem for what concerns positivity and equilibrium points. We then apply our results to study a fractional prey–predator model introduced by Javidi et al.

High-Order Accurate Local Schemes for Fractional Differential Equations

High-order methods inspired by the multi-step Adams methods are proposed for systems of fractional differential equations. The schemes are based on an expansion in a weighted $$L^2$$L2 space. To obtain the schemes this expansion is terminated after $$P+1$$P+1 terms. We study the local truncation error and its behavior with respect to the step-size h and P. Building on this analysis, we develop an error indicator based on the Milne device. Methods with fixed and variable step-size are tested numerically on a number of problems, including problems with known solutions, and a fractional version on the Van der Pol equation.