The use of Jacobi polynomials in the numerical solution of coupled system of fractional differential equations

In this paper, we study shifted Jacobi polynomials and develop a simple but highly accurate scheme for the numerical solution of coupled system of fractional differential equations. We derive some operational matrices of integration and differentiation of fractional order. By the application of these matrices we provide a theoretical treatment to approximate the solutions of the corresponding system. We use Matlab to perform necessary operations. The applicability of the technique is shown with some examples and the results are displayed graphically.     

An alternative use of fuzzy transform with application to a class of delay differential equations

Fuzzy transforms (or F-transforms for short) are an approximation technique recently introduced. The main application is referred to image and data compression. There are really few works devoted to the use of F-transform for solving ordinary differential equations. In the present paper, an F-transform-based Picard-like numerical scheme is proposed in order to solve a class of delay differential equations. For linear cases, the proposed approach leads to a non-recursive approximate solution by means of operational matrices and vectors of known quantities. Numerical results show the good performance of the proposed method against known solutions.     

Numerical solution of nonlinear delay differential equations of fractional variable-order using a novel shifted Jacobi operational matrix

This paper presents the generalized nonlinear delay differential equations of fractional variable-order. In this article, a novel shifted Jacobi operational matrix technique is introduced for solving a class of multi-terms variable-order fractional delay differential equations via reducing the main problem to an algebraic system of equations that can be solved numerically. The suggested technique is successfully developed for the aforementioned problem. Comprehensive numerical experiments are presented to demonstrate the efficiency, generality, accuracy of proposed scheme and the flexibility of this method. The numerical results compared it with other existing methods such as fractional Adams method (FAM), new predictor–corrector method (NPCM), a new approach, Adams–Bashforth–Moulton algorithm and L1 predictor–corrector method (L1-PCM). Comparing the results of these methods as well as comparing the current method (NSJOM) with the exact solution, indicating the efficiency and validity of this method. Note that the procedure is easy to implement and this technique will be considered as a generalization of many numerical schemes. Furthermore, the error and its bound are estimated.

     

A new fractional Chebyshev FDM: an application for solving the fractional differential equations generated by optimisation problem

In this paper, we introduce a new numerical technique which we call fractional Chebyshev finite difference method. The algorithm is based on a combination of the useful properties of Chebyshev polynomial approximation and finite difference method. We implement this technique to solve numerically the non-linear programming problem which are governed by fractional differential equations (FDEs). The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the Caputo fractional derivatives. This operational matrix method can be regarded as a non-uniform finite difference scheme. The error bound for the fractional derivatives is introduced. The application of the method to the generated FDEs leads to algebraic systems which can be solved by an appropriate method. Two numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method. A comparison with the fourth-order Runge–Kutta method is given.     

Stability and non-standard finite difference method of the generalized Chua’s circuit

In this paper, we develop a framework to obtain approximate numerical solutions of the fractional-order Chua’s circuit with Memristor using a non-standard finite difference method. Chaotic response is obtained with fractional-order elements as well as integer-order elements. Stability analysis and the condition of oscillation for the integer-order system are discussed. In addition, the stability analyses for different fractional-order cases are investigated showing a great sensitivity to small order changes indicating the poles’ locations inside the physical s-plane. The Grünwald-Letnikov method is used to approximate the fractional derivatives. Numerical results are presented graphically and reveal that the non-standard finite difference scheme is an effective and convenient method to solve fractional-order chaotic systems, and to validate their stability.     

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.     

An efficient computational method for a class of singularly perturbed delay parabolic partial differential equation

In this paper, a new technique is constructed skillfully in order to solve a class of singularly perturbed delay parabolic partial differential equation. The outer and inner exact solutions of the linear problem can be expressed in the form of series and the outer and inner approximate solutions of the nonlinear problem are given by the iterative formula. Compared with known investigations, the advantages of our method are that the representation of the exact solution is obtained by using a new technique in a new reproducing kernel Hilbert space and the accuracy of numerical computation is higher. Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate that it is simple and effective.     

Numerical Solutions of Fractional Differential Equations by Using Fractional Taylor Basis

In this paper, a new numerical method for solving fractional differential equations (FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.      

Modelling anomalous diffusion using fractional Bloch–Torrey equations on approximate irregular domains

Diffusion-weighted imaging is an in vivo, non-invasive medical diagnosis technique that uses the Brownian motion of water molecules to generate contrast in the image and therefore reveals exquisite details about the complex structures and adjunctive information of its surrounding biological environment. Recent work highlights that the diffusion-induced magnetic resonance imaging signal loss deviates from the classic monoexponential decay. To investigate the underlying mechanism of this deviated signal decay, diffusion is re-examined through the Bloch–Torrey equation by using fractional calculus with respect to both time and space. In this study, we explore the influence of the complex geometrical structure on the diffusion process. An effective implicit alternating direction method implemented on approximate irregular domains is proposed to solve the two-dimensional time–space Riesz fractional partial differential equation with Dirichlet boundary conditions. This scheme is proved to be unconditionally stable and convergent. Numerical examples are given to support our analysis. We then applied the proposed numerical scheme with some decoupling techniques to investigate the magnetisation evolution governed by the time–space fractional Bloch–Torrey equations on irregular domains.     

A sixth-order finite difference WENO scheme for Hamilton–Jacobi equations

In this paper, a sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme is developed to approximate the viscosity solution of the Hamilton–Jacobi equations. This new WENO scheme has the same spatial nodes as the classical fifth-order WENO scheme proposed by Jiang and Peng [Weighted ENO schemes for Hamilton–Jacobi equations, SIAM. J. Sci. Comput. 21 (2000), pp. 2126–2143] but can be as high as sixth-order accurate in smooth region while keeping sharp discontinuous transitions with no spurious oscillations near discontinuities. Extensive numerical experiments in one- and two-dimensional cases are carried out to illustrate the capability of the proposed scheme.     

Finite element approximation of nonlinear transient magnetic problems involving periodic potential drop excitations

This paper deals with the computation of nonlinear 2D transient magnetic fields when the data concerning the electric current sources involve potential drop excitations. In the first part, a mathematical model is stated, which is solved by an implicit time discretization scheme combined with a finite element method for space approximation. The second part focuses on developing a numerical method to compute periodic solutions by determining a suitable initial current which avoids large simulations to reach the steady state. This numerical method leads to solve a nonlinear system of equations which requires to approximate several nonlinear and linear magnetostatic problems. The proposed methods are first validated with an axisymmetric example and sinusoidal source having an analytical solution. Then, we show the saving in computational effort that this methodology offers to approximate practical problems specially with pulse-width modulation (PWM) voltage supply.     

Numerical solution of functional differential equations: a Green's function-based iterative approach

In this article, a recently introduced iterative method based on Green's functions and fixed-point iteration schemes is presented for the approximate solutions of delay and functional differential equations. The approach is especially suited for handling boundary value problems (BVPs). The algorithm is illustrated through a number of examples that confirm the high accuracy and efficiency of the strategy. The results of the test examples show excellent agreement with exact solutions and outperforms other existing numerical iterative schemes.     

Wright分数阶时滞微分方程的离散化过程

引进了一种离散化方法对分数阶时滞微分方程进行离散化求解。首先考察Wright分数阶时滞微分方程;其次分析相应具有分段常数变元的Wright分数阶时滞微分方程,并应用离散化过程对模型进行数值求解;然后根据不动点理论讨论该合成动力系统不动点的稳定性;最后借助MATLAB对模型进行数值仿真,并结合Lyapunov指数、相图、时间序列图、分岔图探讨模型更多复杂的动力学现象。结果显示,提出方法成功对Wright分数阶时滞微分方程进行离散。     

The use of a Legendre pseudospectral viscosity technique to solve a class of nonlinear dynamic Hamilton–Jacobi equations

In this research, we study the problem of finding the approximate solution of a class of Hamilton–Jacobi equations, namely the Eikonal equation. We employ the Legendre pseudospectral viscosity method to solve this problem. This method basically consists of adding a spectral viscosity to the equation. This spectral viscosity, which is sufficiently small to retain the formal spectral accuracy is large enough to stabilize the numerical scheme. Several test problems are considered and the numerical results are given to show the efficiency of the proposed method.     

求解非线性方程组的一种并行算法

提出了一种在分布式环境下求解非线性方程组的并行算法,该算法将Newton迭代法中的Jacobi矩阵进行适当的分裂,使得Newton迭代法具有很好的并行性。并在理论上进行了收敛性分析。在HP rx2600集群上进行的数值实验结果表明并行效率达70%以上。     

Application of shifted Jacobi pseudospectral method for solving (in)finite-horizon min–max optimal control problems with uncertainty

The difficulty of solving the min–max optimal control problems (M-MOCPs) with uncertainty using generalised Euler–Lagrange equations is caused by the combination of split boundary conditions, nonlinear differential equations and the manner in which the final time is treated. In this investigation, the shifted Jacobi pseudospectral method (SJPM) as a numerical technique for solving two-point boundary value problems (TPBVPs) in M-MOCPs for several boundary states is proposed. At first, a novel framework of approximate solutions which satisfied the split boundary conditions automatically for various boundary states is presented. Then, by applying the generalised Euler–Lagrange equations and expanding the required approximate solutions as elements of shifted Jacobi polynomials, finding a solution of TPBVPs in nonlinear M-MOCPs with uncertainty is reduced to the solution of a system of algebraic equations. Moreover, the Jacobi polynomials are particularly useful for boundary value problems in unbounded domain, which allow us to solve infinite- as well as finite and free final time problems by domain truncation method. Some numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. A comparative study between the proposed method and other existing methods shows that the SJPM is simple and accurate.     

A bounded numerical solver for a fractional FitzHugh–Nagumo equation and its high-performance implementation

In this paper, we propose a high-performance implementation of a space-fractional FitzHugh–Nagumo model. Our implementation is based on a positivity- and boundedness-preserving finite-difference model to approximate the solutions of a Riesz space-fractional reaction-diffusion equation. The model generalizes the FitzHugh–Nagumo model. The stability and convergence of the difference scheme are thoroughly discussed. Moreover, we prove the existence and uniqueness of numerical solutions, positivity, boundedness and consistency of the model. The scheme is based on weighted and shifted Grünwald differences. The conjugate gradient method is used then to solve the sparse matrix system. The MPI and PETSc libraries are used for the computational implementation. We investigate the influence of some computer factors on the performance of our implementation and scalability. More precisely, we consider the number of cores, the size of the computation mesh and the orders of the fractional derivatives. Tests are evaluated on a ccNUMA architecture with two CPUs.

     

基于自适应动态规划的一类带有时滞的离散时间非线性系统的最优控制策略

针对一类状态和控制变量均带有时滞的非线性系统的带有二次性能指标函数最优控制问题, 本文提出了一种基于新的迭代自适应动态规划算法的最优控制方案. 通过引进时滞矩阵函数, 应用动态规划理论, 本文获得了最优控制的显式表达式, 然后通过自适应评判技术获得最优控制量. 本文给出了收敛性证明以保证性能指标函数收敛到最优. 为了实现所提出的算法, 本文采用神经网络近似性能指标函数、计算最优控制策略、求解时滞矩阵函数、以及给非线性系统建模. 最后本文给出了两个仿真例子说明所提出的最优策略的有效性.     

A new operational matrix for solving fractional-order differential equations

Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of a class of fractional differential equations. The fractional derivatives are described in the Caputo sense. Our main aim is to generalize the Legendre operational matrix to the fractional calculus. In this approach, a truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem. The method is applied to solve two types of fractional differential equations, linear and nonlinear. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.