Numerical approach to differential equations of fractional order

In this paper, the variational iteration method and the Adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper presents a numerical comparison between the two methods for solving systems of fractional differential equations. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order.     

Solving systems of fractional differential equations using differential transform method

This paper presents approximate analytical solutions for systems of fractional differential equations using the differential transform method. The fractional derivatives are described in the Caputo sense. The application of differential transform method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of systems of fractional differential equations. The solutions of our model equations are calculated in the form of convergent series with easily computable components. Some examples are solved as illustrations, using symbolic computation. The numerical results show that the approach is easy to implement and accurate when applied to systems of fractional differential equations. The method introduces a promising tool for solving many linear and nonlinear fractional differential equations.     

Short memory principle and a predictor–corrector approach for fractional differential equations

Fractional differential equations are increasingly used to model problems in acoustics and thermal systems, rheology and modelling of materials and mechanical systems, signal processing and systems identification, control and robotics, and other areas of application. This paper further analyses the underlying structure of fractional differential equations. From a new point of view, we apprehend the short memory principle of fractional calculus and farther apply a Adams-type predictor–corrector approach for the numerical solution of fractional differential equation. And the detailed error analysis is presented. Combining the short memory principle and the predictor–corrector approach, we gain a good numerical approximation of the true solution of fractional differential equation at reasonable computational cost. A numerical example is provided and compared with the exact analytical solution for illustrating the effectiveness of the short memory principle.     

Analytical solution of the linear fractional differential equation by Adomian decomposition method

In this paper, we consider the n-term linear fractional-order differential equation with constant coefficients and obtain the solution of this kind of fractional differential equations by Adomian decomposition method. With the equivalent transmutation, we show that the solution by Adomian decomposition method is the same as the solution by the Green's function. Finally, we illustrate our result with some examples.     

Explicit methods for fractional differential equations and their stability properties

The use of explicit methods in the numerical treatment of differential equations of fractional order is an area not yet widely investigated. In this paper stability properties of some multistep methods of explicit type are investigated and new methods with larger intervals of stability are proposed. Some numerical experiments are presented in order to validate theoretical results.     

On some explicit Adams multistep methods for fractional differential equations

In this paper we present a family of explicit formulas for the numerical solution of differential equations of fractional order. The proposed methods are obtained by modifying, in a suitable way, Fractional-Adams–Moulton methods and they represent a way for extending classical Adams–Bashforth multistep methods to the fractional case. The attention is hence focused on the investigation of stability properties. Intervals of stability for k$k$-step methods, k=1,…,5$k=1,\dots ,5$, are computed and plots of stability regions in the complex plane are presented.     

Numerical methods for multi-term fractional (arbitrary) orders differential equations

Our main concern here is to give a numerical scheme to solve a nonlinear multi-term fractional (arbitrary) orders differential equation.     

On accurate product integration rules for linear fractional differential equations

This paper addresses the numerical solution of linear fractional differential equations with a forcing term. Competitive and highly accurate Product Integration rules are derived by starting from an equivalent formulation in terms of a Volterra integral equation with a generalized Mittag-Leffler function in the kernel. The error analysis is reported and aspects related to the computational complexity are treated. Numerical tests confirming the theoretical findings are presented.     

A nodal spline interpolant for the Gregory rule of even order

Summary The Gregory rule is a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order. In the literature, the methods of constructing the Gregory rule have, in contrast to Newton-Cotes quadrature,not been based on the integration of an interpolant. In this paper, after first characterizing an even-order Gregory interpolant by means of a generalized Lagrange interpolation operator, we proceed to explicitly construct such an interpolant by employing results from nodal spline interpolation, as established in recent work by the author and C.H. Rohwer. Nonoptimal order error estimates for the Gregory rule of even order are then easily obtained.     

Numerical methods for nonlinear partial differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order.     

A method for solving differential equations of fractional order

In this paper, we consider Caputo type fractional differential equations of order 0<α<1$0<\alpha <1$ with initial condition x(0)=_x0$x\left(0\right)=x_0$. We introduce a technique to find the exact solutions of fractional differential equations by using the solutions of integer order differential equations. Generalization of the technique to finite systems is also given. Finally, we give some examples to illustrate the applications of our results.     

Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients

A class of weighted finite difference methods (WFDMs) for solving a class of initial-boundary value problems of space fractional partial differential equations with variable coefficients is presented. Their stability and convergence properties are considered. It is proven that the WFDMs are unconditionally-stable for , and conditionally-stable for , here r is the weighting parameter subjected to 0≤r≤1. Some convergence results are given. These methods, problems and results generalize the corresponding methods, problems and results given in [7], [8] and [10]. Some numerical examples are provided to show the effectiveness of the methods with different weighting parameters.     

A Fourier method and an extrapolation technique for Stokes’ first problem for a heated generalized second grade fluid with fractional derivative

In this article, we consider Stokes’ first problem for a heated generalized second grade fluid with fractional derivative (SFP-HGSGF). Implicit and explicit numerical approximation schemes for the SFP-HGSGF are presented. The stability and convergence of the numerical schemes are discussed using a Fourier method. In addition, the solvability of the implicit numerical approximation scheme is also analyzed. A Richardson extrapolation technique for improving the order of convergence of the implicit scheme is proposed. Finally, a numerical test is given. The numerical results demonstrate the good performance of our theoretical analysis.     

Analysis of differential equations of fractional order

This paper provides a robust convergence checking method for nonlinear differential equations of fractional order with consideration of homotopy perturbation technique. The differential operators are taken in the Caputo sense. Some theorems to prove the existence and uniqueness of the series solutions are presented. Results show that the proposed theoretical analysis is accurate.     

Rooted tree analysis of the order conditions of row-type scheme for stochastic differential equations

Numerical schemes for initial value problems of stochastic differential equations (SDEs) are considered so as to derive the order conditions of ROW-type schemes in the weak sense. Rooted tree analysis, the well-known useful technique for the counterpart of the ordinary differential equation case, is extended to be applicable to the SDE case. In our analysis, the roots are bi-colored corresponding to the ordinary and stochastic differential terms, whereas the vertices have four kinds of label corresponding to the terms derived from the ROW-schemes. The analysis brings a transparent way for the weak order conditions of the scheme. An example is given for illustration.     

Banach空间中一类分数阶微分方程边值问题

研究Banach空间中一类非线性分数阶微分方程边值问题.构建此类方程的Green函数,利用非紧测度和相关的不动点定理,得到了此类方程的mild解存在的几个充分条件,所得结果改进和推广了一些已有的结论.     

The monotonic property and stability of solutions of fractional differential equations

In this paper we improve on the monotone property of Lemma 1.7.3 in Lakshmikantham et al. (2009) [5] for the case g(t,u)=λu with a nonnegative real number λ. We also investigate the Mittag-Leffler stability of solutions of fractional differential equations by using the fractional comparison principle.     

Numerical methods for the eigenvalue determination of second-order ordinary differential equations

An accurate method for the numerical solution of the eigenvalue problem of second-order ordinary differential equation using the shooting method is presented. The method has three steps. Firstly initial values for the eigenvalue and eigenfunction at both ends are obtained by using the discretized matrix eigenvalue method. Secondly the initial-value problem is solved using new, highly accurate formulas of the linear multistep method. Thirdly the eigenvalue is properly corrected at the matching point. The efficiency of the proposed methods is demonstrated by their applications to bound states for the one-dimensional harmonic oscillator, anharmonic oscillators, the Morse potential, and the modified Pöschl–Teller potential in quantum mechanics.     

Existence of positive solutions of nonlinear fractional delay differential equations

This paper investigates the existence and uniqueness of positive solutions for a class of nonlinear fractional delay differential equations. Using a nonlinear alternative of Leray-Schauder type, we show the existence of positive solutions for the equations in question.     

A one-step 7-stage Hermite-Birkhoff-Taylor ODE solver of order 11

A one-step 7-stage Hermite-Birkhoff-Taylor method of order 11, denoted by HBT(11)7, is constructed for solving nonstiff first-order initial value problems y^′=f(t,y), y(t₀)=y₀. The method adds the derivatives y^′ to y⁽⁶⁾, used in Taylor methods, to a 7-stage Runge-Kutta method of order 6. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution to order 11 leads to Taylor- and Runge-Kutta-type order conditions. These conditions are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The new method has a larger scaled interval of absolute stability than the Dormand-Prince DP87 and a larger unscaled interval of absolute stability than the Taylor method, T11, of order 11. HBT(11)7 is superior to DP87 and T11 in solving several problems often used to test higher-order ODE solvers on the basis of the number of steps, CPU time, and maximum global error. Numerical results show the benefit of adding high-order derivatives to Runge-Kutta methods.