A Numerical Method to Solve Higher-Order Fractional Differential Equations

Almeida, Ricardo; Bastos, Nuno R. O.

doi:10.1007/s00009-015-0550-2

Cited by 12 publications

(11 citation statements)

References 24 publications

(17 reference statements)

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“…There are several different numerical approaches and methods to solve fractional differential equations [2,10,12]. Different discretizations of fractional derivatives are possible, but many of them do not preserve the fundamental properties of the systems, such as stability [15,29].…”

Section: Direct Methodsmentioning

confidence: 99%

A survey on fractional variational calculus

Almeida¹,

Torres²

2019

Basic Theory

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Main results and techniques of the fractional calculus of variations are surveyed. We consider variational problems containing Caputo derivatives and study them using both indirect and direct methods. In particular, we provide necessary optimality conditions of Euler-Lagrange type for the fundamental, higher-order, and isoperimetric problems, and compute approximated solutions based on truncated Grünwald-Letnikov approximations of Caputo derivatives.

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Section: Direct Methodsmentioning

confidence: 99%

A survey on fractional variational calculus

Almeida¹,

Torres²

2019

Basic Theory

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“…Property For u ( x ) ∈ C m [0, 1], then 36

J^{α} D^{α} u (x) = u (x) - \sum_{k = 0}^{m - 1} \frac{1}{(m - k - 1)!} u^{(m - k - 1)} (0) x^{m - k - 1},

and if

u^{false(i false)} false(0 false) = 0, 0 \leq i \leq m - 1

, we have J α D α u ( x ) = u ( x ).…”

Section: Fundamental Conceptsmentioning

confidence: 99%

“…Example Consider the problem 36 :

\{\begin{array}{ll} D^{2.5} u + u = \frac{6!}{normalΓ false(4.5 false)} x^{3.5} + x^{6}, x \in false(0,1 false), \\ u false(0 false) = 0, u^{'} false(0 false) = 0, u^{''} false(0 false) = 0 . \end{array}

The exact solution is u ( x ) = x 6 . The results are shown in Table 1 and in Figures 1 and 2.…”

Section: Numerical Examplesmentioning

confidence: 99%

A new algorithm for fractional differential equation based on fractional order reproducing kernel space

Zhang

Yang

2020

Math Methods in App Sciences

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This paper develops an effective and new method to solve a class of fractional differential equations. The method is based on a fractional order reproducing kernel space. First, depending on some theories, a fractional order reproducing kernel space W [0, 1] is constructed. The fractional order reproducing kernel space is a very suitable space to solve a class of fractional differential equations. Then, we calculate the reproducing kernel R y (x) of the space W [0, 1] skilfully in §3. And convergence order and time complexity of this algorithm are discussed. We prove that the approximate solution u n of (1.1) converges to its exact solution u is not less than the second order. The time complexity of the algorithm is equal to the polynomial time of the third degree. Finally, three experiments support the algorithm strongly from the aspect of theory and technique. KEYWORDS complexity analysis, convergence order, fractional reproducing kernel space, initial value problem of fractional equations MSC CLASSIFICATION 34A45; 65L05; 65L20 1 INTRODUCTION Fractional differential equations(FDEs) have important applications in signal processing, physics, controller design, electric circuits, and so forth. Fractional order modelling minimizes the inaccuracy that arises from the ignorance of significant real parameters. It permits a greater degree of freedom in the model compared to an integer-order system. Thus, in the last two decades, the theory and numerical analysis of FDEs are receiving more and more attention. Many schemes such as fractional linear multistep method, 1 operational matrix method, 2,3 homotype analysis method, 4,5 moving least square reproducing kernel method, 6 predictor-corrector method, 7 dual reciprocity boundary integral equation technique, 8 finite difference method, 9 Adomian decomposition method, 10 extrapolation method, 11 Chebyshev wavelets method, 12 modified fractional Euler method, 13 variational iteration method, 14 and Laplace transform 15 have been developed to determine the numerical solutions of several classes of fractional ordinary differential equations and partial differential equations (PDEs). Recently, Kumar et al 16 used Bernstein wavelet and Euler methods to handle nonlinear fractional predator-prey biological model. In 2020, Kumar et al 17 utilized residual power series and perturbation methods to handle Fokker-Plank equation. Veeresha et al 18 used the q-HATM to acquire the solution of FGNLS equation. Kumar et al 19 applied new amagalation of homotopy analysis method, Laplace transform method, and Adomian polynomial's for fractional discreate KdV equations. In addition, Kumar et al 20 employed homotopy perturbation transform method to obtain approximate analytical solutions of the time-fractional PDEs. Odibat et al 21 analyzed systems of nonlinear FDEs by using homotopy asymptotic method (HAM). Kumar et al 22 suggested two new fractional derivatives, namely, Yang-Gao-Tenreiro

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“…There are many practical implementations of fractional differential equations in the field of science and engineering [5–8]. Fractional derivatives are used to model various physical and engineering systems such as viscoelastic systems, electrode‐electrolyte polarization and dielectric polarization systems [9–11].…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of fractional partial differential equations via Haar wavelet

Zada

Aziz

2020

Numerical Methods Partial

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Haar wavelet collocation method is applied for the numerical solution of fractional partial differential equations. The proposed method is first applied to one‐dimensional fractional partial differential equations and then it is extended to higher‐dimensional fractional partial differential equations as well. Both time‐fractional as well as space–time‐fractional partial differential equations are considered. The fractional order derivatives involved are evaluated using the Caputo definition. The proposed method is semi‐analytic as it involves exact integration of Caputo derivative. The proposed method is widely applicable and robust. The method is tested upon several test problems. The results are computed and presented in the form of maximum absolute errors. The numerical tests confirm the accuracy, efficiency and simple applicability of the proposed method.

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A Numerical Method to Solve Higher-Order Fractional Differential Equations

Cited by 12 publications

References 24 publications

A survey on fractional variational calculus

A survey on fractional variational calculus

A new algorithm for fractional differential equation based on fractional order reproducing kernel space

Numerical solution of fractional partial differential equations via Haar wavelet

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