A High-Order Predictor-Corrector Method for Solving Nonlinear Differential Equations of Fractional Order

Nguyen, Tan Tien; Jang, Bongsoo

doi:10.1515/fca-2017-0023

Cited by 27 publications

(33 citation statements)

References 23 publications

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“…Numerical methods include a difference scheme for time fractional heat equations based on the Crank-Nicholson method [11], a high-order predictor-corrector method [12] and an application of Picard's method [13].…”

Section: Introductionmentioning

confidence: 99%

A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

et al. 2020

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In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.

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Section: Introductionmentioning

confidence: 99%

A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

et al. 2020

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“…In this method, the explicit one-step Adams-Bashforth rule and the implicit one-step Adams-Moulton method are used as predictor and corrector, respectively. Other more recent works include [25,26,27,28]. In this paper, we propose a new predictor-corrector method where an improved version of the Atangana-Seda method of [15,16] is used as the predictor.…”

Section: Introductionmentioning

confidence: 99%

A Newton interpolation based predictor-corrector numerical method for fractional differential equations with an activator-inhibitor case study

Douaifia¹,

Bendoukha²,

Abdelmalek³

2020

Preprint

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This paper presents a new predictor-corrector numerical scheme suitable for fractional differential equations. An improved explicit Atangana-Seda formula is obtained by considering the neglected terms and used as the predictor stage of the proposed method. Numerical formulas are presented that approximate the classical first derivative as well as the Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. Simulation results are used to assess the approximation error of the new method for various differential equations. In addition, a case study is considered where the proposed scheme is used to obtained numerical solutions of the Gierer-Meinhardt activator-inhibitor model with the aim of assessing the system's dynamics.

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“…1. In case of that 0 < α 1 < 1, we transform the FNBVP (1) with a = 0 into a system of FIVPs using Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

“…2. To deal with FIVPs, we adopt high-order predictor-corrector methods (HPCMs) with linear interpolation and quadratic interpolation [1] into Volterra integral equations which are equivalent to FIVPs.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Numerical Approach for Solving Two-Point Fractional Order Nonlinear Boundary Value Problems with Robin Boundary Conditions

Lee

Jang

Kim³

2020

Preprint

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This article proposes new strategies for solving two-point Fractional order Nonlinear Boundary Value Problems (FN-BVPs) with Robin Boundary Conditions (RBCs). In the new numerical schemes, a two-point FNBVP is transformed into a system of Fractional order Initial Value Problems (FIVPs) with unknown Initial Conditions (ICs). To approximate ICs in the system of FIVPs, we develop nonlinear shooting methods based on Newton's method and Halley's method using the RBC at the right end point. To deal with FIVPs in a system, we mainly employ High-order Predictor-Corrector Methods (HPCMs) with linear interpolation and quadratic interpolation [1] into Volterra integral equations which are equivalent to FIVPs. The advantage of proposed schemes with HPCMs is that even though they are designed for solving two-point FNBVPs, they can handle both linear and nonlinear two-point Fractional order Boundary Value Problems (FBVPs) with RBCs and have uniform convergence rates of HPCMs, O(h 2 ) and O(h 3 ) for shooting techniques with Newton's method and Halley's method, respectively. A variety of numerical examples are demonstrated to confirm the effectiveness and performance of the proposed schemes. Also we compare the accuracy and performance of our schemes with another method.

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A High-Order Predictor-Corrector Method for Solving Nonlinear Differential Equations of Fractional Order

Cited by 27 publications

References 23 publications

A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

A Newton interpolation based predictor-corrector numerical method for fractional differential equations with an activator-inhibitor case study

An Efficient Numerical Approach for Solving Two-Point Fractional Order Nonlinear Boundary Value Problems with Robin Boundary Conditions

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