Numerical solutions of nonlinear fractional differential equations by alternative Legendre polynomials

Zhang, Meng; Yi, Mingxu; Huang, Jun; Song, Lei

doi:10.1016/j.amc.2018.04.072

Cited by 26 publications

(20 citation statements)

References 22 publications

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“…Several numerical techniques are available to find approximate solutions to differential equations of mathematical models of engineering problems . One of the techniques available is collocation technique.…”

Section: Introductionmentioning

confidence: 99%

“…Several numerical techniques are available to find approximate solutions to differential equations of mathematical models of engineering problems. [20][21][22] One of the techniques available is collocation technique. A collocation method involves satisfying a differential equation to some tolerance at a finite number of points, called collocation points.…”

Section: Introductionmentioning

confidence: 99%

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Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

Ramezani

2019

Math Methods in App Sciences

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In this work, we present numerical analysis for nonlinear multi‐term time fractional differential equation which involve Caputo‐type fractional derivatives for 0<αi<1,i=1,2,⋯, double-struckN. The proposed method is based on utilization of fractional B‐spline basics in collocation method. The scheme can be readily obtained efficient and quite accurate with less computational work numerical result. The proposal approach transform nonlinear multi‐term time fractional differential equation into a suitable linear system of algebraic equations which can be solved by a suitable numerical method. The numerical experiments will be verify to demonstrate the effectiveness of our method for solving one‐ and two‐dimensional multi‐term time fractional differential equation.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

Ramezani

2019

Math Methods in App Sciences

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“…In recent years, fractional calculus has been playing more and more important roles in physics and other fields. [9][10][11][12][13][14][15] In [16][17][18][19][20][21][22][23][24][25][26][27], many researchers have made great efforts to studying the theory and computation for fractional equations. Laskin derived the space fractional Schrödinger equations in [28,29].…”

Section: Introductionmentioning

confidence: 99%

A Fourier spectral method for the nonlinear coupled space fractional Klein‐Gordon‐Schrödinger equations

et al. 2019

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In this paper, we give an efficient numerical method for the nonlinear coupled space fractional Klein-Gordon-Schrödinger (NCSFKGS) equations, based on the Crank-Nicolson method, the central difference method and the Fourier spectral method. As far as we know, no one has studied the Equations (5)-(6) in our paper, these equations are different from those in [39] which only considers space fractional Schrödinger equation while the Klein-Gordon equation is classical, here, we consider the two equations which are both space fractional. In this paper, the Crank-Nicolson method and the central difference method are used to discretize the space fractional Schrödinger equation and the space fractional Klein-Gordon equation in time direction, respectively. The Fourier spectral method is used to discretize the NCSFKGS equations in space direction. This numerical method conserves the mass and energy in the discrete level. The convergence of the numerical method is of second order accuracy in time and spectral accuracy in space. Rigorous proofs are developed here for the conservation laws and the convergence of the numerical method. Numerical experiments are presented to confirm the theoretical results and they proved that our numerical method is very efficient (it only takes a few seconds to get the numerical solutions). K E Y W O R D Sconvergence analysis, fast Fourier transform, Fourier spectral method, mass and energy conservation laws, nonlinear coupled space fractional Klein-Gordon-Schrödinger equations

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“…Rahimkhani et al 23 presented generalized fractional-order Bernoulli wavelet functions to obtain the numerical solution of fractional-order pantograph differential equations in a large interval. Meng et al 20 presented numerical techniques for solving initial value problems of nonlinear fractional differential equations based on alternative Legendre polynomials, and carried out the error analysis of the obtained method. Jhinga and Daftardar-Gejji 11 constructed a new finite difference predictorcorrector method to solve nonlinear fractional differential equations along with its error and stability analysis, and this method can be extended for systems of fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

A superlinear convergence scheme for nonlinear fractional differential equations and its fast implement

Gong

Zhang

Guo

et al. 2019

Int. J. Model. Simul. Sci. Comput.

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In this paper, we first construct an efficient scheme for nonlinear Caputo fractional differential equations with the initial value and the fractional degree [Formula: see text]. Then, the unconditional stability and the superlinear convergence with the order [Formula: see text] of the proposed scheme are strictly proved and discussed. Due to the nonlocal property of fractional operators, the new scheme is time-consuming for long-time simulations. Thus, a fast implement of the proposed scheme is presented based on the sum-of-exponentials (SOE) approximation for the kernel [Formula: see text] on the interval [Formula: see text] in the Riemann–Liouville integral, where [Formula: see text] is the stepsize. Some numerical experiments are provided to support the theoretical results of the new scheme and demonstrate the computational performance of its fast implement.

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Numerical solutions of nonlinear fractional differential equations by alternative Legendre polynomials

Cited by 26 publications

References 22 publications

Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

A Fourier spectral method for the nonlinear coupled space fractional Klein‐Gordon‐Schrödinger equations

A superlinear convergence scheme for nonlinear fractional differential equations and its fast implement

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