Orthogonal spline collocation method for the two-dimensional fractional sub-diffusion equation

Yang, Xuehua; Zhang, Haixiang; Xu, Dan

doi:10.1016/j.jcp.2013.09.016

Cited by 54 publications

(29 citation statements)

References 41 publications

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“…Now, it would be straight to get (81) from (87) by substituting n by U n . By (38), the stated result for α = 3/2 follows analogously. Thus, we complete the proof.…”

Section: Theorem 5 Suppose That the Exact Solution U Of The System (1mentioning

confidence: 61%

“…Therefore, there have been growing interests recently in developing numerical methods for solving fractional differential equations. Until now, various numerical methods are given for solving fractional differential equations such as finite difference methods [8,19,22,41], spectral methods [18,20], collocation methods [38], finite element methods [5,9], etc.…”

Section: Introductionmentioning

confidence: 99%

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Galerkin finite element method for nonlinear fractional Schrödinger equations

2016

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In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.

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“…Now, it would be straight to get (81) from (87) by substituting n by U n . By (38), the stated result for α = 3/2 follows analogously. Thus, we complete the proof.…”

Section: Theorem 5 Suppose That the Exact Solution U Of The System (1mentioning

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Galerkin finite element method for nonlinear fractional Schrödinger equations

2016

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“…The classical diffusion equation is an unavoidable candidate in various fields because it illustrates the behavior of the collective motion of micro-particles in a material resulting from the random movement of each micro-particle. The classical diffusion equation is represented as follows [36][37][38][39][40][59][60][61] :…”

Section: Fractional Model Of Diffusion Equation With New Fcmentioning

confidence: 99%

“…[27][28][29][30][31][32][33][34][35] Several numerical and analytical methods including nonpolynomial cubic spline methods, finite difference method, the Adomian decomposition method, homotopy perturbation method, variational iteration method, homotopy analysis method, and differential transform method have been developed for solving linear or nonlinear nonhomogeneous fractional partial differential equations. [36][37][38][39][40][41][42][43][44][45][46][47][48] Recent research focuses on the combination of various methods with the Laplace transform to solve the linear and nonlinear partial differential equations of arbitrary order. Among these, a few coupled methods with Laplace transform is given as Kumar et al 49 Two new FCs based on the nonsingular kernel with normalized sinc function and Robotnov fractional exponential function by Yang, Gao, Terneiro Machado, and Baleanu and Yang,Abdel-Aty, and Cattani are dicussed.…”

mentioning

confidence: 99%

A new Rabotnov fractional‐exponential function‐based fractional derivative for diffusion equation under external force

et al. 2020

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This work suggested a new generalized fractional derivative which is producing different kinds of singular and nonsingular fractional derivatives based on different types of kernels. Two new fractional derivatives, namely Yang‐Gao‐Tenreiro Machado‐Baleanu and Yang‐Abdel‐Aty‐Cattani based on the nonsingular kernels of normalized sinc function and Rabotnov fractional‐exponential function are discussed. Further, we presented some interesting and new properties of both proposed fractional derivatives with some integral transform. The coupling of homotopy perturbation and Laplace transform method is implemented to find the analytical solution of the new Yang‐Abdel‐Aty‐Cattani fractional diffusion equation which converges to the exact solution in term of Prabhaker function. The obtained results in this work are more accurate and proposed that the new Yang‐Abdel‐Aty‐Cattani fractional derivative is an efficient tool for finding the solutions of other nonlinear problems arising in science and engineering.

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“…Spectral methods [29][30][31][32][33][34][35][36][37] are widely used as numerical techniques for solving different kinds of problems. The main advantage of spectral numerical methods over finite element and finite difference [38][39][40][41][42] methods is its efficiency in achieving a high level of accuracy.…”

Section: Introductionmentioning

confidence: 99%

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

2015

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A spectral shifted Legendre Gauss-Lobatto collocation method is developed and analyzed to solve numerically one-dimensional two-sided space fractional Boussinesq (SFB) equation with non-classical boundary conditions. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials PL,n(x), x ∈ [0, L] is assumed, for the function and its space-fractional derivatives occurring in the two-sided SFB equation. The Legendre-Gauss-Lobatto quadrature rule is established to treat the non-local conservation conditions, and then the problem with its non-local conservation conditions is reduced to a system of ordinary differential equations (ODEs) in time. Thereby, the expansion coefficients are then determined by reducing the two-sided SFB with its boundary and initial conditions to a system of ODEs for these coefficients. This system may be solved numerically in a step-by-step manner by using implicit Runge-Kutta method of order four. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.Mathematics Subject Classification. 34A08, 65M70, 33C45.

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Orthogonal spline collocation method for the two-dimensional fractional sub-diffusion equation

Cited by 54 publications

References 41 publications

Galerkin finite element method for nonlinear fractional Schrödinger equations

Galerkin finite element method for nonlinear fractional Schrödinger equations

A new Rabotnov fractional‐exponential function‐based fractional derivative for diffusion equation under external force

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

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