a Nonstandard Finite Difference Scheme for Two-Sided Space-Fractional Partial Differential Equations
23 May 2012-International Journal of Bifurcation and Chaos (World Scientific Publishing Company)-Vol. 22, Iss: 04, pp 1250079
TL;DR: The Mickens nonstandard discretization method is applied to solve a class of initial-boundary value fractional partial differential equations with variable coefficients on a finite domain, and thereby increase the accuracy of the solutions.
Abstract: In this paper, we apply the Mickens nonstandard discretization method to solve a class of initial-boundary value fractional partial differential equations with variable coefficients on a finite domain, and thereby increase the accuracy of the solutions. We examine the case when a left-handed and a right-handed fractional spatial derivative may be present in the partial differential equation. Two numerical examples using this method are presented and compared successfully with the exact analytical solutions.
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TL;DR: In this article, a new unconditionally stable implicit difference method, derived from the weighted and shifted Grunwald formula, converges with the second-order accuracy in both time and space variables.
Abstract: In this paper we intend to establish fast numerical approaches to solve a class of initial-boundary problem of time-space fractional convection–diffusion equations. We present a new unconditionally stable implicit difference method, which is derived from the weighted and shifted Grunwald formula, and converges with the second-order accuracy in both time and space variables. Then, we show that the discretizations lead to Toeplitz-like systems of linear equations that can be efficiently solved by Krylov subspace solvers with suitable circulant preconditioners. Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from $${\mathcal {O}}(N^2)$$
to $${\mathcal {O}}(N)$$
and the computational complexity from $${\mathcal {O}}(N^3)$$
to $${\mathcal {O}}(N\log N)$$
in each iterative step, where N is the number of grid nodes. Extensive numerical examples are reported to support our theoretical findings and show the utility of these methods over traditional direct solvers of the implicit difference method, in terms of computational cost and memory requirements.
94 citations
TL;DR: In this article, an implicit difference method based on two-sided weighted shifted Grunwald formulae is proposed with a discussion of the stability and convergence in both time and space, and it converges with second order accuracy in both space and time.
Abstract: In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of time-space fractional convection-diffusion equations (TSFCDEs). To start with, an implicit difference method based on two-sided weighted shifted Grunwald formulae is proposed with a discussion of the stability and convergence. We construct an implicit difference scheme (IDS) and show that it converges with second order accuracy in both time and space. Then, we develop fast solution methods for handling the resulting system of linear equation with the Toeplitz matrix. The fast Krylov subspace solvers with suitable circulant preconditioners are designed to deal with the resulting Toeplitz linear systems. Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ and the computational complexity from $O(N^3)$ to $O(N\log N)$ in each iterative step, where $N$ is the number of grid nodes. Extensive numerical example runs show the utility of these methods over the traditional direct solvers of the implicit difference methods, in terms of computational cost and memory requirements.
76 citations
TL;DR: This paper reviews many recent developments and further applications of nonstandard finite difference (NSFD) methods encountered in the past decade and gives a detailed account on various definitions/notions of NSFD methods appeared in the literature in past two decades.
Abstract: In this paper, we review many recent developments and further applications of nonstandard finite difference (NSFD) methods encountered in the past decade. In particular, it is a follow up article of the one published in 2005 [K.C. Patidar, On the use of non-standard finite difference methods, J. Differ. Equ. Appl. 11 (2005), pp. 735–758]. It also includes those research contributions in this field that are very significant and published prior to the above article but were not included in the above paper simply because we did not have access to them when we wrote the above article. We also give a detailed account on various definitions/notions of NSFD methods appeared in the literature in past two decades. All contributions are listed chronologically except that in some instances we have grouped certain works to show connectivity in those fields. While categorizing these research contributions, we considered a number of different application areas. Moreover, due to space limitations, firstly, we have not i...
73 citations
TL;DR: In this paper, the operational matrices of the left Caputo fractional derivative, right Caputo derivative and Riemann-Liouville fractional integral for shifted Legendre polynomials are presented.
Abstract: In this paper we present the operational matrices of the left Caputo fractional derivative, right Caputo fractional derivative and Riemann–Liouville fractional integral for shifted Legendre polynom...
50 citations
Cites background from "a Nonstandard Finite Difference Sch..."
...Consider the following two-sided FPDE (Momani et al., 2012),...
[...]
...Consider the following two-sided FPDE (Momani et al., 2012), @uðx, tÞ @t ¼ @uðx, tÞ @x þ cþðxÞ @1:7uðx, tÞ @þx1:7 þ c ðxÞ @1:7uðx, tÞ @ x1:7 þ qðx, tÞ ð4:2Þ on a finite domain 0< x< 2, with the diffusion coefficient cþðxÞ ¼ ð1:3Þx1:7 c ðxÞ ¼ ð1:3Þð2 xÞ1:7 ðxÞ ¼ x the source function qðx, tÞ ¼ 1 299…...
[...]
...1) reduces to two-sided space fractional advection–dispersion equation (Momani et al., 2012; Pang and Sun, 2012; Tian et al., 2012; Wang et al., 2010)...
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...…two-sided space FDE (Meerschaert and Tadjeran, 2006) @uðx, tÞ @t ¼ cþðxÞ @ uðx, tÞ @þx þ c ðxÞ @ uðx, tÞ @ x þ qðx, tÞ ð3:8Þ If ¼ 1, Equation (3.1) reduces to two-sided space fractional advection–dispersion equation (Momani et al., 2012; Pang and Sun, 2012; Tian et al., 2012; Wang et al.,…...
[...]
TL;DR: The results show that the new system exhibits a rich variety of dynamical behaviors such as limit cycles, chaos and transient phenomena where fractional-order derivative represents a key parameter in determining system qualitative behavior.
Abstract: This paper presents an analytical framework to investigate the dynamical behavior of a new fractional-order hyperchaotic circuit system. A sufficient condition for existence, uniqueness and continuous dependence on initial conditions of the solution of the proposed system is derived. The local stability of all the system’s equilibrium points are discussed using fractional Routh–Hurwitz test. Then the analytical conditions for the existence of a pitchfork bifurcation in this system with fractional-order parameter less than 1/3 are provided. Conditions for the existence of Hopf bifurcation in this system are also investigated. The dynamics of discretized form of our fractional-order hyperchaotic system are explored. Chaos control is also achieved in discretized system using delay feedback control technique. The numerical simulation are presented to confirm our theoretical analysis via phase portraits, bifurcation diagrams and Lyapunov exponents. A text encryption algorithm is presented based on the proposed fractional-order system. The results show that the new system exhibits a rich variety of dynamical behaviors such as limit cycles, chaos and transient phenomena where fractional-order derivative represents a key parameter in determining system qualitative behavior.
48 citations
Cites methods from "a Nonstandard Finite Difference Sch..."
...However, as one of numerical schemes, the nonstandard finite-difference scheme is well known and has been applied to various problems in science for example [Mickens, 1994; Dimitrov & Kojouharov, 2006; Anguelov et al., 2009; Momani et al., 2012; Ongun et al., 2013]....
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121 citations