Stability and convergence of the space fractional variable-order Schrödinger equation
© Atangana and Cloot; licensee Springer. 2013
Received: 22 February 2013
Accepted: 20 March 2013
Published: 28 March 2013
The space fractional Schrödinger equation was further extended to the concept of space fractional variable-order derivative. The generalized equation is very difficult to handle analytically. We solved the generalized equation numerically via the Crank-Nicholson scheme. The stability and the convergence of the space fractional variable-order Schrödinger equation were presented in detail.
In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of some physical system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. Fractional-order derivatives can be dated back to the seventeenth century . It has primarily developed as a pure notional field of mathematics since its appearance. However, fractional-order differential systems have been proved to be useful in physics, engineering and even financial analysis in the last few decades . The fractional-order dynamical systems consist of viscoelastic systems , dielectric polarization , electrode-electrolyte polarization , electromagnetic waves , quantitative finance  and quantum evolution of complex systems . In particular, in fractional quantum evolution fields, there are many results obtained by some scholars (see in [8–17]). For instance, Laskin put up the space fractional quantum mechanics in 2000 [9–12], and the Schrödinger equation with space fractional derivative was also studied [13–16]. In the same way, the time fractional Schrödinger equation was also discussed in the works [15–17]. However, many researchers in the field of mathematics and physics paid attention to study physical problems described by the variable order derivative (see [18–22]). As a result of important differences between fractional order differential equations (FODE) and variable-order differential equations (VFODE), most characteristics or conclusions of the FODE systems can sometimes be extended to the case of the VFODE systems. Recently, many efforts have been devoted to the study of chaotic dynamics of variable-order differential systems [23–28]. Many current results about variable-order chaotic systems, however, are accomplished only by numerical simulations. In this work we further investigate the possibility of extension of the fractional Schrödinger equation to the concept of variable-order time fractional. The stability and the convergence of the new equation will be investigated in detail.
2 Problem formulation
For the readers that are not acquainted with the concept of the variable-order derivative, we start this section and we present the basic definition of this derivative.
2.1 Variable-order differential operator
The above derivative is called the Caputo variable-order differential operator; in addition, the derivative of the constant is zero.
2.2 Modification of the Schrödinger equation
The above equation will be called the space fractional variable-order Schrödinger equation. This modified equation cannot be solved analytically, therefore, in the following section we present, the discussion underpinning the numerical solution via the Crank-Nicholson scheme.
3 Numerical solution of the modified equation
Environmental phenomena, such as space fractional variable-order Schrödinger equation, are highly complex phenomena, which do not lend themselves readily to analysis of analytical models. The discussion presented in this section will therefore be devoted to the derivation of numerical solution to the space fractional variable-order Schrödinger equation (2.3).
Numerical methods yield approximate solutions to the governing equation through the discretization of space and time. Within the discretized problem domain, the variable internal properties, boundaries, and stresses of the system are approximated. Deterministic, distributed-parameter, numerical models can relax the rigid idealized conditions of analytical models or lumped-parameter models, and they can therefore be more realistic and flexible for simulating fields conditions. The finite difference schemes for constant-order time or space fractional diffusion equations have been extensively considered in the literature; see, for instance, the work done in [14–19]. For constant-order time fractional diffusion equations, an implicit difference approximation scheme was presented in . The weighted average finite difference method was initiated in . Podlubny proposed the matrix approach for fractional diffusion equations  and Hanert proposed a flexible numerical scheme for the discretization of the space-time fractional diffusion equation . Recently, Zhuang well thought out the numerical schemes for the variable-order (VO) space fractional advection-dispersion equation . Lin studied the explicit scheme for the VO nonlinear space fractional diffusion equation . Recently, Atangana and Botha presented the stability and the convergence of the generalized time-fractional variable-order groundwater flow equation .
3.1 Crank-Nicholson scheme
Before performing the numerical methods, we assume Equation (2.3) has a unique and sufficiently smooth solution. To establish the numerical schemes for the above equation, we let , , , , , , h is the step and τ is the time size, M and N are grid points.
It is important to point out that the quadrature formula (3.3) does not provide the values of the time fractional derivative at which are not required by the implicit finite difference and the Crank-Nicholson method schemes that follow.
where is the truncation term. Thus, according to Equation (3.4), the numerical method is consistent, first order correct in time and second order correct in space.
With the inclusion of the boundary conditions: , . It is important to note that, Equation (3.7) requires, at each time step, to solve a tri-diagonal system of linear equations where the right-hand side utilizes all the history of the computed solution up to that time.
Our next concern here is to show that the stability of the fractional numerical schemes can be analyzed very successfully with no trouble and powerfully with the distinguished Von Neumann method of non-fractional partial differential equations .
4 Stability analysis of space fractional variable-order Schrödinger equation
In this section, we will analyze the stability conditions of the Crank-Nicholson scheme for the space fractional variable-order Schrödinger equation.
To achieve this, we make use of the recurrence technique on the natural number k.
The condition is true for .
and the proof is completed.
5 Convergence analysis of space fractional variable-order Schrödinger equation
and the proof is completed.
An interested reader can find the solvability of the Crank-Nicholson scheme in the work done in . Therefore, the details of the proof will not be presented in this paper.
We paid attention to study a possible generalization of the Schrödinger equation to the concept of space fractional variable-order derivative. The Laplace operator in the Schrödinger equation was replaced by the fractional variable Laplace operator. Since the new equation cannot be solved analytically, it was solved numerically via the Crank-Nicholson technique.
We presented in detail the stability and the convergence of this problem.
Authors would like to thank the editor for his valuable time spared to access this manuscript and for his valuable comments toward the enhancement of this paper. Also, the authors would like to thank the anonymous referee for their valuable comments and suggestions toward the enhancement of this manuscript.
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