Solving a nonlinear fractional Schrödinger equation using cubic B-splines

We study the inhomogeneous nonlinear time-fractional Schrödinger equation for linear potential, where the order of fractional time derivative parameter α varies between 0 < α < 1. First, we begin from the original Schrödinger equation, and then by the Caputo fractional derivative method in natural units, we introduce the fractional time-derivative Schrödinger equation. Moreover, by applying a finite-difference formula to time discretization and cubic B-splines for the spatial variable, we approximate the inhomogeneous nonlinear time-fractional Schrödinger equation; the simplicity of implementation and less computational cost can be mentioned as the main advantages of this method. In addition, we prove the convergence of the method and compute the order of the mentioned equations by getting an upper bound and using some theorems. Finally, having solved some examples by using the cubic B-splines for the spatial variable, we show the plots of approximate and exact solutions with the noisy data in figures.


Introduction
The most famous equation in quantum mechanics that can explain the behavior of particles in Hilbert spaces is the Schrödinger equation. This equation is obtained in a different form in quantum physics, for example, in canonical quantization of the quantum mechanics, time evolution of the wave function leads to the Schrödinger equation. Also, Feynman [1] applied a path integral approach for a Gaussian probability distribution function to achieve the Schrödinger equation. The mathematical appearance of this equation is similar to a diffusion equation that can be derived by taking into account probability distributions.
Recently, the word fractional has been widely used in various sciences, especially, physics and mathematics. This concept for the first time was introduced by Mandelbrot [2] in many research items. Feynman's path integral first introduced the concept of fractal into quantum mechanics.
The fractional derivative Schrödinger equation was generalized by Laskin [3][4][5][6][7] two decades ago. It is an extension of the Feynman path integral formalism. This prominent formalism of quantum mechanics is a basis for the fractional quantum mechanics. As we know, the Schrödinger equation has two parts: the first part contains the first-order time derivative, and the second part has the second-order space derivative. Laskin [3][4][5][6][7] used the time-fractional derivative Schrödinger equation and showed that the connection between the statistical form of the quantum path was of the same nature as basic equations of quantum mechanics. In the Laskin scenario, the time-fractional derivative Schrödinger equation correlates with consideration of the non-Markovian evolutions. Also, he introduced the space-fractional derivative by the Levy distributions for all the possible paths. Furthermore, it has been shown that the fractional Hamiltonian and parity remain conserved, and by the time-fractional derivative the spectrum of energy levels of the hydrogen atom and the harmonic oscillator are computed. He, in his paper, investigated the energy levels for the fractional three-dimensional Coulomb potential Bohr atom; see [3][4][5][6][7]. The fractional Laplacian-Schrödinger equation for initial value problems was investigated by Hu and Kallianpur [8], in which the solutions are represented as probability. Also, the Green function in quantum scattering and the barrier influence were computed by Guo and Xu [9].
Some of the properties of time-fractional derivation nonlinear Schrödinger equation, such as fractal oscillator, were investigated by Naber [10]. This process was done by replacing the first order of time derivatives with a Caputo fractional derivative [11]; meanwhile, the second-order space derivatives remained unchanged. In [10], the order of the time-fractional Schrödinger derivative is 0 < α < 1, and with this order, the Schrödinger equation for a free particle in box and a finite potential well was solved. After that, the Schrödinger equation with both space and time-fractional derivatives was developed and computed by Wang and Xu [12] for free particle and an infinite rectangular potential well. The solution of Coulomb potential, linear potential, and δ-potential for the fractional Schrödinger equation were investigated in [13].
In this paper, we study the Schrödinger equation based on fractal time. This equation was discussed in [3-7, 10, 14-21]. In paper [14], according to Caputo's fractional derivatives, Planck mass and constant were represented by fractal equations whose dimensions are also fractal quantities. By this technique, they have shown that the time-dependent fractal Schrödinger differential equation for a particle in the potential field exactly matches the standard form of the equation. In [10], the Schrödinger fractal time wave function for free particle and wells was obtained by using the Mittag-Leffler function with the complex argument, and the eigenvalues corresponding to this function were also shown. In paper [3][4][5][6][7] also presented by Laskin, a Schrödinger time-independent fractal equation was considered and applications of this equation such as determining the shape of the Schrödinger wave function and its exact solution, the wave function and the eigenvalues for infinite potential wells, and in particular the values for a linear potential field, were obtained for 0 < α < 1. In all of these papers, the Schrödinger fractional time differential equation was investigated, and for different potentials, the wave function and specific differential equation values were specified. In this paper, we solve the general form of a heterogeneous fractal time differential equation corresponding to a heterogeneous fractal time Schrödinger equation using the B-spline method [22] and obtain numerical answers for such a function. To make this more concrete, we solve the fractal time-dependent Schrödinger equation for a linear potential field with this method and present the exact solutions.

The inhomogeneous nonlinear time-fractional Schrödinger equation
One way to do a fractal form of the time-fractional derivation of nonlinear Schrödinger equation is to apply the natural units = c = 1 for a wave function. Some people such as [10] used Planck's units for our models, both of which are the same. The definition of Plank's units [23] in terms of three theoretical physics constants G, , c, where G states for gravitational constant, is Planck's constant, and c denotes the speed of light, is as follows: where the last equation denotes the energy in Planck's scale. Indeed, in natural units, L pl = T pl = 1 M pl = 1 E pl . In our model, for the construction of the time-fractional derivative for a nonlinear Schrödinger equation in natural units, we start from the primary definition of this equation. The Schrödinger equation in one-dimensional space-time was introduced in [24] as follows: where V (x) indicates the potential of the system. In quantum mechanics literature, various potential functions are usually chosen to describe the models. In this work, to construct a nonlinear time-derivative fractional Schrödinger equation, we consider Eq. (2) in terms of natural units as follows: Now, we apply the Caputo fractional derivation method to introduce nonlinear Schrödinger equation in one-dimensional spaces as follows: In Eq. (4), D ν t denotes the Caputo fractional time derivative of order 0 < ν < 1. The main intention of this paper is to consider the abstract time-fractional evolution equation (4) on a potential well, where ∂ ν Ψ ∂t ν denotes the Caputo fractional derivative. Now, we consider the motion of a particle in a linear potential field [25] which is generated by an external field. The potential arising from this field reads as follows: where F is the force that affects the particle in the external field. Thus one can apply an example of inhomogeneous nonlinear time-fractional Schrödinger equation for a linear potential given by The Schrödinger equation is the basic equation in quantum mechanics. This equation is the generator of probability function producing the temporal and local evaluation of a mechanical system. In this article, we consider the time-fractional Schrödinger equation where δ has a real constant. Also, the initial, boundary, and overspecified conditions, respectively, read as follows: Therefore, we have By substitution of (9) into (7), the coupled real differential equations are obtained as follows: For any positive integer m, the Caputo fractional derivatives of order α read as follows: If h = 1 N is a step size in the x-axis and t = T M is a step size in the t-axis, then, for any point of (x i , t k ), we have By using the discretization of the time-fractional derivative term in [26], we have where u k i has a numerical approximation of u(x i , t k ) and

Implementation of the cubic B-spline functions
In this section, we introduce a set of nodes over [0, 1] such as We extend the sets as
Also, by using Definition 3.1, the value and derivatives of B i (x) at the nodes x i s are given by If c i and d i are unknown time-dependent quantities, which should be determined, then By discretizing the time derivative of Eq. (10) and using the finite difference, we have where t is the time step. By applying (15) and (16) at the point x = x m and using (14), we have Then we have System (17) consists of 2(N + 1) equations in 2(N + 3) unknown coefficients. In addition, by imposing the boundary conditions (8), we complete the following equations: where x s = x * , 1 ≤ s ≤ N -1. This system can be rewritten as where The matrices M 1 and O have the same size (N + 3) × (N + 3), where O is the zero matrix and By using the boundary conditions in (8), the initial vector C 0 can be obtained as where Since the matrix A is singular and ill-posed, the estimate of X 0 by (20) will be unstable. In this work, we adapt the Tikhonov regularization method (TRM) to solve matrix equations (18) and (20), which are given by By using the first-and second-order TRM, the matrices R (1) and R (2) are given by [27] where M = 2(N + 3). Therefore for equations (18) and (20), by using TRM, we have To compute the equation and to determine a suitable value of σ , we use the generalized cross-validation (GCV) scheme (see [28][29][30]).

Convergence analysis
In this section, we discuss the convergence of scheme (7). For this purpose, we introduce on the interval [a, b] as = {a = x 0 < x 1 < · · · < x n = b}.
Also, the spline function is interpolated, the values of the function u ∈ C 4 [a, b] named S . First we begin with a useful theorem that was proved by Stoer and Bulirsch [31].
Theorem 4.2 The collocation approximations U n (x) and V n (x) for the solutions u n (x) and v n (x) of problem (7) for any constant of μ > 0 satisfy the following error estimate: Proof LetÛ n (x, t) and letV n (x, t) be the computed splines for U n (x, t) and V n (x, t) defined in (15) such that Following (18), forÛ n andV n , we have wherê T , and X n -1 = g 1 (t n ), X n N+1 = q 1 (t n ), R n -1 = g 2 (t n ), R n N+1 = q 2 (t n ).
By subtracting (18) and (25), we have where and for every 0 ≤ m ≤ N , we have By using the Cauchy-Schwarz inequality, we have Then, after simplification and differentiation, we have By using Theorem 4.1, we have After simplification, we get So we can rewrite (28) as follows: Similarly, we have From (27), Eq. (29) is deduced By applying (26) and (21), we have Using relation (30) and taking the infinity norm, we find where We can easily see that Therefore, according to (23) and (32), we obtain (u n -U n , v n -V n ) ∞ ≤ λ 0 Lh 4 + 10M 1 h 2 + λ 0 Lh 4 + 10M 1 h 2 = h 2 2λ 0 Lh 2 + 20M 1 .

Figure 4
The plots of approximation and exact solutions of p 1 (t) and p 2 (t) for Example 5.2 with the noisy data with the initial and boundary condition as follows: The exact solution is Also, R(x) is obtained by the conditions of the problem.
Similar to the previous examples, the plots of approximation and exact solutions between of p 1 (t), p 2 (t) and u(x, t), v(x, t) for Example 5.2 with noisy data, are drawn in Fig. 4 and Fig. 5, respectively. Also, the plots of error of u(x, t) and v(x, t) for Example 5.2 with the noisy data is drawn in Fig. 6. In addition, the numerical solutions of Example 5.2 with α = 0.75, is shown in Table 2.

Conclusion
It is a fact that the time evolution of the system or the wave function in quantum mechanics is described by the Schrödinger equation. By applying a finite-difference formula to time discretization and cubic B-splines for the spatial variable, we got a numerical method for solving the Schrödinger equation, the simplification of implementation and lower computational cost can be mentioned as its main advantages. We proved the convergence of the method by getting an upper bound and using some theorems. We also computed the order of the mentioned equations in the convergence analysis section. Finally, in the illustrative examples section, we showed two examples with the approximate solutions, which were obtained by using the cubic B-splines for the spatial variable. The plots of approximate and exact solutions with the noisy data are presented in figures.