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An Implementation of Haar Wavelet Based Method for Numerical IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 6, NO. 1, JANUARY 2019 177 An Implementation of Haar Wavelet Based Method for Numerical Treatment of Time-fractional Schrodinger¨ and Coupled Schrodinger¨ Systems Najeeb Alam Khan and Tooba Hameed Abstract—The objective of this paper is to solve the time- equations, such as the variational iteration method [3], differ- fractional Schrodinger¨ and coupled Schrodinger¨ differential ential transform method [4], homotopy analysis method [5], equations (TFSE) with appropriate initial conditions by using the Jacobi spectral tau and collocation method [6]¡[8], Laplace Haar wavelet approximation. For the most part, this endeavor is made to enlarge the pertinence of the Haar wavelet method transform method [9], homotopy perturbation method [10], to solve a coupled system of time-fractional partial differential Adomian decomposition method [11], [12], high-order finite equations. As a general rule, piecewise constant approximation of element methods [13] and many others [14]¡[19]. The scope a function at different resolutions is presentational characteristic and distinct aspects of fractional calculus have been written of Haar wavelet method through which it converts the differential by many authors in [20]¡[22]. equation into the Sylvester equation that can be further simplified easily. Study of the TFSE is theoretical and experimental research In the past few years, there has been an extensive attraction and it also helps in the development of automation science, in employing the spectral method [23]¡[25] for numerically physics, and engineering as well. Illustratively, several test prob- solving the copious type of differential and integral equations. lems are discussed to draw an effective conclusion, supported The spectral methods have an exponential quota of conver- by the graphical and tabulated results of included examples, to gence and high level of efficiency. Spectral methods are to reveal the proficiency and adaptability of the method. express the approximate solution of the problem in term of Index Terms—Fractional calculus, Haar wavelets, operational a finite sum of certain basis functions and then selection of matrix, wavelets. coefficients in order to reduce the difference between the exact and approximate solutions as much as possible. The spectral collocation method is a distinct type of spectral methods, I. INTRODUCTION that is more relevant and extensively used to solve most of N recent decades, fractional calculus (calculus of integrals differential equations [26]. I and derivatives of any arbitrary real order) has attained For the reason of the distinctive attributes of wavelet theory appreciable fame and importance due to its manifest uses in in representing continuous functions in the form of discontin- apparently diverse and outspread fields of science. Certainly, uous functions [27], its applications as a mathematical tool is it provides potentially helpful tools for solving integral and widely expanding nowadays. Besides image processing and differential equations and many other problems of mathemati- signal decomposition it is also used to assess many other cal physics. The fractional differential equations have become mathematical problems, such as differential and integral equa- crucial research field essentially due to their immense range of tions. Wavelets comprise the incremental conception between utilization in engineering, fluid mechanics, physics, chemistry, two consecutive levels of resolution, called multi-resolution. biology, viscoelasticity, etc. Numerous mathematicians and The first component of multi-resolution analysis is vector physicists have been studying the properties of fractional cal- spaces. For each vector space, another vector space of higher culus [1], [2] and have established several methods for accurate resolution is found and this continues until the final image or analytical and numerical solutions of fractional differential signal is executed. The basis of each of these vector spaces acts as the scaling function for the wavelets. Each vector space Manuscript received November 30, 2015; revised February 9, 2016; ac- having an orthogonal component and a basis function is said cepted March 3, 2016. Recommended by Associate Editor Antonio Visioli. to be the wavelet [28]. (Corresponding author: Najeeb Alam Khan.) Citation: N. A. Khan and T. Hameed, “An implementation of Haar wavelet Up till now, a number of wavelet families have been based method for numerical treatment of time-fractional Schrodinger¨ and presented by different authors, but among all Haar wavelet coupled Schrodinger¨ systems,” IEEE/CAA J. Autom. Sinica, vol. 6, no. 1, are considered to be the easiest wavelets family. Haar wavelet pp. 177¡187, Jan. 2019. N. A. Khan and T. Hameed are with the Department of Mathematics, was introduced in 1910 by Hungarian mathematician Alfred University of Karachi, Karachi-75270, Pakistan (e-mail: [email protected]; Haar. These wavelets are obtained from Daubechies wavelets [email protected]). of order 1, which consist of piecewise constant functions on Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. the real axis that can take only three values, ¡1, 0 and 1. Here Digital Object Identifier 10.1109/JAS.2016.7510193 we are using collocation method, by increasing the level of 178 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 6, NO. 1, JANUARY 2019 ( ¡(®+1) t®¡º ; m > ® ¸ m ¡ 1 resolution, collocation points are also increasing and level of º ¡(®¡º+1) Dt f(t) = accuracy too. Haar wavelet collocation method is extensively 0; if ® 2 f0; 1; 2; : : : ; m ¡ 1g: used due to its constructive ability of being smooth, fast, con- (3) venient and being computationally attractive [29]. In addition, it has the competency to reduce the computations for solving In the following, some main computational properties and differential equations by converting them into some system relations of fractional integral and differential operators are of algebraic equations. The main advantages of the proposed defined as ® ¯ ®+¯ ¯ ® algorithm are, its simple application and no residual or product 1) It It f(t) = It f(t) = It It f(t) operational matrix is required. The method is well addressed @¯ 2) I®f(x; t) = I®¡¯f(x; t) in [22], [29]¡[32]. @t¯ t t The time-fractional Schrodinger¨ equation (T-FSE) differs n¡1 @® X tk @kf(x; t) j from the standard Schrodinger¨ equation. The first-order time 3) f(x; t) = f(x; t) ¡ t=0 @t® k! @tk derivative is replaced by a fractional derivative, it makes k=0 the problem overall in time. It describes, how the quantum nX¡1 k state (physical situation) of a quantum system changes with = f(x; t) + ³k(x)t (4) time, soliton dispersion, deep water waves, molecular orbital k=0 k theory and the potential energy of a hydrogen-like atom 1 @ f(x;t)jt=0 where ³k(x) = ¡ k! @tk . For more details see [1]. (fractional “Bohr atom”). The aim of this work is to explore the numerical solutions of the time-fractional Schrodinger¨ B. Haar Wavelets and Function Approximation equations by using Haar wavelet method. Due to the large Basis of Haar wavelets is obtained with a multi-resolution number of applications of the Schrodinger¨ equation in different of piecewise constant functions. Let the interval x 2 [0; 1) be aspects of quantum mechanics and engineering, many attempts divided into 2m subintervals of equal length, where m = 2j have been exercised on analytical and numerical methods to and J is maximal level of resolution. Next, two parameters are calculate the approximate solution of T-FSE. Some of them introduced, j = 0; 1; 2;:::;J and k = 0; 1; 2; : : : ; m¡1, such are studied [6], [10], [18], [33]¡[36], and enumerated here for that the wavelet number i satisfies the relation i = k + m + 1. better perception of the presented analysis. Also, the existence The ith Haar wavelet can be determined as and uniqueness of solutions of fractional Schrodinger¨ equation 8 >1; x 2 [# ;# ) have been proved by multiple authors [35], [37], [38]. < 1 2 hi(x) = ¡1; x 2 [#2;#3) (5) :> 0; elsewhere II. PRELIMINARIES where # = k=m, # = (k + 0:5)=m, # = (k + 1)=m. In this section, some notations and properties of fractional 1 2 3 For the case i = 1, corresponding scaling function can be calculus, the basis of Haar function approximation for partial defined as: differential equation and solution of Haar by multi-resolution ( analysis are given that will help us in exploring the main theme 1; x 2 [#1;#3) h1 = (6) of the paper. 0; elsewhere: Here, we consider the wavelet-collocation method, therefore A. Riemann-Liouville Differential and Integral Operator collocation points are generated by using, m l ¡ 0:5 Assume º > 0, m = dºe and f(x; t) 2 C ([0; 1] £ [0; 1]) xl = ; l = 1; 2; 3;:::; 2m: (7) then the partial Caputo fractional derivative of f(x; t) with 2m respect to t is defined as The Haar system forms an orthonormal basis for the Hilbert space f(t) 2 L2(0; 1). We may consider the inner product ( m º m¡º @ expansion of f(t) 2 L ([0; 1)) in Haar series [31] as @ It @tm f(x; t) 2 f(x; t) = m (1) @tº @ f(x; t) JX¡1 2Xj ¡1 @tm T f(t) ¼ hf; 'i'(t) + hf; hj;kihj;k(t) = C H(t) º where It is the Riemann-Liouville fractional integral, given j=0 i=0 as (8) Z t J º 1 º¡1 where C is 1 £ 2 coefficient vector and H(t) = [h0(t); It f(t) = (t ¡ ') f(')d' T ¡(º) 0 h1(t); : : : ; hm¡1(t)] . Also, a function of two variables can 0 be expanded by Haar wavelets [32] as It f(t) = f(t) (2) mX¡1 mX¡1 º @º T u(x; t) ¼ ui;jhi(x)hj(t) = H (x) £ U £ H(t) we use the notation Dt in replacement of @tº for the Caputo fractional derivative.
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