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An Implementation of Haar Wavelet Based Method for Numerical Treatment of Time-Fractional Schrodinger¨ and Coupled Schrodinger¨ Systems Najeeb Alam Khan, Tooba Hameed

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