Inter national J our nal of Electrical and Computer Engineering (IJECE) V ol. 9, No. 5, October 2019, pp. 3720 3731 ISSN: 2088-8708, DOI: 10.11591/ijece.v9i5.pp3720-3731 r 3720 Computational Sinc-scheme f or extracting analytical solution f or the model K uramoto-Si v ashinsk y equation Kamel Al-Khaled, Issam Ab u-Irwaq Department of Mathematics and Statistics, Jordan Uni v ersity of Science and T echnology , P .O.Box 3030, Irbid 22110, Jordan Article Inf o Article history: Recei v ed No v 30, 2018 Re vised Apr 9, 2019 Accepted Apr 9, 2019 K eyw ords: K uramoto-si v ashinsk y equation Sinc-g alerkin Sinc-collocation Fix ed-point iteration ABSTRA CT The present article is designed to supply tw o dif ferent numerical solutions for solv- ing K uramoto-S i v ashinsk y equation. W e ha v e made an attempt to de v elop a numeri- cal solution via the use of Sinc-Galerkin method for K uramoto-Si v ashinsk y equation, Sinc approximations to both deri v ati v es and indefinite inte grals reduce the solution to an e xplicit system of algebraic equations. The fix ed point theory is used to pro v e the con v er gence of the proposed methods. F or comparison purposes, a combination of a Cra nk-Nicolson formula in the tim e direction, with the Sinc-collocation in the space direction is presented, where the deri v ati v es in the space v ariabl e are replaced by the necessary matrices to produce a system of algebraic equations. In addition, we present numerical e xamples and comparisons to support the v alidity of these proposed methods. Copyright c 201x Insitute of Advanced Engineeering and Science . All rights r eserved. Corresponding A uthor: Kamel Al-Khaled, Jordan Uni v ersity of Science and T echnology , Irbid, P .O.Box 3030, Jordan. Phone: 00962795010519 Email: kamel@just.edu.jo 1. INTR ODUCTION The K uramoto-Si v ashinsk y equation (that we abbre viate as K-S), is a simple one-dimensional partial dif ferential equation that e xhibiting a particular comple x dynamical beha vior under some conditions. It arises as an amplitude equation in long-w a v e, weakly nonlinear stability in a great v ariety of applications. F or e xample, it arises in concentration w a v es in chemically reacting systems [1], in flame propag ation and reaction comb us- tion [2]. In its simplest form, the equation is gi v en by subject to the initial condition u t + uu x + u xx + u xxxx = 0 ; x 2 R ; t > 0 (1) u ( x; 0) = f ( x ) ; x 2 R (2) W e seek a real-v alued function u = u ( x; t ) , defined on R R + 0 satisfying (1) with f : R ! R is suf ficiently smooth function satisfying some decay conditions. The u xx term in (1) is responsible for an instability at lar ge scales, where is a positi v e constant; the dissipati v e u xxxx term pro vides damping at small scales, and the positi v e constant playing the role of viscosity; and the non-linear term uu x (which has the same form as that in the Bur gers or one-dimensional Na vier Stok es equations) stabilizes by transferring ener gy between lar ge and small scales. The K-S equation, where it models cellular instabilities, pattern formation, turb ulence phenomena and transition to chaos. A complete account of the numerical literature on these approx- imations are mentioned in [3, 4, 5, 6, 7, 8]. Mesh-free methods are the topic of recent research in man y areas J ournal homepage: http://iaescor e .com/journals/inde x.php/IJECE Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 r 3721 of computational science and approximation theory [9]. Ov er the past se v eral years mesh-free approximation methods ha v e found their w ay into man y dif ferent application areas ranging from engineering applications to the numeri cal solution of dif ferential equations. A meshless method does not require grid, and only mak es use of a set of scattered collocation points. In [10, 11, 12], the authors proposed a mesh-free collocation method and formulate a simple classical radial basis functions for the numerical solution of the KdV equation, coupled KdV equations and the K-S equation. 2. RELA TED W ORKS In recent years v ari ous methods ha v e been presented to find approximate solutions for K-S equation. In [13], the quintic B-spline collocation method is implemented to find numerical solution of the K uramoto- Si v ashinsk y equation. In [14] a numerical technique based on the finite dif ference and collocation methods is presented for the generalized K uramoto-Si v ashinsk y equation. Chebyshe v spectral collocation methods are used in [15] to find approximate solutions for the generalized K-S equation. In [16], authors analyzed and im- plemented a fully discrete schemes for periodic initial v alue problems for a general class of dispersi v ely modi- fied K uramoto-Si v ashinsk y equations. T ime discretizations are constructed using linearly implicit schemes and spectral methods are used for the spatial discretization. 3. RESEARCH METHOD In this paper , we approximate the solution of the K-S equation (1) subject to the initial condition (2) using Sinc-Galerkin method, which b uilds an approximate solution v alid on the entire spatial domain and on a small interv al in the time domain. Another benefit of the Sinc methodology is that the scheme presented automatically handles singularities occur at the boundaries with ease. F or more details about Sinc solutions of analytic problems with singularties, see [17]. The main idea is to replace deri v ati v es and inte grals by their discrete Sinc approximation. Also we solv e the K-S solution (1) in the re gion a x b; t 0 , by discretizing in time through the use of Crank-Nicolson scheme, and in space by Sinc-collocation method. The layout of the paper is as follo ws: In section 2 , we gi v e the rele v ant properties of Sinc function such as notations, definitions and some theorem that we need. In section 3 , we de v eloped the Sinc-Galerkin method for solving K-S equations, and a con v er gence proof is also gi v en. In section 4 , we disc u s s the mesh-free method together with the Sinc-collocation discretization of the K-S equation. Finally , numerical e xperiments are presented and some comparisons are made in section 5 . Some concluding remarks are gi v en in the final section. 4. PR OPOSED METHOD The goal of this section is to recall notations and definitions of the Sinc function that will be used in this paper . These are discussed in [3, 4], and mainly , we will recall section 2 of [6, 18, 19]. The Sinc function is defined on the whole real line R by sinc ( x ) = sin( x ) x ; x 2 R : (3) 4.1. Pr eliminaries Recall that a radial basis function is a function whose v alue depends only on the distance of its input to a central point. F or a series of nodes equally spaced h apart, the Sinc function can be written as a radial basis function: S ( j ; h )( k h ) = (0) j k = 8 < : 1 ; k = j 0 ; k 6 = j (4) Let ( 1) k j = 1 2 + k j , where k j = Z k j 0 sin( t ) t dt: W e define a matrix I ( 1) whose ( k ; j ) th entry is gi v en by ( 1) k j . If a function f ( x ) is defined on the real line, then for h > 0 , the series Computational Sinc-sc heme for e xtr acting analytical solution... (Kamel Al-Khaled) Evaluation Warning : The document was created with Spire.PDF for Python.
3722 r ISSN: 2088-8708 C ( f ; h )( x ) = 1 X j = 1 f ( j h )sinc x j h h ; j = 0 ; 1 ; ::: is called the Whittak er cardinal e xpansion, which has been e xtensi v ely studies in [4, 3]. In practice, we need to use a finite number of terms in the abo v e series, say j = N ; :::; N , where N is the number of sinc grid points. F or a restricted class of functions kno wn as the P aly-W einer class, which are entire functions, the sinc interpolation and quadrature formulae are e xact [4]. A less restricted class of functions that are analytic only on an infinite strip containing the real line, and that allo w specific gro wth restrictions has e xponentially decaying absolute errors in the sinc approximation. Definition 1 Let D d denote the infinite strip domain of width 2 d; d > 0 , given by D d = f w = u + iv : j v j < d = 2 g T o construct approximation on the interv al = (0 ; T 0 ) , which is our time interv al in this paper , we consider the conformal map ( t ) = ln( t T 0 t ) , the map carries the e ye-shaped re gion D = n z = x + iy : arg z T 0 z < d = 2 o : onto the infinite strip D d . F or the si nc method, the basis functions on the interv al at z 2 D are deri v ed from the composite translated sinc functions S j ( z ) = S ( j ; h ) ( z ) = s in c ( z ) j h h : The function z = 1 ( w ) = T 0 exp( w ) 1+exp( w ) is an in v erse mapping of the w = . W e define the range of 1 on the real line as = f 1 ( y ) 2 D : 1 < y < 1g = (0 ; T 0 ) : The sinc grid points z k 2 in D will be denoted by t k , because the y are real, and is gi v en by t k = 1 ( k h ) = T 0 exp( k h ) 1 + e x p( k h ) ; k = 0 ; 1 ; 2 ; ::: T o construct an approximation in the interv al ( 1 ; 1 ) , which is our space domain in this part, we replace by ( x ) = x . T o further e xplain of the sinc method, an important class of functions is denoted by L ( D ) . The properties of the functions in L ( D ) and detailed discussion are gi v en in [4]. W e recall the follo wing definition follo wed by tw o Theorems for our purpose. Definition 2 Let L ( D ) be the class of all analytic functions f in D , for whic h ther e is a number C 0 suc h that, for ( z ) = exp(( z )) , we have j f ( z ) j C 0 j ( z ) j [1 + j ( z ) j ] 2 ; 8 z 2 D : If x is on the arc , we obtain the follo wing theorem Theor em 1 Let f ( x ) 2 L ( D ) , a positive constant, then taking h x = p d= ( N x ) it follows that sup x 2 f ( n ) ( x ) N x X j = N x f ( x j ) S ( n ) j ( x ) C 1 N n +1 2 x exp( p d N x ) for n = 0 ; 1 ; :::; m with C 1 a constant independent of N x . Int J Elec & Comp Eng, V ol. 9, No. 5, October 2019 : 3720 – 3731 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 r 3723 In the ne xt Theorem, we shall gi v e a general formula for approximating the inte gral R a F ( u ) du; 2 . T o this end, we state the follo wing result, which we will use to approximate the obtained inte gral equation. Theor em 2 Let F ( t ) 0 ( t ) 2 L ( D ) , with 0 < 1 , ( 1) j k be defined as abo ve , N t be posi tive inte g er , and h t be selected as h t = p d= ( N t ) , then ther e e xists a positive constant C 2 independent of N t , suc h that Z t k a F ( t ) dt h t N t X j = N t ( 1) j k F ( t k ) 0 ( t k ) C 2 exp( p d N t ) The sinc method requires that the deri v ati v es of sinc functions be e v aluated at the nodes. T echnical calculations pro vide the follo wing results that will be useful in formulating the discrete system [4, 3], and these quantities are delineated by ( m ) j k = h m d m dx m [ S j ( x )] x = x k , where (0) j k = 8 < : 1 ; j = k 0 ; j 6 = k ; (2) j k = 8 > < > : 2 3 ; j = k 2( 1) k j ( k j ) 2 ; j 6 = k and, (4) j k = 8 > < > : 4 5 ; j = k 4( 1) k j [6 ( k j 1) 2 2 ] ( k j 1) 4 ; j 6 = k Then we define the m m T oeplitz matrices I ( q ) ; q = 0 ; 1 ; 2 ; 4 whose j k th entry is ( q ) j k ; q = 0 ; 1 ; 2 ; 4 . The matrix I (0) is the identity matrix. Also we define the diagonal matrix D ( g ( x )) = diag [ g ( x N ) ; :::; g ( x 0 ) ; :::; g ( x N )] T . 4.2. Implementation of the method The objecti v e of this section is to construct a solut ion to the K-S equation using the Sinc-Galerkin method. Inte grate equation (1) in section 2 with respect to t , we get the V olterra inte gral equation u ( x; t ) = Z t 0 u ( x; ) u x ( x; ) + u xx ( x; ) + u xxxx ( x; ) d + f ( x ) (5) with the assumption that the initial condition f ( x ) 2 L ( D ) . T o obtain direct discretization of equation (5), and since our domain is ( x; t ) 2 R (0 ; T 0 ) , the rele v ant maps are defined as follo ws. In the space direction, we choose ( x ) = x , which maps the infinite strip D d onto it self. In the time direction, we choose the map ( t ) = ln( t= ( T 0 t )) that carries the re gion D onto D d . Define the basis elements for ( 1 ; 1 ) ; (0 ; T 0 ) to be S ( m; h x ) ( x ) ; m = N x ; :::; N x , S ( k ; h t ) ( t ) ; k = N t ; :::; N t , res pecti v ely . The mesh sizes h x and h t represent the mesh sizes in the infinite strip D d for the uniform grid f ih x g ; 1 < i < 1 and f j h t g ; 1 < j < 1 . The Sinc grid points x i 2 ( 1 ; 1 ) in D d and t j 2 (0 ; T 0 ) in D are in v erse images of equispaced grid points, i.e., x i = 1 ( ih x ) = ih x and t j = 1 ( j h t ) = T 0 exp( j h t ) 1+exp( j h t ) . In equation (5), we carry out Sinc approximations of u x ( x; t ) ; u xx ( x; t ) and u xxxx ( x; t ) ; to proceed, we use Theorem 1, and the replacement of the deri v ati v es with respect to x by its approximation yields u x ( x; t ) 1 h x I (1) m x u ( x i ; t ) ; u xx ( x; t ) 1 h 2 x I (2) m x u ( x i ; t ) ; and u xxxx ( x; t ) 1 h 4 x I (4) m x u ( x i ; t ) : where m x = 2 N x + 1 . In equation (5), e v aluating all functions at the x nodes, and replacing the deri v ati v e by its appropriate approximation, we obtain the V olterra inte gral equation u ( t ) = Z t 0 u ( ) A 1 u ( ) + A 2 u ( ) + A 4 u ( ) d + f 0 (6) where A 1 = 1 h x I (1) m x ; A 2 = 1 h 2 x I (2) m x , and Computational Sinc-sc heme for e xtr acting analytical solution... (Kamel Al-Khaled) Evaluation Warning : The document was created with Spire.PDF for Python.
3724 r ISSN: 2088-8708 A 4 = 1 h 4 x I (4) m x (7) with u ( t ) = ( u N x ; :::; u N x ) T , where u i ( t ) = u ( x i ; t ) and f 0 = ( f ( z N x ) ; :::; f ( x N x ) : W e ne xt collocate with respect to the t v ariable via the use of Theorem 2, with the matri x B = h t I ( 1) m t D 1 0 ; m t = 2 N t + 1 and the nodes t j = 1 ( j h t ) for j = N t ; :::; N t . Define the matrix F 0 = [ f ( x i ; 0)] : Then the solution of equation (6) in matrix form is gi v en by the rectangular m x m t matrix U = [ u ij ] , U = U A 1 U + A 2 U + A 4 U B T + F 0 (8) where the notation denotes the Hadamard m atrix multiplication. Note that in our discretization, we are taking the time nodes as ro ws and the space nodes as columns, so the matrix U A 1 U + A 2 U + A 4 U forms the v ector nodes for the inte gral in (6). In equation (8) the v ector F 0 has the same dimensions as the v ector U , and e v ery column of F 0 consists of the same v ector f 0 . T o solv e the system in equation (8), one idea is to produce a sequence of iterations that con v er ges to the e xact solution of the K-S equation. Equation (8) can be written as U = G ( U ) + F 0 (9) where G ( U ) = U A 1 U + A 2 U + A 4 U B T : By s electing an initial approximation U 0 , we iterate the continuous map G repeatedly via the formula U ( n +1) = G ( U ( n ) ) + F 0 ; n = 0 ; 1 ; 2 ; ::: 4.3. V alidation of Sinc-Galerkin method Throughout this subsection, we use the notation U = [ u ( x i ; t j )] to denote the m x m t matrix of node v alues of the function u ( x; t ) , and so on for other functions. Set, K ( x; t ) = R t 0 M ( ) d , where M ( ) = u ( x; ) u x ( x; ) + u xx ( x; ) + u xxxx ( x; ) . Using Theorem 2, the approximation of the inte gral in matrix form has an error as k [ K ( x i ; t j )] B U k C 2 exp( p d N t ) (10) In the ne xt Theorem, we sho w that our approximation produce an error of e xponential order Theor em 3 Let u ( x; t ) be the e xact solution of the K S equation, and let U be the matrix defined as in (9). Then for N x ; N t > 16 d , ther e is a constant C independent of N x ; N t suc h that sup ( x i ;t j ) k u ( x; t ) U k C N 2 exp( p d N ) ; wher e N = min f N x ; N t g . Pr oof: Ev aluate the inte gral equation (5) at the nodes ( x i ; t j ) where i = N x ; :::; N x ; j = N t ; :::; N t , we get u ( x i ; t j ) = Z t j 0 u ( x i ; ) u x ( x i ; ) + u xx ( x i ; ) + u xxxx ( x i ; ) d + f ( x i ) (11) T o approximate the abo v e inte gral, we use the definite inte gral formula Theorem 2, and define a matrix B = h t I ( 1) m t D 1 0 , with m t = 2 N t + 1 , we obtain U = U U x + U xx + U xxxx B T + F 0 + C 2 exp( p d N t ) : (12) W ith the use of the approximations mentioned in Theorem 1, we obtain Error = U + U A 1 U + A 2 U + A 4 U B T F 0 = C 2 exp( p d N t ) + C 12 N 2 x B T exp( p d N x ) : Int J Elec & Comp Eng, V ol. 9, No. 5, October 2019 : 3720 – 3731 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 r 3725 But from the definition of the matrix B , and si nce 1 0 ( t ) = t ( T 0 t ) T 0 , which has its maximum at T 0 = 4 , therefore, B = T ~ B . It is kno wn that k I ( 1) m t k 1 : 1 , (see, [4]), s o the matrix B can be written as B = T 0 ~ B where each entry in the matrix ~ B is bounded by 1 : 1 h t T 0 = 4 . W ith all of these bounds, the error term can be bounded as Error = U U A 1 U A 2 U A 4 U B T F 0 C N 2 exp( p d N ) for some constant C . Finally choose N = min f N x ; N t g , and notice that the function N 2 exp( p d N ) is decreasing when N > 16 dN . 4.4. Fixed-point iteration W e no w t ak e up the e xistence proof of the solution of the discrete system by fix ed-point it eration. The idea is to produce a sequence that con v er ge to the solution of the K-S equation. By selecting an initial approximation U 0 we iterate the continuous map G repeatedly via the formula U n +1 = G ( U n ) + F 0 ; n = 0 ; 1 ; 2 ; ::: (13) Since G ( U n ) = U n A 1 U n + A 2 U n + A 4 U n B T , where B = T 0 ~ B for some bounded matrix ~ B . Choose T 0 suf ficiently small such that k G ( U ) k k and k dG ( U ) k < k , for an y U in an y gi v en fix ed ball B about the origin, where k is a constant with 0 < k < 1 . No w k U n +1 U n k = k G ( U n ) G ( U n 1 ) k k k U n U n 1 k . . . k n k U 1 U 0 k which implies that k U n +1 U 0 k k n k U 1 U 0 k + k n 1 k U 1 U 0 k + ::: + k U 1 U 0 k 1 1 k k U 1 U 0 k and for positi v e inte ger p , we ha v e k U n + p U n k k p 1 k k U 1 U 0 k so, with a choice of k 2 (0 ; 1) and U 0 ; U 1 ; :::; we see that all iterates will remain in the ball B = f V : k V k 1 1 k k U 1 U 0 kg : Also there is an inte ger N such that k U n + p U n k < for all n > N , and for an y p . Therefore the sequence f U n g is a Cauch y sequence, and hence con v er ges to some U where lim n !1 U n = U . F or uniqueness, suppose there are tw o distinct solutions, say U and U ? , then using k dG ( U ) k < k , we ha v e k U ? U k = k G ( U ? ) G ( U ) k k k U ? U k for k = 1 = 2 , we arri v e at a contradiction, this sho ws that the solution is unique. W ith the notation as abo v e we ha v e pro v ed the Theorem. Theor em 4 Given a constant R > 0 , ther e is a constant T 0 > 0 suc h that if k U 1 U 0 k < R = 2 , then the solution (12) has a unique solution. Mor eo ver , the iter ation sc heme (13) with U 0 = 0 con ver g es to this unique solution. Computational Sinc-sc heme for e xtr acting analytical solution... (Kamel Al-Khaled) Evaluation Warning : The document was created with Spire.PDF for Python.
3726 r ISSN: 2088-8708 4.5. V alidation of Mesh-Fr ee Sinc-collocation method Consider the nonlinear K-S equati on in (1), subject to the initial condition (2), and the boundary conditions u ( a; t ) = 0 ; u ( b; t ) = 0 ; u x ( a; t ) = 0 ; u x ( b; x ) = 0 ; t > 0 : (14) T o implement the Sinc-collocation method, follo wing [20], we discretize time deri v ati v e of the non- linear K-S equation using the Crank-Nicolson scheme, and space deri v ati v es by the weighted ( = 1 = 2) scheme successi v e tw o time le v els n and n + 1 u n +1 u n t + ( uu x ) n +1 + ( uu x ) n 2 + ( u xx ) n +1 + ( u xx ) n 2 + ( u xxxx ) n +1 + ( u xxxx ) n 2 = 0 (15) where u n = u ( x; t n ) is the v alue of the solution at the n th time step, and t n = t n 1 + t , where t is a time step size. The nonlinear term ( uu x ) n +1 must be linearized before continuing. This can be accomplis h e d by using the follo wing formula which obtained by applying the T aylor e xpansion, as follo ws ( u x ) n +1 ( u x ) n + t u n +1 x u n x t + O ( t 2 ) ; ( uu x ) n +1 ( uu x ) n + t h ( u t ) n u n x + ( u ) n u n xt i + O ( t 2 ) which can be simplified to ( uu x ) n +1 ( uu x ) n + t h u n x ( u ) n +1 ( u ) n t + u n u n +1 x u n x t i + O ( t 2 ) (16) Finally , we arri v e at the linearization ( uu x ) n +1 ( u ) n +1 u n x + u n +1 x u n u n u n x (17) So equation (15) can be re written as u n +1 u n t + u n u n +1 x + u n +1 u n x ) 2 + ( u xx ) n +1 + ( u xx ) n 2 + ( u xxxx ) n +1 + ( u xxxx ) n 2 = 0 (18) Rearranging equation (18), we get u n +1 + t 2 u n u n +1 x + u n +1 u n x + ( u xx ) n +1 + ( u xxxx ) n +1 = u n t 2 ( u xx ) n + ( u xxxx ) n (19) where u n are the n th iterates of the approximate solutions. No w the space v ariable is discretized upon the use of Sinc-collocation at the points f x 1 = a; :::; x i + a + ( i 1) h; :::; x N = b g ; h = j b a j N 1 (20) The solution of equation (15) is interpolated and approximated by means of the Sinc functions as u n ( x ) = N X j =0 u n j S j ( x ) ; S j ( x ) = sinc x ( j 1) h a h (21) The unkno wn parameters u j in equation (21) are to be determined by collocation method. Therefore, for each collocation point x i in (20), equation (21) can be written as u n ( x i ) = N X j =0 u n j S j ( x i ) ; i = 1 ; :::; N : (22) Substituting equation (22) into equation (18), for the interior points x i ; i = 1 ; :::; N 1 , we get Int J Elec & Comp Eng, V ol. 9, No. 5, October 2019 : 3720 – 3731 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 r 3727 P N j =0 u n +1 j S j ( x i ) + t 2 h P N j =0 u n j S j ( x i ) P N j =0 u n +1 j S 0 j ( x i )+ P N j =0 u n +1 j S j ( x i ) P N j =0 u n j S 0 j ( x i ) + P N j =0 u n +1 j S 00 j ( x i ) + P N j =0 u n +1 j S 0000 j ( x i ) i = P N j =0 u n j S j ( x i ) t 2 h P N j =0 u n j S 00 j ( x i ) + P N j =0 u n j S 0000 j ( x i ) i (23) The boundary conditions reads as N X j =0 u n +1 j S j ( x 0 ) = N X j =0 u n +1 j S 0 j ( x 0 ) = 0 ; N X j =0 u n +1 j S j ( x N ) = N X j =0 u n +1 j S 0 j ( x N ) = 0 : (24) The system (23) and (24) contain N + 1 equations with N + 1 unkno wns u n j which can be easily solv ed by Gaussian elimination method. Once the v alues of u n j are obtained then the solution for u can be deri v ed from equation (21). No w we switch to a matrix representation of Equations (23) and (24). Define the matrices f N n 1 g ij = h N X ` =0 (0) ` u n ` ( x i ) i (1) j ( x i ) ; f N n 2 g ij = h N X ` =0 (1) ` u n ` ( x i ) i (0) j ( x i ) The abo v e tw o equations are v alid for i = 1 ; :::; N 2 , and in the ro ws 1 ; N 1 , and N the boundary conditions (24) hold true. So in matrix form equations (23), (24) can be written as h I (0) + t 2 ( N n 1 + N n 2 ) + t 2 ( I (2) + I (4) ) i u n +1 = h I (0) t 2 ( I (2) + I (4) ) i u n + F n +1 (25) If the condition of the left hand side of equation (25) is small, then u n for an y time t k can be easily calculated from the initial condition. F or stability analysis of the Sinc-collocation solution, the e v olution of error can be written as h I (0) + t 2 ( N n 1 + N n 2 ) + t 2 ( I (2) + I (4) ) i e n +1 = h I (0) t 2 ( I (2) + I (4) ) i e n (26) where e n = j u n exact u n appr ox j , where u n exact and u n appr ox are the e xact and Sinc-collocation approx- imated solutions at time t k respecti v ely . Equation (26) can be written as e n +1 = M 1 N e n (27) where M 1 N = h I (0) + t 2 ( N n 1 + N n 2 ) + t 2 ( I (2) + I (4) ) i 1 h I (0) t 2 ( I (2) + I (4) ) i : The scheme is considered numerically stable if ( M 1 N ) 1 , where ( : ) denoted the spectral radius. Stability is assured if 1 0 : 5 t ( 2 + 4 ) 1 + 0 : 5 t ( N 1 + N 2 ) + 0 : 5 t ( 2 + 4 ) 1 (28) where 2 ; 4 ; N 1 ; N 2 are the eigen v alues for the matrices I (2) ; I (4) ; N n 1 ; N n 2 respecti v ely . In order to study stability of Sinc methods, we should find some bound for the eigen v alues of the matrices appeared in Equation (28). F or a bound for the T oeplit z matrices I (2) and I (4) , we refer the reader to [17] where well-kno wn results for upper and lo wer bounds are establ ished. Whi le for other tw o matrices N 1 and N 2 , the eigen v alues depends on the choice of the parameter N , that is already tak en into account for the scheme. The stability of the scheme Computational Sinc-sc heme for e xtr acting analytical solution... (Kamel Al-Khaled) Evaluation Warning : The document was created with Spire.PDF for Python.
3728 r ISSN: 2088-8708 and conditioning of the component matrices of the matrix M 1 N depend on the weight parameter and the minimum distance between an y tw o collocation points h in the domain set [ a; b ] . Remark I: In the pre vious sections, we sho wed ho w to replace the deri v ati v es and inte grals by the necessary matrices if the boundary conditions are homogeneous. In particular , the Sinc methodology presented in this pa- per is still applicable to equation (1) with non-homogeneous boundary conditions. The non-homogeneous boundary conditions can be transformed to homogenous boundary conditions by the change of v ariables w ( x; t ) = u ( x; t ) ( x; t ) , where ( x; t ) is an analytic function that is defined as in Lemma 5 : 1 of [21]. 5. RESUL TS AND AN AL YSIS Choosing e xamples with kno wn solutions allo ws for a more complete error analysis. In order to assess the adv antages of the proposed methods, Sinc-Galerkin method o v er the mesh-free Sinc-collocation m ethod in terms of accurac y and ef ficienc y for solving K-S equation, we ha v e applied the tw o methods to tw o dif ferent e xamples. F or the numerical results: Sinc-Galerkin method(SGM): W e apply the Sinc-Galerkin method which has the matrix form (9), In all cal- culations, we ha v e used d = 2 ; = 1 2 ; N x = N t = 64 , and the step-sizes h x ; h t can be determined by h x = p d= ( N x ) ; h t = p d= ( N t ) . One adv antages of the Sinc method is that it automatically determines the graded mesh. Mesh-fr ee Sinc-collocationn method(SCM): W e also solv e the K-S equation using the mesh-free Sinc- collocation method (25). In our computational w ork, we tak e time step sizes t = 0 : 05 through the interv al [ 5 ; 5] and N = 100 for the set of collocation points as in equation (20). The step-size h is the minimum distance between an y tw o points in equation (20). The computations associated with the tw o e xamples were performed using Mathematica. Example 1 Consider the equation u t + uu x + 2 u xx + u xxxx = 0 (29) Where we set = 2 and = 1 into equation (1). This problem has e xact solution [14] u ( x; t ) = 1 + 60 19 ( 38 2 + 2) tanh + 120 3 tanh 3 where = x + t and = 1 2 p 22 = 19 . W e will use this solution, e v aluated at t = 0 , as the initial condition, also we e xtract the required boundary conditions from the e xact solution on the interv al [ 10 ; 10] . From the numerical results in T able 1 , it can be seen that the approximate sol ution (using either method) is quite close to the e xact solution. This sho ws the approximate solution is ef ficienc y . A surf ace plot of the numerical solution is sho wn in Figure 1 using Sinc-Galerkin method. T able 1. Comparison results for Example 1 t x Exact Sinc-Galerkin Mesh-Free-Collocation 0 : 25 6 5 : 04757 -5.04758 -5.04758 4 3 : 58451 -3.58448 -3.58449 2 2 : 68033 2.68962 2.68963 0 5 : 32919 -5.32005 -5.32010 2 2 : 90157 -2.90110 -2.90113 4 0 : 89126 0.891233 0.891232 6 1 : 46202 1.46202 1.46202 0 : 5 6 4 : 91406 -4.91410 -4.91408 4 2 : 64236 -2.64271 -2.64246 2 3 : 46494 3.46144 3.46140 0 7 : 08063 -7.08060 -7.08061 2 1 : 4381 -1.43859 -1.43862 4 1 : 14153 1.14138 1.14128 6 1 : 49241 1.49252 1.49250 Int J Elec & Comp Eng, V ol. 9, No. 5, October 2019 : 3720 – 3731 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 r 3729 - 10 - 5 0 5 10 x 0.0 0.1 0.2 t - 5 0 Figure 1. The Sinc-Galerkin solution on 10 < x < 10 ; 0 < t < 0 : 25 , for Example 1 Remark II: W e ha v e sho wn in Theorem 1 that the problem has a local solution in the interv al (0 ; T ) pro vided that T is suf ficiently small. It might be possible that the scheme will di v er ge for lar ge T . T o run the scheme, we find a smaller time interv al say (0 ; T 1 ) , in which the scheme will con v er ge, and solving Equation (1) using the gi v en initial condition (2). Then we find a T 2 > 0 and solv e the system o v er the interv al ( T 1 ; T 2 ) , where the initial condition no w is the solution found in the interv al (0 ; T 1 ) e v aluated at t = T 1 . This means that the system so f ar has a solution in the interv al (0 ; T 2 ) . Continuing in this w ay , we generate a sequence T 1 ; T 2 ; T 3 ; to get for (1) defined for all 0 < t < T such that T 1 T 2 T 3 ::: T . Example 2 As a second e xample , we consider equation (1) with = = 1 , subject to the Gaussian initial condition u ( x; 0) = e xp ( x 2 ) (30) with boundary conditions u ( 5 ; t ) = 0 ; u (5 ; t ) = 0 ; u x (5 ; t ) = 0 ; u x (5 ; x ) = 0 ; t > 0 : The K-S equation subject to the Gaussian initial condition (30) e xhibiting the chaotic beha vior o v er a finite spatial domain. The numerical results are presented in Figures 2 and 3 . A surf ace plot of the numerical solution is sho wn in Figure 4 using Sinc-Galerkin method, and Figure 5 using Mesh-free Sinc-collocation. - 3 - 2 - 1 1 2 3 x 0.2 0.4 0.6 0.8 1.0 u (a) - 3 - 2 - 1 1 2 3 x - 100 - 50 50 100 u (b) Figure 2. (a) The chaotic solution with Gaussian initial condition at t = 1 for Example 2 by Sinc-Galerkin, (b) The chaotic solution with Gaussian initial condition at t = 3 for Example 2 by Mesh-Free, with N = 160 Computational Sinc-sc heme for e xtr acting analytical solution... (Kamel Al-Khaled) Evaluation Warning : The document was created with Spire.PDF for Python.