sinc-galerkin 2

Electronic Transactions on Numerical Analysis. Volume 23, pp. 129-140, 2006.Copyright ©2006, Kent State University. ISSN 1068-9613.ETNA Kent State University [email protected] THE SINC-GALERKIN METHOD FOR SOLVING SINGULARLY-PERTURBED REACTION-DIFFUSION PROBLEM /A3 MOHAMED EL-GAMEL /DD Abstract. One of the new techniques used in solving boundary-value problems involving partial differential equations is the Sinc-Galerkin method. In this paper we solve the singularly-perturbed reaction-diffusion problemusing the Sinc-Galerkin method. The scheme is tested on four problems and a comparison with finite elementmethods and the method of reduction of order is made. It is show that the Sinc-Galerkin method yields better results. Key words. Sinc function, Sinc-Galerkin, singularly perturbed, reaction-diffusion, numerical solutions. AMS subject classifications. 65N30,65T60, 35K05, 42C40 1. Introduction. There is a vast literature on numerical solutions of boundary-value problems involving ordinary and partial differential equations. Some of the well-known tech- niques used in solving these problems are the finite differences, finite elements, and multigrid methods. The Galerkin method is another type of numerical technique used to solve partial differ- ential equations with boundary or initial conditions. In this method the solution is assumedto be in a Hilbert space/C0with inner product /CW /BN /CX /BNand one seeks an approximate solution to the problem in the form Ꜷ /B4 /DC /B5/BP /C8/C6/CZ /BP/BD /CP/CZ /AW/CZ /B4 /DC /B5 /BNwhere /CU /AW/CZ /B4 /DC /B5 /CV /C6/CZ /BP/BDis a basis for an /C6- dimensional subspace of functions, /CB/BMThe functions /AW/CZ /B4 /DC /B5 /BN/CZ /BP/BD /BN /BE /BN /A1/A1/A1 /BN/C6 /BNare called test functions and the space /CBis called the test space. There are number of hybrids of the Galerkin method that use different types of test func- tions. In the last decade or so, Sinc functions have been used in many applications, includ- ing numerical solutions of ordinary and partial differential equations. In the Sinc-Galerkin method, the test functions are translates of the sinc function, /CB /B4 /DC /B5 /BP /D7/CX/D2 /AP /DC/BP/AP /DC/BM The sinc method, which was introduced and developed by F. Stenger more than twenty years ago [ 23], is based on the Whittaker-Shannon-Kotel’nikov sampling theorem for entire functions. This method, which uses entire functions as bases, has many advantages over classical methods that use polynomials as bases. For example, in the presence of singularities, it gives a much better rate of convergence and accuracy than polynomial methods. In recent years, a lot of attention has been devoted to the study of the Sinc-Galerkin method to investigate various scientific models. The efficiency of the method has been for- mally proved by many researchers [ 3,5,6,7,15,18,22,24]. For more details of the Sinc- Galerkin method see[ 16,23] and the references therein. In this paper, we will consider the Sinc-Galerkin method for the singularly perturbed elliptic boundary value problem/A0 /AF /AI/BS /BE/D9 /BS/DC /BE /B7 /BS /BE/D9 /BS/DD /BE /AJ/B7 /CP/D9 /BP /CU /B4 /DC/BN /DD /B5 in Ꜳ/BP /B4 /BC /BN /BD/B5 /A2 /B4/BC /BN /BD/B5 /BN (1.1) and/D9 /BP/BC /BNon /BS Ꜳ /BN (1.2) /A3Received January 19, 2005. Accepted for publication February 25, 2006. Recommended by F. Stenger./DDDepartment of Mathematical Sciences, Faculty of Engineering, Mansoura University, Egypt. (gamel [email protected] ). 129 ETNA Kent State University [email protected] 130 M. EL-GAMEL where /BC /BO/AF /AS /BD, /CPis a finite constant and /CU /B4 /DC/BN /DD /B5is analytic in Ꜳ. We shall assume that the solution /D9 /B4 /DC/BN /DD /B5is analytic in Ꜳ. For more details about the formulation of this equation, see Roos [ 20]. The numerical solution of singularly perturbed boundary value problems has recently received much attention. In fact, singularly perturbed problems appear in many branches of engineering, such as fluid mechanics, heat transfer, and problems in structural mechanics posed over thin domains. For more details see Roos [ 20]. Theorems that list conditions for the existence and uniqueness of solutions of such problems are thoroughly discussed in [ 20]. This kind of problem has been investigated by many researchers [ 1,2,4,8,9,10,13,17,19,21]. The organization of the paper is as follows. In section 2, we review some basic facts about the sinc approximation that are necessary for the formulation of the discrete system. In section 3, the Sinc-Galerkin method is developed for linear second-order singularly perturbedboundary value problems with homogeneous boundary condition. The Sinc-Galerkin method is developed for nonlinear second-order singularly perturbed boundary value problems in section/BG. Section /BHaddresses singularly perturbed reaction diffusion equations, which will be solved using the Sinc-Galerkin method. Finally, some numerical examples will be presented in section 6, followed by the conclusions. 2. Sinc Interpolation. The goal of this section is to recall notation and definitions of the Sinc function, state some known results, and derive useful formulas that are important for this paper. First we denote the set of all integers, the set of all real numbers, and the set of all complex numbers by /CA, /CIand /BV, respectively. The sinc function is defined on /CAby sinc /B4 /DC /B5/BP /D7/CX/D2/B4 /AP/DC /B5 /AP/DC /BN /A0/BD /BO/DC/BO /BD /BM For /CW/BQ /BC, the translated sinc functions with evenly spaced nodes are given as/CB /B4 /CZ/BN /CW /B5/B4 /DC /B5/BPsinc /AI/DC /A0 /CZ/CW /CW /AJ/BNk /BP/BC /BN /A6 /BD /BN /A6 /BE /BN/BM/BM/BM /BM If /CUis defined on /CA, then for /CW/BQ /BCthe series/BV /B4 /CU/BN /CW /B5/BP /BD/CG/CZ /BP /A0/BD /CU /B4 /CW/CZ /B5sinc /AI/DC /A0 /CW/CZ /CW /AJ is called the Whittaker cardinal expansion of /CUwhenever this series converges. The properties of Whittaker cardinal expansions have been studied and are thoroughly surveyed in [ 23]. These properties are derived in the infinite strip /BW/CSof the complex plane where for /CS/BQ /BC/BW/CS /BP /D2/AG /BP Ꜵ /B7 /CX/AH /BM /CY /AH /CY /BO/CS /AK /AP /BE /D3/BM Approximations can be constructed for infinite, semi-finite, and finite intervals. To construct approximations on the interval /B4/BC /BN /BD/B5which are used in this paper, consider the conformal mapꜶ /B4 /DE /B5/BP /D0 /D2 /AI/DE /BD /A0 /DE /AJ/BN which maps the eye-shaped region,/BW /BP /AQ/DE /BP /DC /B7 /CX/DD /BM /AC/AC /AC /AC arg /AI/DE /BD /A0 /DE /AJ /AC/AC /AC /AC /BO/CS /AK /AP /BE /AR/BN ETNA Kent State University [email protected] THE SINC-GALERKIN METHOD FOR SINGULARLY-PERTURBED REACTION-DIFFUSION 131 onto the infinite strip /BW/CS. The sinc-Galerkin method requires that the derivatives of composite sinc functions be evaluated at the nodes. We need the following lemma. LEMMA 2.1. [ 23]Let Ꜷbe a conformal one-to one map of the arbitrary simply con- nected domain /BWonto /BW/CS. ThenÆ /B4/BC/B5/CY/CZ /BP/CJ /CB /B4 /CY/BN /CW /B5 Æ Ꜷ /B4 /DC /B5/CL /CY/DC /BP /DC/CZ /BP /AQ/BD /BN if /CY /BP /CZ/BN/BC /BN if /CY /BI/BP /CZ/BN(2.1)Æ /B4/BD/B5/CY/CZ /BP /CW /CS /CSꜶ /CJ /CB /B4 /CY/BN /CW /B5 Æ Ꜷ /B4 /DC /B5/CL /CY/DC /BP /DC/CZ /BP /AQ/BC /BN if /CY /BP /CZ,/B4 /A0 /BD/B5 /CZ /A0 /CY /CZ /A0 /CY /BN if /CY /BI/BP /CZ,(2.2) andÆ /B4/BE/B5/CY/CZ /BP /CW /BE /CS /BE /CSꜶ /BE /CJ /CB /B4 /CY/BN /CW /B5 Æ Ꜷ /B4 /DC /B5/CL /CY/DC /BP /DC/CZ /BP /B4/A0 /AP /BE /BF /BN if /CY /BP /CZ,/A0 /BE/B4 /A0 /BD/B5 /CZ /A0 /CY /B4 /CZ /A0 /CY /B5 /BE /BN if /CY /BI/BP /CZ.(2.3) In equations ( 2.1)-(2.3) /CWis step size and /DC/CZis a sinc grid point given by/DC/CZ /BP Ꜷ /A0 /BD/B4 /CZ/CW /B5/BP /CT /CZ/CW /BD/B7 /CT /CZ/CW /BM 3. Linear Second Order Singularly Perturbed Boundary Value Problem. In this section, we shall study the Sinc-Galerkin scheme for the singularly perturbed boundary value problem/AF/D9 /BC/BC/B7 /CP /B4 /DC /B5 /D9 /BC/B7 /CQ /B4 /DC /B5 /D9 /BP /CU /B4 /DC /B5 for /BC /BO/DC /BO /BD (3.1) subject to boundary conditions/D9 /B4/BC/B5 /BP /BC /BN /D9 /B4/BD/B5 /BP /BC /BM (3.2) We assume an approximate solution of the form/D9/D1 /B4 /DC /B5/BP /C6/CG/CY /BP /A0 /C5 /CR/CY /CB/CY /B4 /DC /B5 /BN /D1 /BP /C5 /B7 /C6 /B7/BD /BM (3.3) where /CB/CY /B4 /DC /B5is the function /CB /B4 /CY/BN /CW /B5 Æ Ꜷ /B4 /DC /B5for some fixed step size /CW. The unknown coef- ficients /CU /CR/CY /CV /C6/A0 /C5in (3.3) are determined by orthogonalizing the residual with respect to the basis functions /CU /CB/CZ /CV /C6/CZ /BP /A0 /C5, i.e.,/CW /AF/D9 /BC/BC/BN/CB/CZ /CX /B7 /CW /CP /B4 /DC /B5 /D9 /BC/BN/CB/CZ /CX /B7 /CW /CQ /B4 /DC /B5 /D9/BN /CB/CZ /CX /BP /CW /CU/BN /CB/CZ /CX /BN /CZ /BP /A0 /C5 /BN/BM/BM/BM/BN/C6 /BM (3.4) The weighted inner product /CW /BN /CXis taken to be/CW /CV /B4 /DC /B5 /BN/CU /B4 /DC /B5 /CX /BP /CI/BD/BC /CV /B4 /DC /B5 /CU /B4 /DC /B5 /DB /B4 /DC /B5 /CS/DC/BM Here /DB /B4 /DC /B5plays the role of a weight function which is chosen depending on the boundary conditions, the domain, and the differential equation. For the case of second order boundary value problems, it is convenient to take/DB /B4 /DC /B5/BP /BD Ꜷ /BC/B4 /DC /B5 /BM ETNA Kent State University [email protected] 132 M. EL-GAMEL A complete discussion of the choice of the weight function can be found in [ 16,23]. The method of approximating the integrals in ( 3.4) begins by integrating by parts to transfer all derivatives from /D9to /CB/CZ. We need the following theorem. THEOREM 3.1. The following relations hold/CW /AF/D9 /BC/BC/BN/CB/CZ /CX/AP /CW /C6/CG/CY /BP /A0 /C5 /BE/CG/CX /BP/BC /D9 /B4 /DC/CY /B5 Ꜷ /BC/B4 /DC/CY /B5 /CW /CX Æ /B4 /CX /B5/CZ/CY /CV/BE /BN/CX /B4 /DC/CY /B5 /BN (3.5) for some functions /CV/BE /BN/CXto be determined. Proof . The inner product with sinc basis element is given by/CW /AF/D9 /BC/BC/BN/CB/CZ /CX /BP /CI/BD/BC /AF/D9 /BC/BC/B4 /DC /B5 /CB/CZ /B4 /DC /B5 /DB /B4 /DC /B5 /CS/DC/BM (3.6) This expression contains the second derivative of /D9, but the desired result is the variable /D9 with no derivatives. Integrating by parts to remove second derivatives from the dependent variable /D9leads to the equality,/CW /AF/D9 /BC/BC/B4 /DC /B5 /BN/CB/CZ /B4 /DC /B5 /CX /BP /BU/DC /B7 /CI/BD/BC /D9 /B4 /DC /B5/B4 /AF/CB/CZ /B4 /DC /B5 /DB /B4 /DC /B5/B5 /BC/BC/CS/DC/BN (3.7) where the boundary term/BU/DC /BP Ꜽ/BD/CG/CX /BP/BC /B4 /A0 /BD/B5 /CX/D9 /B4/BD /A0 /CX /B5/B4 /AF/CB/CZ /DB /B5 /B4 /CX /B5 /AZ/BD/DC /BP/BC /BP/BC /BM Setting/CS /D2 /CSꜶ /D2 /CJ /CB/CZ /B4 /DC /B5/CL /BP /CB /B4 /D2 /B5/CZ /B4 /DC /B5 /BN /BC /AK /D2 /AK /BE /BN and noting that/CS /CS/DC /CJ /CB/CZ /B4 /DC /B5/CL /BP /CB /B4/BD/B5/CZ /B4 /DC /B5 Ꜷ /BC/B4 /DC /B5 /BN by expanding the derivatives under the integral in ( 3.7), we obtain/CW /AF/D9 /BC/BC/BN/CB/CZ /B4 /DC /B5 /CX /BP /CI/BD/BC /AW/BE/CG/CX /BP/BC /D9 /B4 /DC /B5 /CB /B4 /CX /B5/CZ /B4 /DC /B5 /CV/BE /BN/CX /AX/CS/DC /BN (3.8) where/CV/BE /BN /BE /B4 /DC /B5/BP /AF/DB /B4 Ꜷ /BC/B5 /BE/BN /CV/BE /BN /BC /B4 /DC /B5/BP /AF/DB /BC/BC/BN and/CV/BE /BN /BD /B4 /DC /B5/BP /AF/DB /B4 Ꜷ /B5 /BC/BC/B7/BE /AF/DB /BCꜶ /BC/BM Applying the Sinc quadrature rule to the right-hand side of ( 3.8) and deleting the error terms yields ( 3.5). ETNA Kent State University [email protected] THE SINC-GALERKIN METHOD FOR SINGULARLY-PERTURBED REACTION-DIFFUSION 133 The approximation of the last three inner products on the right-hand side of ( 3.4) has been thoroughly treated in [ 16]. We will list them for convenience:/CW /CP /B4 /DC /B5 /D9 /BC/BN/CB/CZ /CX/AP /A0 /CW /C6/CG/CY /BP /A0 /C5 /D9 /B4 /DC/CY /B5 Ꜷ /BC/B4 /DC/CY /B5 /AK/BD /CW Æ /B4/BD/B5/CZ/CY /B4 /CP/DB /B5 Ꜷ /BC/B7 Æ /B4/BC/B5/CZ/CY /B4 /CP/DB /B5 /BC /AL (3.9) and/CW /BZ/BN /CB/CZ /CX/AP /CW /BZ /B4 /DC/CZ /B5 /DB /B4 /DC/CZ /B5 Ꜷ /BC/B4 /DC/CZ /B5 /BN (3.10) where /BZis /CQ /B4 /DC /B5 /D9or /CU, respectively. Replacing each term of ( 3.4) with the approximation defined in ( 3.5), (3.9), and ( 3.10), respectively, and replacing /D9 /B4 /DC/CY /B5by /CR/CYand dividing by /CW, we obtain the following theorem. THEOREM 3.2. If the assumed approximate solution of the boundary-value problem (3.1)-(3.2)i s( 3.3), then the discrete Sinc-Galerkin system for the determination of the un- known coefficients /CU /CR/CY /CV /C6/CY /BP /A0 /C5is given by/C6/CG/CY /BP /A0 /C5 /BE/CG/CX /BP/BC /BD /CW /CX Æ /B4 /CX /B5/CZ/CY /CV/BE /BN/CX /B4 /DC/CY /B5 Ꜷ /BC/B4 /DC/CY /B5 /CR/CY /A0 /C6/CG/CY /BP /A0 /C5 /AK/BD /CW Æ /B4/BD/B5/CZ/CY /B4 /CP /B4 /DC/CY /B5 /DB /B5/B7 Æ /B4/BC/B5/CZ/CY /B4 /CP /B4 /DC/CY /B5 /DB /B5 /BC Ꜷ /BC/B4 /DC/CY /B5 /AL/CR/CY (3.11)/B7 /CQ /B4 /DC/CZ /B5 /DB /B4 /DC/CZ /B5 Ꜷ /BC/B4 /DC/CZ /B5 /CR/CZ /BP /CU /B4 /DC/CZ /B5 /DB /B4 /DC/CZ /B5 Ꜷ /BC/B4 /DC/CZ /B5/CZ /BP /A0 /C5/BN /A1/A1/A1 /BN/C6 /BM The following notation will be necessary for writing down the system. Let /BW /B4 /CV /B5be the/D1 /A2 /D1diagonal matrix D /B4 /CV /B5/BP /BC/BU/BU/BU/BS /CV /B4 /DC/A0 /C5 /B5/CV /B4 /DC/A0 /C5 /B7/BD /B5 .../CV /B4 /DC/C6 /B5 /BD/BV/BV/BV/BT /BM Define the /D1 /A2 /D1matrices I /B4 /D4 /B5(see [ 11]) for /BC /AK /D4 /AK /BEby/C1 /B4 /D4 /B5/BP /CWÆ /B4 /D4 /B5/CY/CZ /CX/BNj /BN/CZ /BP /A0 /C5 /BN/BM/BM/BM /BN/C6 /BM Letcbe the /D1-vector with /CY-th component given by /CR/CY, and 1be an /D1-vector each of whose components is /BD. In this notation the system in ( 3.11) takes the matrix form A /BC/BU/BU/BU/BS /CR/A0 /C5/CR/A0 /C5 /B7/BD .../CR/C6 /BD/BV/BV/BV/BT /BP/A2 /BN (3.12) where A /BP /BE/CG/CY /BP/BC /BD /CW /CYI /B4 /CY /B5D /AI/CV/BE /BN/CY Ꜷ /BC /AJ/A0 /AK/BD /CWI /B4/BD/B5D /B4 /CP/DB /B5/B7D /AI/B4 /CP/DB /B5 /BC Ꜷ /BC /AJ/AL/B7D /AI/CQ/DB Ꜷ /BC /AJ/BN (3.13) ETNA Kent State University [email protected] 134 M. EL-GAMEL and/A2/BP /BW /AI/DB/CU Ꜷ /BC /AJ 1 /BM (3.14) Now we have a linear system of /D1equations of the /D1unknown coefficients, namely,/CU /CR/CY /BN/CY /BP /A0 /C5 /BN/BM/BM/BM /BN/C6 /CV. We can obtain the coefficients of the approximate solution by solving this linear system. This system ( 3.12) may be easily solved by a variety of methods. In this paper we use the /C9/CAmethod [ 12]. The solution c=( /CR/A0 /C5, /CR/A0 /C5 /B7/BD, ... /CR/C6 /B5 /CCgives the coefficients in the Sinc-Galerkin approximation /D9/D1 /B4 /DC /B5of /D9 /B4 /DC /B5. 4. Non-Linear Second Order Singularly Perturbed Boundary Value Problem. Con- sider a nonlinear, second order BVP of the form/AF/D9 /BC/BC/B7 /CP /B4 /DC /B5 /D9 /BC/B7 /CQ /B4 /DC /B5 /D9 /B7 /AR /B4 /DC /B5 /C9 /B4 /D9 /B5/BP /CU /B4 /DC /B5 for /BC /BO/DC/BO /BD, (4.1) subject to boundary conditions/D9 /B4 /BC /B5/BP/BC /BN /D9 /B4/BD/B5 /BP /BC /BN (4.2) where /C9 /B4 /D9 /B5may be a polynomial, a rational function, or an exponential. Due to the large number of different possibilities, our work will be focused mainly on the following forms/C9 /B4 /D9 /B5:/BD /BM /C9 /B4 /D9 /B5/BP /D9 /D2/BN /D2/BQ /BD /BN/BE /BM /C9 /B4 /D9 /B5 /BP /CT/DC/D4 /B4 /A6 /D9 /B5 /BN /CR/D3/D7 /B4 /D9 /B5 /BN /D7/CX/D2/B4 /D9 /B5 /BN /D7/CX/D2/CW/B4 /D9 /B5 /BN /CR/D3/D7/CW /B4 /D9 /B5 /BN/BM/BM/BM /BN/BF /BM /C9 /B4 /D9 /B5/BP /BD /B4/BD /A6 /D9 /B5 /D2 /BN /BD /B4/BD /A6 /D9 /BE/B5 /D2 /BN /BD /B4 /D9 /BE/A6 /BD/B5 /D2 /BN /D2 /BI/BP/BC /BN or any analytic function of /D9which has a power series expansion. We start with the case /C9 /B4 /D9 /B5/BP /D9 /D2, where /D2is a non-negative integer, or a fraction. The approximate solution for /D9 /B4 /DC /B5is represented by the formula/D9/D1 /B4 /DC /B5/BP /C6/CG/CY /BP /A0 /C5 /CR/CY /CB/CY /B4 /DC /B5 /BN /D1 /BP /C5 /B7 /C6 /B7/BD /BM (4.3) The unknown coefficients /CR/CYin equation ( 4.3) are determined by orthogonalizing the residual with respect to the basis functions, i.e.,/CW /AF/D9 /BC/BC/BN/CB/CZ /CX /B7 /CW /CP /B4 /DC /B5 /D9 /BC/BN/CB/CZ /CX /B7 /CW /CQ /B4 /DC /B5 /D9/BN /CB/CZ /CX /B7 /CW /AR/D9 /D2/BN/CB/CZ /CX /BP /CW /CU/BN /CB/CZ /CX /BM (4.4) We need the following lemma LEMMA 4.1. [ 7]The following relations hold/CW /AR /B4 /DC /B5 /D9 /D2/BN/CB/CZ /CX/AP /CW /DB /B4 /DC/CZ /B5 /D9 /D2/B4 /DC/CZ /B5 /AR /B4 /DC/CZ /B5 Ꜷ /BC/B4 /DC/CZ /B5 /BM (4.5) Replacing each term of ( 4.4) with the approximations defined in ( 3.5),(3.9),(3.10), (4.5), and replacing /D9 /B4 /DC/CY)b y /CR/CYand dividing by /CW, we obtain the following theorem. THEOREM 4.2. If the assumed approximate solution of the boundary-value problem (4.1)and ( 4.2)i s( 4.3), then the discrete Sinc-Galerkin system for the determination of the ETNA Kent State University [email protected] THE SINC-GALERKIN METHOD FOR SINGULARLY-PERTURBED REACTION-DIFFUSION 135 unknown coefficients /CU /CR/CY /CV /C6/CY /BP /A0 /C5is given by/C6/CG/CY /BP /A0 /C5 /BE/CG/CX /BP/BC /BD /CW /CX Æ /B4 /CX /B5/CZ/CY /CV/BE /BN/CX /B4 /DC/CY /B5 Ꜷ /BC/B4 /DC/CY /B5 /CR/CY /A0 /C6/CG/CY /BP /A0 /C5 /AK/BD /CW Æ /B4/BD/B5/CZ/CY /B4 /CP /B4 /DC/CY /B5 /DB /B5/B7 Æ /B4/BC/B5/CZ/CY /B4 /CP /B4 /DC/CY /B5 /DB /B5 /BC Ꜷ /BC/B4 /DC/CY /B5 /AL/CR/CY (4.6)/B7 /CQ /B4 /DC/CZ /B5 /DB /B4 /DC/CZ /B5 Ꜷ /BC/B4 /DC/CZ /B5 /CR/CZ /B7 /DB /B4 /DC/CZ /B5 /AR /B4 /DC/CZ /B5 Ꜷ /BC/B4 /DC/CZ /B5 /CR /D2/CZ /BP /CU /B4 /DC/CZ /B5 /DB /B4 /DC/CZ /B5 Ꜷ /BC/B4 /DC/CZ /B5/CZ /BP /A0 /C5/BN /A1/A1/A1 /BN/C6 /BM Proof . Combine Lemma 4.1 and Theorem 3.2. Using the notation in the previous section, let c /D2be the m-vector with j-th component given by /CR /D2/CY. The system in ( 4.6) takes the matrix form Ac /B7Ec /D2/BP/A2 /BN (4.7) where E /BPD /AI/AR/DB Ꜷ /BC /AJ (4.8) with Aand /A2defined by equations ( 3.13) and ( 3.14), respectively. Now we have a nonlinear system of /D1 /BP /C5 /B7 /C6 /B7/BDequations of the /D1unknown coefficients, namely, /CU /CR/CY /CV /C6/CY /BP /A0 /C5. We can obtain the coefficients of the approximate solution by solving this nonlinear system by Newton’s method [7]. The solution c /BP/B4 /CR/A0 /C5 /BN/BM/BM/BM /BN/CR/C6 /B5 /CC gives the coefficients in the Sinc-Galerkin approximation /D9/D1 /B4 /DC /B5of /D9 /B4 /DC /B5. 5. Singularly Perturbed Reaction-Diffusion Equation. In this section the Sinc- Galerkin method is given for the approximate solution of the equations ( 1.1)-(1.2). The assumed approximate solution takes the form/D9/D1/DC /BN/D1/DD /B4 /DC/BN /DD /B5/BP /C6/DD/CG/CY /BP /A0 /C5/DD /C6/DC/CG/CX /BP /A0 /C5/DC /D9/CX/CY /CB/CX/CY /B4 /DC/BN /DD /B5 /BN (5.1) where /D1/DC /BP /C5/DC /B7 /C6/DC /B7/BD /BN /D1/DD /BP /C5/DD /B7 /C6/DD /B7/BD /BMThe basis functions /CU /CB/CX/CY /B4 /DC/BN /DD /B5 /CVfor/A0 /C5/DC /AK /CX /AK /C6/DC /BN /A0 /C5/DD /AK /CY /AK /C6/DDare given as the product of basis functions. In this paper we take/CB/CX/CY /B4 /DC/BN /DD /B5/BP /CB/CX /B4 /DC /B5 /CB/CY /B4 /DD /B5/BP /CJ /CB /B4 /CX/BN /CW/DC /B5 Æ Ꜷ /B4 /DC /B5/CL/CJ /CB /B4 /CY/BN /CW/DD /B5 Æ /AD /B4 /DD /B5/CL /BM The unknown coefficients /CU /D9/CX/CY /CVin equation ( 5.1) are determined by orthogonalizing the residual with respect to the functions /CU /CB/CZ/D0 /B4 /DC/BN /DD /B5 /CV /BN /A0 /C5/DC /AK /CZ /AK /C6/DC /BN /A0 /C5/DD /AK /D0 /AK /C6/DD. This yields the discrete Galerkin system/A0/CW /AF/D9/DC/DC /BN/CB/CZ /CB/D0 /CX/A0/CW /AF/D9/DD/DD /BN/CB/CZ /CB/D0 /CX /B7 /CW /CP/D9 /BN/CB/CZ /CB/D0 /CX /BP /CW /CU/BN /CB/CZ /CB/D0 /CX /BM (5.2) We need the following lemma. LEMMA 5.1. The following relations hold/CW /AF/D9/DC/DC /BN/CB/CZ /CB/D0 /CX /BP /CW/DC /CW/DD /AT /B4 /DD/D0 /B5 Ꜷ /BC/BE /B4 /DD/D0 /B5 /C6/DC/CG/CX /BP /A0 /C5/DC /BE/CG/CY /BP/BC /D9 /B4 /DC/CX /BN/DD/D0 /B5 Ꜷ /BC/BD /B4 /DC/CX /B5 /AK/BD /CW /CY/DC Æ /B4 /CY /B5/CZ/CX /AQ/CY /AL/BN (5.3) ETNA Kent State University [email protected] 136 M. EL-GAMEL/CW /AF/D9/DD/DD /BN/CB/CZ /CB/D0 /CX /BP /CW/DC /CW/DD /DB /B4 /DC/CZ /B5 Ꜷ /BC/BD /B4 /DC/CZ /B5 /C6/DD/CG/CX /BP /A0 /C5/DD /BE/CG/CY /BP/BC /D9 /B4 /DC/CZ /BN/DD/CX /B5 Ꜷ /BC/BE /B4 /DD/CX /B5 Ꜽ/BD /CW /CY/DD Æ /B4 /CY /B5/CZ/CX /AH/CY /AZ/BN (5.4)/CW /CP/D9 /BN/CB/CZ /CB/D0 /CX /BP /CP/CW/DC /CW/DD /DB /B4 /DC/CZ /B5 /D9 /B4 /DC/CZ /BN/DD/D0 /B5 /AT /B4 /DD/D0 /B5 Ꜷ/BD /B4 /DC/CZ /B5 Ꜷ/BE /B4 /DD/D0 /B5 /BN (5.5) and/CW /CU/BN /CB/CZ /CB/D0 /CX /BP /CW/DC /CW/DD /DB /B4 /DC/CZ /B5 /CU /B4 /DC/CZ /BN/DD/D0 /B5 /AT /B4 /DD/D0 /B5 Ꜷ/BD /B4 /DC/CZ /B5 Ꜷ/BE /B4 /DD/D0 /B5 /BN (5.6) where/AQ/BE /BP /AF /B4 Ꜷ /BC/BD /B5 /BE/DB/BN /AQ/BD /BP /AFꜶ /BC/BD /DB /B7/BE /AFꜶ /BC/BD /DB /BC/BN /AQ/BC /BP /AF/DB /BC/BC/BN and/AH/BE /BP /AF /B4 Ꜷ /BC/BE /B5 /BE/AT/BN /AH/BD /BP /AFꜶ /BC/BE /AT /B7/BE /AFꜶ /BC/BE /AT /BC/BN /AH/BC /BP /AF/AT /BC/BC/BM Replacing each term of ( 5.2) with the approximation defined in ( 5.3)–(5.6) and replacing/D9 /B4 /DC/CZ /BN/DD/D0 /B5by /D9/CZ/D0and dividing by /CW/DC /CW/DDwe obtain the following theorem. THEOREM 5.2. If the assumed approximate solution of the problem ( 1.1)-(1.2)i s(5.1), then the discrete Sinc-Galerkin system for the determination of the unknown coefficients/CU /D9/CZ/CY /BN /A0 /C5/DC /BO/CZ /BO /C6/DC /BN /A0 /C5/D8 /BO/CY /BO /C6/D8 /CVis given by/A0 /AT /B4 /DD/D0 /B5 Ꜷ /BC/BE /B4 /DD/D0 /B5 /C6/DC/CG/CX /BP /A0 /C5/DC /BE/CG/CY /BP/BC /D9 /B4 /DC/CX /BN/DD/D0 /B5 Ꜷ /BC/BD /B4 /DC/CX /B5 /AK/BD /CW /CY/DC Æ /B4 /CY /B5/CZ/CX /AQ/CY /AL (5.7)/A0 /DB /B4 /DC/CZ /B5 Ꜷ /BC/BD /B4 /DC/CZ /B5 /C6/DD/CG/CX /BP /A0 /C5/DD /BE/CG/CY /BP/BC /D9 /B4 /DC/CZ /BN/DD/CX /B5 Ꜷ /BC/BE /B4 /DD/CX /B5 Ꜽ/BD /CW /CY/DD Æ /B4 /CY /B5/CZ/CX /AH/CY /AZ/B7/CP /DB /B4 /DC/CZ /B5 /D9 /B4 /DC/CZ /BN/DD/D0 /B5 /AT /B4 /DD/D0 /B5 Ꜷ/BD /B4 /DC/CZ /B5 Ꜷ/BE /B4 /DD/D0 /B5 /BP /DB /B4 /DC/CZ /B5 /CU /B4 /DC/CZ /BN/DD/D0 /B5 /AT /B4 /DD/D0 /B5 Ꜷ/BD /B4 /DC/CZ /B5 Ꜷ/BE /B4 /DD/D0 /B5 /BM Introducing the notation of Toeplitz matrices in equation ( 5.7) leads to the matrix form/A0 /BE/BG /BE/CG/CY /BP/BC /BD /CW /CY/DC /C1 /B4 /CY /B5D /AI/AQ/CY Ꜷ /BC/BD /DB /AJ /BF/BHD /B4 Ꜷ /BC/BD /B5D /B4 /DB /B5UD /AI/AT Ꜷ /BC/BE /AJ/A0D /AI/DB Ꜷ /BC/BD /AJ UD /B4 /AT /B5D /B4 Ꜷ /BC/BE /B5 /BE/BG /BD/CG/CY /BP/BC /BD /CW /CY/DD /C1 /B4 /CY /B5D /AI/AH/CY Ꜷ /BC/BE /AT /AJ /BF/BH /D8/B7 /CPD /AI/DB Ꜷ /BC/BD /AJ UD /AI/AT Ꜷ /BC/BE /AJ/BPD /AI/DB Ꜷ /BC/BD /AJ FD /AI/AT Ꜷ /BC/BE /AJ/BM Note that /CJ /CL /D8, denotes the transpose of the matrix /CJ /CL. Premultiplying by D /B4 Ꜷ /BC/BD /B4 /DC/CZ /B5/B5and postmultiplying by D /B4 Ꜷ /BC/BE /B4 /DD/D0 /B5/B5yields the equivalent system AX /B7XB /BPG /BN (5.8) ETNA Kent State University [email protected] THE SINC-GALERKIN METHOD FOR SINGULARLY-PERTURBED REACTION-DIFFUSION 137 where A /BPA1 /B7 /CPI /BN A1 /BP /A0D /B4 Ꜷ /BC/BD /B5 /BE/BG /BE/CG/CY /BP/BC /BD /CW /CY/DCI /B4 /CY /B5D /AI/AQ/CY Ꜷ /BC/BD /DB /AJ /BF/BHD /B4 Ꜷ /BC/BD /B5 /BN B /BP /A0D /B4 Ꜷ /BC/BE /B5 /BE/BG /BE/CG/CY /BP/BC /BD /CW /CY/DDI /B4 /CY /B5D /AI/AH/CY Ꜷ /BC/BE /AT /AJ /BF/BH /D8 D /B4 Ꜷ /BC/BE /B5 /BN G /BPD /B4 /DB /B5FD /B4 /AT /B5 /BN and X /BPD /B4 /DB /B5UD /B4 /AT /B5 /BM The four matrices A /BNB /BNXandGhave dimension /D1/DC /A2 /D1/DC, /D1/DD /A2 /D1/DD, /D1/DC /A2 /D1/DDand /D1/DC /A2 /D1/DD, respectively. Lastly, the /D1/DC /A2 /D1/DDmatrices UandFhave /CZ/D0-th entries given by /D9/CZ/D0and/CU /B4 /DC/CZ /BN/DD/D0 /B5/BP /CU /B4 /CT /CZ/CW /BD/B7 /CT /CZ/CW /BN /CT /D0/CW /BD/B7 /CT /D0/CW /B5, respectively. To obtain the approximate solution of equation ( 5.1), we need to solve the system for Uwhich requires solving ( 5.8) for X. To solve ( 5.8), see[ 5]. 6. Numerical Examples. In this section, four examples will be tested by using the Sinc- Galerkin method discussed above. For purposes of comparison, contrast and performance,examples with known solutions were chosen. For the sake of comparison only, we will discuss the first and the third examples that were investigated by Reddy [ 19] and Navon [ 14]. In all examples,/CSis taken to be /AP/BP /BEand we report absolute error which is defined as absolute error /BP /AC/AC/D9exact solution /A0 /CDSinc-Galerkin /AC/AC EXAMPLE 6.1. [ 19] Consider the boundary-value problem/AF /CS /BE/D9 /CS/DC /BE /B7 /CS/D9 /CS/DC /BP/BD /B7 /BE /DC/BN /BC /AK /DC /AK /BD /BN and/D9 /B4/BC/B5 /BP /BD /BN /D9 /B4/BD/B5 /BP /BD /BN whose exact solution is/D9 /B4 /DC /B5/BP /DC /B4 /DC /B7/BD /A0 /BE /AF /B5/B7 /B4/BE /AF /A0 /BD/B5 /B4/BD /A0 /CT /A0 /DC/BP/AF/B5 /BD /A0 /CT /A0 /BD /BP/AF /BM The parameters /AB /BP /AC /BP /BD /BEand /C6 /BP/BD /BC /BC are used. Maximum absolute error are tabulated in Table 6.1for Sinc-Galerkin together with the analogous results of Reddy [ 19], who use the method of reduction of order. ETNA Kent State University [email protected] 138 M. EL-GAMEL TABLE 6.1 Maximum absolute error for Example 6.1/AF The method of reduc- tion of order [ 19] Sinc-Galerkin method /BD/BC /A0 /BF 0.213E-02 0.358E-08/BD/BC /A0 /BG 0.112E-04 0.271E-08/BD/BC /A0 /BI 0.797E-10 EXAMPLE 6.2. Consider the boundary-value problem/AF /CS /BE/D9 /CS/DC /BE /B7/BE /CS/D9 /CS/DC /B7 /D9 /BE/BP /AI/A0 /BD /AF /B7 /CT /A0 /DC/BP/AF /AJ/CT /A0 /DC/BP/AF/BN /BC /AK /DC /AK /BD /BN and/D9 /B4/BC/B5 /BP /BD /BN /D9 /B4/BD/B5 /BP /CT /A0 /BD /BP/AF whose exact solution is/D9 /B4 /DC /B5/BP /CT /A0 /DC/BP/AF/BM The parameters /AB /BP /AC /BP /BD /BEand /C6 /BP/BD /BC /BC are used. Maximum absolute error at different /AF are tabulated in Table 6.2. TABLE 6.2 Maximum absolute error for Example 6.2/AF Maximum absolute error /BD/BC /A0 /BE 0.135E-07/BD/BC /A0 /BG 0.432E-08/BD/BC /A0 /BI 0.297E-10 EXAMPLE 6.3. [ 14] Consider the boundary-value problem/A0 /AF /AI/BS /BE/D9 /BS/DC /BE /B7 /BS /BE/D9 /BS/DD /BE /AJ/B7/BE /D9 /BP /CU /B4 /DC/BN /DD /B5 /BN and/D9 /B4/BC /BN/DD /B5/BP /D9 /B4/BD /BN/DD /B5/BP /D9 /B4 /DC/BN /BC/B5 /BP /D9 /B4 /DC/BN /BD /B5/BP/BC /BN where /CU /B4 /DC/BN /DD /B5is chosen such that the solution is/D9 /B4 /DC/BN /DD /B5/BP /AI/BD /A0 /CT /A0 /DC/BP/AF/B7 /CT /A0 /B4/BD /A0 /DC /B5 /BP/AF /BD /A0 /CT /A0 /BD /BP/AF /AJ/AI/BD /A0 /CT /A0 /DD/BP /AF/B7 /CT /A0 /B4/BD /A0 /DD /B5 /BP/AF /BD /A0 /CT /A0 /BD /BP/AF /AJ/BM The parameters /C5/DC /BP /C6/DC /BP /C5/DD /BP /C6/DD /BP /BD/BC/BCand /AB /BP /AC /BP /BF /BEare used for the Sinc- Galerkin solution. Table 6.3exhibits a comparison between the errors obtained by using Sinc-Galerkin method and errors of Li and Navon [ 14], who use the Finite Element method. EXAMPLE 6.4. Consider the boundary-value problem/A0 /AF /AI/BS /BE/D9 /BS/DC /BE /B7 /BS /BE/D9 /BS/DD /BE /AJ/A0 /D9 /BP /CU /B4 /DC/BN /DD /B5 /BN ETNA Kent State University [email protected] THE SINC-GALERKIN METHOD FOR SINGULARLY-PERTURBED REACTION-DIFFUSION 139 TABLE 6.3/CZ /D9/CT/DC/CP/CR/D8 /A0 /D9/BT/D4/D4 /CZ/C4 /BEfor Example 6.3/AF The Finite Element Method [ 14] with/C6 /BP/BH /BF /BE /BL Sinc-Galerkin Method /BD/BC /A0 /BE 1.978E-03 0.387E-05/BD/BC /A0 /BF 0.457E-03 0.321E-05/BD/BC /A0 /BH 0.704E-03 0.112E-06/BD/BC /A0 /BI 0.317E-04 0.137E-06/BD/BC /A0 /BJ 0.136E-04 0.133E-06 and/D9 /B4/BC /BN/DD /B5/BP /D9 /B4/BD /BN/DD /B5/BP /D9 /B4 /DC/BN /BC/B5 /BP /D9 /B4 /DC/BN /BD /B5/BP/BC /BN where /CU /B4 /DC/BN /DD /B5is chosen such that the solution is/D9 /B4 /DC/BN /DD /B5/BP /DC/DD /D0/D2 /DC /D0/D2 /DD/BM The parameters /C5/DC /BP /C6/DC /BP /C5/DD /BP /C6/DD /BP /BD/BC/BCand /AB /BP /AC /BP /BF /BEare used for the Sinc- Galerkin solution. 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