Numerical solutions of singularly perturbed problems(SPP) are given by using variants of finite elements method. Both Galerkin and subdomain Galerkin methods based on quadratic B-spline functions are applied to the SPP over the geometrically graded mesh. Results of some text problems are compared with analytical solutions of the SPP

Dag, Idris

Ersoy Hepson, Ozlem

Applied Mathematics & Information Sciences

arXiv

Arab Journals Platform

Appl. Math. Inf. Sci.12, No. 3, 495-500 (2018) 495Applied Mathematics & Information SciencesAn International Journalhttp://dx.doi.org/10.18576/amis/120303Numerical Solution of Singularly Perturbed Problems viaboth Galerkin and Subdomain Galerkin methodsOzlem Ersoy Hepson1,∗ and Idris Dag21 Mathematics-Computer Department, Eskişehir Osmangazi University, Eskişehir, Turkey2 Computer Engineering Department, Eskişehir Osmangazi University, Eskişehir, TurkeyReceived: 4 Jan. 2018, Revised: 2 Mar. 2018, Accepted: 9 Mar.2018Published online: 1 May 2018Abstract: Numerical solutions of singularly perturbed problems(SPP) are given by using variants of finite elements method. BothGalerkin and subdomain Galerkin methods based on quadraticB-spline functions are applied to the SPP over the geometrically gradedmesh. Results of some text problems are compared with analytical solutions of the SPP.Keywords: Subdomain Galerkin method, graded mesh, spline, singularly pe turbed problem.1 IntroductionDifferential equations including a small parametersεmultiplied with the highest order derivative are known asthe singularly perturbed problem and these equationsmodels many real life problems so that it can be used inmany areas of science and engineering such as fluiddynamics, quantum mechanics, control theory, chemicalscience etc. It has the form:−εu′′+ρ(x)u′+ s(x)u = f (x), x ∈ [0,1] (1)with boundary conditionsu(0) = u0 andu(1) = un, u0, un ∈ R (2)where 0< ε ≤ 1, ρ(x), s(x), f (x) are sufficiently smoothfunctions.When the singular perturbation parameter becomessmaller, the problem possess the boundary layer on thenarrow right hand side of the problem region. Thus asmall parameter of the mathematical model cause asudden change of dependent boundaries near boundaries.Occurrence of sharp boundary layers asε tend to infinityfails to yield analytical solutions of the boundary layerproblem. In addition, classical numerical methods leadsto happening of the oscillations in the compute solutions.Therefor it is of interest for scientist to be developed newnumerical methods to tackle the boundary layer for thesingularly perturbed differential equations. In order tomodel the boundary layer, one way is to use finer mesh onsome part of the region on where boundary layer behaviorexist. Two of the special meshes known as Shishkin meshand Bakhvalov meshes have been widely used to haveolutions of the singularly perturbed problem. Bakhvalovgrid is based on mesh generating functions and used themto develop numerical methods for solving the problems.Shishkin mesh is a special piece-wise uniform mesh inwhich splitting of the problem domain is conductedaccording to predetermined boundary layer location andtransition of the submesh is achieved with specialdefinition.B-spline related numerical methods have beenconstructed to have solution of the singularly perturbedproblems in some studies[4,5,6,7,8,9]. Both the cubicB-spline and quadratic B-spline collocation method is setup for the SPP on using the geometrically graded mesh[1,2]. Also, Galerkin method is set up to find numericalsolutions of SPP using cubic B-splines [3]. In this studysolution of the SPP is given by using combination of thequadratic B-splines defined over the graded mesh.Quadratic B-spline Galerkin algorithm and quadraticB-spline subdomain algorithm are defined for gettingsolution of the SPP over the graded mesh. Thus very finemesh are used over interval on which the rapid changesolution exist and larger mesh is employed on the portionof the domain when solution is quite smooth. A meshparameter is used to arrange the fineness of mesh throughthe end boundary to have the higher accuracy of the SPP.∗ Corresponding author e-mail:ozersoy@ogu.edu.trc© 2018 NSPNatural Sciences Publishing Cor.496 O. E. Hepson, I. Dag: Numerical solution of singularly perturbed problems...2 Quadratic B-splinesConsider a partition of the problem domain[0,1] with thegrid pointsxk, such that0= x0 < x1 < · · ·< xN = 1wherexk+1 = xk+hk andhk is the size of interval[xk,xk+1]with relationhk = τhk−1. Whereτ is mesh ratio constant.Sinceh0+ h1+ · · ·+ hN−1 = 1it can be concluded thath0 =11+ τ + τ2+ · · ·+ τN−1.If the mesh ratioτ is taken unity then the partition willbe uniform. To make the mesh size smaller at the rightboundary,τ must be selected asτ < 1 Conversely, To getfiner mesh at the left boundary,τ must be selected asτ >1.Mentioned selection ofτ will be done by experimentally.2.1 Quadratic B-spline Galerkin methodThe QB-splines over the geometrically graded mesh[10]can be expressed such as:Qk−1QkQk+1=1h2k(hk − ξ )2τ,h2k +2hkτξ − (1+ τ)ξ2,ξ 2(3)where ξ = x − xk and 0≤ ξ ≤ hk. Each quadraticB-spline Qk and its derivatives vanish outside of theinterval [xk−1,xk+2] and thus an element is covered bythree consecutive QB-splines. The set of the QB-splines{Q−1,Q0, ...,QN} also constitutes a basis for thefunctions defined in[a,b] [11]. Let U(x, t) be approximatesolution tou(x, t) defined asU =N∑k=−1γkQk (4)whereγk are unknown parameters. By the substitution ofthe value ofQk at the knotsxk in Eq.(4), the nodal valueUand its derivativeU ′ are expressed in terms ofγk byUk =U(xk) = τγk−1+ γk,U ′k =U′(xk) =2τhk(γk − γk−1).(5)To carry on the Galerkin procedure, one can multiply Eq.(1), by the wight functionv(x) and integrate over theinterval[xk,xk+1]xk+1∫xk(−εv′U ′(x)+ vρ(x)U ′+ vs(x)U(x))dx− εvU ′(x)xk+1|xk−xk+1∫xkv f (x)dx = 0.If weight functions are selected asQ j, j = k−1,k,k +1and (4) is substituted in equation above, we obtain thesystem of algebraic equationsk+1∑j=k−1[−εhk∫0(φ ′i φ ′j +ρφiφ ′j + sφiφ j)dξ ]γ j − εφiφ ′jhk|0γ j−hk∫0φi f (xk + ξ )dξ = 0(6)Then following values can be computed.ai j =hk∫0φ ′i φ ′jdξ , ri j = φiφ ′jhk|0,bi j =hk∫0φiφ ′jdξ , fi =hk∫0φi f (xk + ξ )dξ ,ci j =hk∫0φiφ jdξ ,Wherei, j = k−1,k,k+1 andA(k) =23hk2τ2 τ(1− τ) −ττ(1− τ) 2(1− τ + τ2) τ −2−τ τ −2 2B(k) =−τ2216τ (3τ −1)τ6− 16τ (3τ +5)12τ2− 1256τ +12−τ616τ −1212C(k) = hk15τ2hk130τhk (4τ +9)τ30130τhk (4τ +9)815τ2+ 115 τ +815310τ +215τ30310τ +21515hkR(k) =1hk2τ2 0 02τ(3τ +2) −4τ 2τ0 0 0andF (k) =1h2kϑ1 ϑ1 ϑ1ϑ2 ϑ2 ϑ2ϑ3 ϑ3 ϑ3This system is expressed in notational form:ϑ1 = −τ(2hk −2ehk + h2k +2)ϑ2 = 2(τ +1)(1− ehk)+2hk(τ + ehk)− h2k(1− τehk)ϑ3 =(ehk(h2k −2hk +2)−2)Using the local matricesA(i), B(i), C(i), R(i) and F (i),equation (6) can be represented as(−εA(i)+ρB(i)+ sC(i)− εR(i))γ(i) = F (i)whereγ(i) = (γ(i)k−1,γ(i)k ,γ(i)k+1,γ(i)ki+2), F(i) = ( f (i)k−1, f(i)k , f(i)k+1, f(i)k+2)c© 2018 NSPNatural Sciences Publishing Cor.Appl. Math. Inf. Sci.12, No. 3, 495-500 (2018) /www.naturalspublishing.com/Journals.asp 497By combining the local matrices which is defined on[xi,xi+1], i = 0, . . . ,N − 1, the global system in the rangeof [x0,xN ] can be defined as follows.(−εA+ρB+ sC− εR)γ = F (7)The matrix ofA isσ (0)0,0 σ(0)0,1 σ(0)0,2σ (0)1,0 σ∗(1)1,1 σ∗(1)1,2 σ∗(1)1,3σ (0)2,0 σ∗(1)2,1 σ∗(2)2,2 σ∗(2)2,3 σ∗(2)2,4. . .. . .. . .. . .. . .σ∗(i−1)i,i−2 σ∗(i)i,i−1 σ∗(i+1)i,i σ∗(i+1)i,i+1 σ∗(i+1)i,i+2. . .. . .. . .. . .. . .σ∗(n−2)n−1,n−3 σ∗(n−1)n−1,n−2 σ∗(n)n−1,n−1 σ∗(n)n−1,n σ(n)n−1,n+1σ∗(n−1)n,n−2 σ∗(n−1)n,n−1 σ∗(n−1)n,n σ(n)n,n+1σ (n)n+1,n−1 σ(n)n+1,n a(n)n+1,n+1Whereσ∗(1)1,1 = σ(0)1,1 +σ(1)1,1 ,σ∗(1)1,2 = σ(0)1,2 +σ(1)1,2 ,σ∗(1)2,1 = σ(0)2,1 +σ(1)2,1 ,σ∗(2)2,2 = σ(0)2,2 +σ(1)2,2 +σ(2)2,2 ,σ∗(2)2,3 = σ(2)2,3 +σ(1)2,3 ,σ∗(i)i,i−1 = σ(i−1)i,i−1 +σ(i)i,i−1,σ∗(i)i,i = σ(i−1)i,i +σ(i)i,i +σ(i+1)i,i ,σ∗(i)i,i+1 = σ(i)i,i+1+σ(i+1)i,i+1 ,σ∗(n−1)n−1,n−2 = σ(n−2)n−1,n−2+σ(n−1)n−1,n−2,σ∗(n)n−1,n−1 = σ(n−2)n−1,n−1+σ(n−1)n−1,n−1+σ(n)n−1,n−1,σ∗(n)n−1,n = σ(n−1)n−1,n +σ(n)n−1,n,σ∗(n−1)n,n−1 = σ(n−1)n,n−1 +σ(n)n,n−1,B, C, R matrices are obtained similarly. Also the matrix ofF is computed as follows.F =f 00f 01 + f11f 02 + f12 + f22...f i−1i + fii + fi+1i...f n−1n−2 + fn−2n−2 + fn−3n−2f n−1n−1 + fn−2n−1f n−1nThe system (7) consist ofN +1 linear equation inN +3unknown parameters. To get a unique solution anadditional two constraints are needed. The followingequations can be written by using boundary conditionsγ−1 =u0− γ0τ, γN = un − τγN−1.From these equations,γ−1 andγN can be eliminated fromthe system. This system is solved with Matlab programusing five banded Thomas algorithm.2.2 Quadratic B-spline Subdomain GalerkinmethodIf equation (1), multiplied by weight functionVn havingthe formVn ={1, xk ≤ x < xk+10, other caseis integrated over the interval[xk,xk+1] then integrate formof the equation is obtainedxn∫x0[−εU ′′(x)+ρU ′(x)+ sU(x)− f (x)]dx = 0integrating by part is applied to reduce the second orderterm in equation (1)−εU ′(x)xk+1|xk+ρU(x)xk+1|xk+sxk+1∫xkU(x)dx =xk+1∫xkf (x)dxSubstituting Eq. (4) into nodal value and its derivativeyield the system of equations[−ε(k+1∑J=k−1φ ′ihk|0)γ j)+ρ(xk + ξ )(k+1∑J=k−1φihk|0γ j)+∫s(xk + ξ )k+1∑J=k−1xk+1∫xkφiγ jdx=xk+1∫xkf (xk + ξ )dx(8)Performing integration an evaluation gives(−2τεhk+ τ(13hks(x)−ρ(x)))γk−1+(2εhk(1+ τ)+ (τ−1)ρ(x)+23hk (τ +1)s(x))γk+(−2εhk+ρ(x)+13hks(x))γk+1 =xk+1∫xkf (x)dx(9)The equation (9) can be converted the following matricessystem;BX = F (10)whereB =κ01 κ02 κ03κ11 κ12 κ13κ21 κ22 κ23κ31 κ32 κ32. . .. . .. . .κn1 κn2 κn3c© 2018 NSPNatural Sciences Publishing Cor.498 O. E. Hepson, I. Dag: Numerical solution of singularly perturbed problems...andκk1 =−2τεhk− τρk +13τhksk,κk2 =2εhk(1+ τ)+ (τ −1)ρk +23hk (τ +1)skκk3 =−2εhk+ρk +13hkskX =[γ−1, γ0, γ1, · · · , γn−1, γn]TF =[f0, f1, f2, · · · , fn−1, fn]Tfk = f (xk), k = 0,1, · · · ,NThe system (10) consist ofN +1 linear equation inN +3unknown parametersdn+1 = (γn+1−1 ,γn+10 , . . . ,γn+1N+1). Tomake solvable the system, boundary conditionsu(0) = u0and u(1) = un are used to find two additional linearequations:γ−1 =u0− γ0τ, γN = un − τγN−1 (11)(11) can be used to eliminateδ−1,δN+1 from the system(10) which then becomes the solvable matrix equation.Three banded Thomas algorithm is used to solve thesystem.3 Numerical ResultsThe discrete maximum norm measuring the error betweenthe numerical and the analytical solution is defined asL∞ = |u−U |∞ = maxk|uk −Unk |whereunk andUnk represent exact and numerical solutionsat the n.th time level, respectively. In order to minimizethe error, the scan to determine the optimum parameterτcan be searched the interval(0,1) , because of theboundary layers are at the right boundary in this example.Our example is−εu′′+ u′ = exp(x),u(0) = u(1) = 0with the exact solutionu(x) =[ex −1− e1−1/ε +(e−1)e(x−1)/ε1− e−1/ε]/(1− ε).taken from [12].Solution profiles are shown in Figs. 1-4 for theexample. These figures are depicted forN = 20 andε = 0.01,0.001. In order to observe the achievement ofFig. 1: Uniform mesh forε = 0.01 Galerkin methodFig. 2: Graded mesh forε = 0.01 Galerkin methodthe proposed methods, analytical solutions and obtainedresults are illustrated together in all figures. Continuousline represents the exact solutions and the dash linesindicates approximate solutions in Figures .Visualsolutions are depicted in Fig. 1-2 for QG method and inFig. 3-4 for QSG method. Since boundary layer occurs onthe right side of solution domain, use of uniform meshgive oscillations on the right side. When the appropriategraded mesh is used, no oscillations is observed in Fig.2-4.c© 2018 NSPNatural Sciences Publishing Cor.Appl. Math. Inf. Sci.12, No. 3, 495-500 (2018) /www.naturalspublishing.com/Journals.asp 499Fig. 3: Uniform mesh forε = 0.01 subdomain Galerkin methodFig. 4: Graded mesh forε = 0.01 subdomain Galerkin method4 ConclusionThe QG and QSG methods based on quadratic B-splinefunctions are derived for the numerical solutions of theSPP. Difficulties occur from the modelling of theboundary layers in numerical methods are tried toovercome over the geometrically graded mesh viaB-splines. We observe that taking suitably free parametersτ for the quadratic B-splines gives the best solutions. Wesee that QB-splines over the geometrically graded meshproduce slightly better results than the QB-splines overthe unit mesh for the SPP. And also the simplicity of theadaptation of B-splines can be noted as advantages ofproposed numerical methods. Thus, to obtain thenumerical solution of the differential equations havingboundary layers, QG and QSG methods are advisable.AcknowledgementThis study was partially presented at 3rd InternationalConference on Pure and Applied Sciences, Dubai, 2017.References[1] I. Dag and A. Sahin, Numerical solution of singularlyperturbed problems, International Journal of NonlinearScience Vol.8, No.1, pp.3 2-39 (2009).[2] I.Dag and A. Sahin, Numerical solution of the Burgers’equation over geometrically graded mesh, Kybernetes. Vol.36, pp. 721-735 (2007).[3] O. E. Hepson and I. Dag, A cubic subdomain Galerkinmethod over the geometrically graded mesh to the singularlyperturbed problem, Aip Conference Proceedings. Vol. 1926,No. 020015, pp. 1-5 (2018).[4] M. K. Kadalbajoo and P. Arora, B-splines with artificialviscosity for solving singularly perturbed boundary valueproblems, Mathematical and Computer Modelling Vol. 52,pp. 654–666, (2010).[5] M.K.Kadalbajoo and P. Arora, B-spline collocation methodfor the singular-perturbation problem using artificialviscosity, Computers & Mathematics with Applications,Vol. 57, No. 4, pp. 650-663, (2009).[6] J. Chang, Q.Yang and L.Zhao, Solving SingularPerturbation Problems by B-Spline and ArtificialViscosity Method, International Conference on InformationComputing and Applications ICICA, Vol. 2011, pp.742-749, (2011).[7] D. Radunovic, Multiresolution exponential B-splinesand singularly perturbed boundary problem, NumericalAlgorithms, Vol. 47, No. 2, pp. 191–210, (2008).[8] Y. Gupta and M. Kumar, B-Spline Based NumericalAlgorithm for Singularly Perturbed Problem of FourthOrder, American Journal of Computational and AppliedMathematics Vol. 2, No.2, pp. 29-32, (2012).[9] B. Lin, K.T. Li and Z.X. Cheng, B-Spline Solution of aSingularly Perturbed Boundary Value Problem Arising inBiology, Chaos, Solitons & Fractals, Vol. 42, pp. 2934-2948,(2009).[10] I. Dag, Studies of B-spline finite elements, PhD Thesis,University College of North Wales, Bangor, UK, (1994).[11] P.M. Prenter, Splines and variational methods, J.Wiley, NewYork, (1975).[12] J. Lorenz, Combination of initial and boundary valuemethods for a class of singular perturbation problems,in: P.W. Hemker, J.J.H. Miller (Eds.), Numerical Analysisof Singular Perturbation Problems, Academic Press, NewYork, pp. 295-315, (1979).c© 2018 NSPNatural Sciences Publishing Cor.500 O. E. Hepson, I. Dag: Numerical solution of singularly perturbed problems...Ozlem Ersoy Hepsonis teaching assistant atthe Mathematics-ComputerDepartment, Art and ScienceFaculty, Eskişehir OsmangaziUniversity, in Turkey.Dr. Ersoy Hepson receivedher B.Sc. in 2009 and M.Scin 2012 from the EskişehirOsmangazi University. In2015, she earned Ph.D. in Applied Mathematicsfrom Mathematics-Computer Department, EskişehirOsmangazi University. Her research interests are thefinite element method, numerical solutions of partialdifferential equations using splines.Idris Dag was bornin Bolu, 1963. He receivedBsc from Ankara University,1985, and M.S. fromthe Erciyes University,1987 in Turkey and PhD fromUniversity of Wales , BangorCollege , 1984, in UK. Heworked as a research assistantfrom 1986 to 1988 at theUniversity of Erciyes. He is a Professor of the numericalanalysis, working as both research and teaching staff atthe University of Eskişehir Osmangazi since 1994. Hisresearch interest includes the numerical solutions of thedifferential equations using both spline finite elementsmethod and differential quadrature methods.c© 2018 NSPNatural Sciences Publishing Cor.

English

Numerical Solution of Singularly Perturbed Problems via both Galerkin and Subdomain Galerkin methods

Ozlem Ersoy Hepson

Idris Dag

Crossref

Numerical Solution of Singularly Perturbed Problems via both Galerkin and Subdomain Galerkin methods

Abstract

Similar works

Full text

Available Versions

Arab Journals Platform

Crossref