NON STANDARD FITTED FINITE DIFFERENCE METHOD FOR SINGULAR PERTURBATION PROBLEMS USING CUBIC SPLINE

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1 Global and Stocastic Analysis Vol. 4 No. 1, January 2017, 1-10 NON STANDARD FITTED FINITE DIFFERENCE METHOD FOR SINGULAR PERTURBATION PROBLEMS USING CUBIC SPLINE K. PHANEENDRA AND E. SIVA PRASAD Abstract. Tis paper presents a class of accurate non standard finite difference metod based on cubic spline function on uniform mes for te numerical solution of a second order singularly perturbed two point boundary value problems associate wit boundary layer teory. A fitting factor known to be artificial viscosity was introduced in discretization equation of non standard finite difference sceme and its value was obtained from te teory of singular perturbations. We ave solved te discretization equation using discrete invariant imbedding. Te proposed sceme is analyzed for Convergence. Te numerical results for several test examples demonstrate te efficiency of te proposed metod. 1. Introduction Singularly perturbed two point boundary value problems occurs frequently in many areas of applied matematics and matematical pysics suc as fluid mecanics, elasticity, optimal control, cemical-reactor teory, aerodynamics, reaction diffusion process, geopysics and many oter related areas. Solutions of tis type of equations exibit boundary layer penomena; tat is, te solution of tese problems varies rapidly in some parts and varies slowly in some oter parts. Te numerical treatment of singularly perturbed differential equations is far from trivial and also gives major computational difficulties due to te presence of boundary and/or interior layers. Standard numerical metods fail to approximate teir solutions because of te boundary layer beavior. So to capture te layers, two classical approaces are used. Te first approac is to design a special mes known as Siskin mes [15] or graded meses [3] and te second approac is a fitting factor is introduced in te existing finite difference metods known as fitted operator metod [9] to control te beavior in te boundary layer. Tis paper uses te second approac to solve te singularly perturbed two point boundary value problems. To describe tis metod, we consider singularly perturbed two point boundary value problems of te form y x + pxy x + qxyx = rx, 0 x Matematics Subject Classification. Primary 65L11; Secondary 65L12. Key words and prases. Singularly perturbed boundary value problem, Fitting factor, Cubic spline, Tridiagonal system, Invariant imbedding algoritm. 1 1

2 2 K. PHANEENDRA AND E. SIVA PRASAD wit boundary conditions y0 = α, y1 = β 1.2 were 0 < << 1, px, qx and rx are bounded continuous functions in 0, 1, and α, β are finite constants. Tis type of problem was solved asymptotically by Bellman [1], Bender and Orszag [2], Kevorkian and Cole [6], Nayfe [10], O Malley [11] and numerically by Kreiss [7], Miller [9], Kadalbajoo and Kumar [5], Rasidinia etc. [14], Lin and Vancouver [8], Reddy [4, 13]. In tis paper, a fitted non standard finite difference sceme using cubic spline functions on a uniform mes was presented for solving linearly singularly perturbed two-point boundary value problems wit boundary layer at one end point. In section 2, cubic spline was discussed briefly. In section 3, numerical metod using cubic spline was presented. In section 4, convergence analysis of te metod was discussed. To justify te proposed metod, test examples are illustrated in section 5. Finally, in section 6, summary and conclusions of te metod are given. 2. Description of te metod Discretize te interval [0,1] into N equal subintervals of mes size = 1 N, so tat x i = i, i =0, 1, 2,..., N wit 0 = x 0, 1 = x N. Let yx be te exact solution and y i be an approximation to yx i obtained by te polynomial cubic splines i x passing troug te points x i, y i andx i+1, y i+1. From te teory of te Splines, S i x satisfies not only te interpolatory conditions at x i andx i+1 but also te continuity of first derivative at te common nodes x i, y i are fulfilled. For eac i t segment, te cubic polynomial spline function S i xas te form S i x = a i + b i x x i + c i x x i 2 + d i x x i 3, i = 0, 1, 2,..., N were a i, b i, c i and d i are constants. A cubic spline function Sx of class C 2 [a, b] interpolates yx at te grid pointsx i for i = 0,1,2...,N. To develop expressions for te four coefficients of Eq. 2.1 in terms ofy i,y i+1,m i and M i+1. We first define S i x i = y i,s i x i+1 = y i+1,s i x i = M i,s i x i+1 = M i+1. From algebraic manipulation, we obtain te following expression: a i = y i, b i = y i+1 y i M i+1 M i, c i = M i 6 2, d i = M i+1 M i 6 Were i = 0, 1, 2...N-1. Using te continuity of cubic spline Sx and its first derivative atx i, y i, tat iss i 1 x i = S i x i, we obtain te following relations for i =1,2,...,N-1. M i+1 + 4M i + M i 1 = 6 2 y i+1 2y i + y i Numerical Sceme 3.1. Left end boundary layer. We assume tat qx 0, px M > 0 trougout te interval [0, 1], were M is positive constant. Under tese assumptions, Eq. 1.1 as unique solution and display boundary layer at x = 0 for small values of. 2

3 NON STANDARD FDM FOR SPP USING CUBIC SPLINE 3 From te teory of singular perturbation it is known tat te solution of Eq. 1.1 and Eq. 1.2 is of te form O Malley [11] yx = y 0 x + p0 px α y 00 e x 0 px pxdx qx + O 3.1 were y 0 x is te solution of reduced problem pxy 0x + qxy 0 x = rx,wit y 0 1 = β. Expanding px and qx about te point 0 up to first term using Taylor s series, Eq. 3.1 gives From Eq. 3.2 we ave i.e., terefore yx = y 0 x + α y 0 0 e yx i = y 0 x i + α y 0 0 e yi = y 0 i + α y 0 0 e lim yi = y 00 + α y 0 0 e 0 p0 p0 p0 q0 p0x + O 3.2 q0 p0x i + O, q0 p0i + O, p 2 0 q0 iρ p0 + O, wereρ = 3.3 At te grid pointx i, te proposed differential equation Eq. 1.1 may be discretized by M i = rx i px i y ix qx i yx i Te non standard finite differences of y i 1, y i+1 and y i are y i 1 y i+1 + 4y i 3y i 1 y i+1 3y i+1 4y i + y i 1 y i y i+1 y i 1. Substituting te values of M i, M i 1 and M i+1 along wit above differences in Eq. 2.2, we get 6 y 2 i+1 2y i + y i 1 = pi+1 + 2pi q i 1 + 3pi 1 y i 1 + 2pi+1 4q i 2pi 1 y i + 3pi+1 2pi q i 1 + pi 1 + r i+1 + 4r i + r i 1 Introducing fitting factor σρ in Eq. 3.4, we get 6σρ 2 y i+1 2y i + y i 1 = + 3pi+1 2pi q i 1 + pi 1 pi+1 + 2pi q i 1 + 3pi 1 + r i+1 + 4r i + r i 1 y i 1 + 2pi q i 2pi 1 y i 3.5 3

4 4 K. PHANEENDRA AND E. SIVA PRASAD Multiplying Eq. 3.5 by and taking limit as 0, we get σ p0 lim yi + 1 2yi + yi 1 = lim yi 1 yi ρ By substituting Eq. 3.3 in to Eq. 3.6, we get σ = ρ 4 p0 cot p0 2 q0 p0 ρ is te required fitting factor in te left end boundary layer. From Eq.3.5 we get te following tridiagonal system were E i y i 1 + F i y i + G i y i+1 = H i for i = 1, 2,..., N E i = σ+ 4 p i p i 12 p i q i 1, F i = 2σ 3 p i p i q i G i = σ + 12 p i 1 3 p i 4 p i q i+1, H i = [r i r i + r i+1 ] px i = p i,qx i = q i, rx i = r i, for i = 0, 1,...,N. We solve te tridiagonal system 3.8, using metod of invariant imbedding algoritm Rigt-end boundary layer. Furter, if we assume tat qx 0, px M < 0 trougout te interval [0, 1] were M is negative constant. Under tese assumptions, 1.1 as unique solution and display boundary layer at x =1 for small values of. From te teory of singular perturbation it is known tat te solution of Eq. 1.1 and Eq. 1.2 is of te form yx = y 0 x + p1 px α y 01 e x 0 px pxdx qx + O 3.9 were y 0 x is te solution of pxy 0x + qxy 0 x = rx,wit y 0 0 = α. Expanding px and qx about te point 1 up to first term using Taylor s series, Eq. 3.9 gives yx = y 0 x + α y 0 1 e yx i = y 0 x i + α y 0 1 e i.e., yi = y 0 i + α y 0 1 e p1 p1 p1 q1 p11 x + O 3.10 q1 p11 x i + O, q1 p11 i + O, terefore lim 00 + α y 0 1 e 0 p 2 1 q1 p1 1 iρ + O 3.11 were ρ = Now, we consider cubic spline finite difference sceme and introduce te fitting 4

5 NON STANDARD FDM FOR SPP USING CUBIC SPLINE 5 factor σρ as: 6σρ 2 y i+1 2y i + y i 1 = + 3pi+1 2pi q i 1 + pi 1 pi+1 + 2pi q i 1 + 3pi 1 y i 1 + 2pi+1 + r i+1 + 4r i + r i 1 Multiplying 3.12 by and taking limit as 0, we get 4q i 2pi 1 y i 3.12 σ p0 lim yi + 1 2yi + yi 1 = lim yi 1 yi ρ By substituting Eq in to Eq we get σ = ρ 4 p0 cot p1 2 q1 p1 ρ 2 is te fitting factor in te rigt end boundary layer. From Eq.3.12 we get te following tridiagonal system were 3.14 E i y i 1 + F i y i + G i y i+1 = H i for i = 1, 2,..., N E i = σ+ 4 p i p i 12 p i q i 1, F i = 2σ 3 p i p i q i G i = σ + 12 p i 1 3 p i 4 p i q i+1, H i = [r i r i + r i+1 ] px i = p i,qx i = q i, rx i = r i, for i = 0, 1,...,N To solve te above system 3.15, we used metod of invariant imbedding algoritm. 4. Convergence Analysis Incorporating te given boundary conditions we obtain te matrix equation as D + P Y + Q + T = σ σ σ 2σ σ were D = [ σ, 2σ, σ] = 0 σ 2σ σ σ 2σ and v 1 w z 2 v 2 w P = [z i, v i, w i ] = 0 z 3 v 3 w z N 1 v N 1 5

6 6 K. PHANEENDRA AND E. SIVA PRASAD were z i = 4 p i p i 12 p i q i 1, v i = 3 p i p i q w i = 12 p i 1 3 p i 4 p i q i+1fori = 1, 2, 3, 4...N 1 and Q = [q 1 + σ + z 1 α, q 2, q 3,..., q N 2, q N 1 + σ + w N 1 β] T were q i = 2 6 r i 1 + 4r i + r i+1 i = 1, 2, 3, 4,..., N 1 T = 0 4 and Y = [Y 1, Y 2, Y 3,..., Y N 1 ] T, T = [T 1, T 2,..., T N 1 ] T, O = [0, 0,..., 0] T are associated vectors of equation 4.1. Let y = [y 1, y 2,..., y N 1 ] T = Y wic satisfies te equation D + P y + Q = Lete i = y i Y i, i = 11N 1be te discretization error so tat E = [e 1, e 2,..., e N 1 ] T = y Y. Subtracting 4.1 from 4.2, we obtain te error equation D + P E = T 4.3 Let px C 1 and qx C 2 were C 1, C 2 are positive constants. If P i,j be te i, j t element of te matrix D + P, ten P i,i+1 σ + 3 C C 2, i = 1, 2,..., N 2 P i,i 1 σ + 3 C C 2i = 2, 3,..., N 1 Tus for sufficiently small i i.e. as 0 we ave P i,i+1 < σ, i = 1, 2,...,.N 2 P i,i 1 < σ, i = 2, 3,..., N Hence D + P is irreducible see Ref. [16]. Let S i be te sum of te elements of te it row of te matrix D + P, ten we ave S i = σ 4 p i 1 3 p i + 12 p i q i + q i+1 for i = 1 S i = 2 6 qi-1 + 4q i + q i+1 for i = 2, 3,..., N 2 S i = σ 12 p i p i + 4 p i q i + q i 1 for i = N 1 Let C 1 = min px and 1 i N C 1 = max px, C 2 1 i N = min qx and 1 i N C 2 = max qx. 1 i N Since 0 < << 1 and O,it is verify tat for sufficiently small, D + P is monotone [16, 17]. Hence D + P 1 exists and D + P 1 0. Tus from Eq.4.3, we ave E D + P 1 T 4.5 6

7 NON STANDARD FDM FOR SPP USING CUBIC SPLINE 7 For sufficiently small, we ave Let D + P 1 i,k D + P 1 = Since D + P 1 i,k Hence Furtermore, N 1 k=1 S i > 2 C 2 for i = 1 S i > 2 C 2 for i = n 1 and S i > 2 C 2 for i = 2, 3,..., n 2. be te i, kt element of D + P 1 and we define N 1 max 1 i N 1 k=1 N 1 0 and k=1 D + P 1 i,k and T = max T, i N 1 D + P 1 i,k.s k = 1 for i = 1, 2, 3,..., N 1. D + P 1 i,k 1 S i < 1 2 C 2, i = D + P 1 i,k 1 S i < 1 2 C 2, i = N D + P 1 i,k 1 min 2 i N 2 S i By te elp of Eqs , from 4.5, we obtain Hence te metod is second order convergence. < 1 2 C 2 for i = N E O Numerical examples Tis section presents test examples to demonstrate te efficiency of te metod computationally. We consider two numerical examples wit left-end boundary layer and one problem wit rigt-end boundary layer of te underlying interval were considered. Tese problems were cosen because tey ave been widely discussed in te literature and exact solutions were available for comparison. Te maximum absolute errors wit and witout fitting factor were presented to support te given metod. Example 1. [2], Consider te boundary value problem from Bender and Orszag y x + y x yx = 0; x [0, 1] wit y0=1 and y1=1. Clearly tis problem as a boundary layer at x = 1. Te exact solution is given by yx = [em2 1 e m1x + 1 e m1 e m2x ] e m2 e m1 were m 1 = /2 and m 2 = /2. Table 1 gives te maximum absolute errors for different values of and wit and witout fitting factor. 7

8 8 K. PHANEENDRA AND E. SIVA PRASAD Example 2. Consider te following non-omogeneous problem from fluid dynamics for fluid of small viscosity y x + y x = 1 + 2x; x [0, 1] wit y0 = 0 and y1 = 1. Te exact solution is given by yx = xx e x/ 1 e 1/. Table 2 gives te maximum absolute errors for different values of and wit and witout fitting factor. Example 3. Consider te boundary problem y x y x 1 + yx = 0; x [0, 1] wit y0 = 1 + exp 1 + / and y1 = 1 + 1/e Clearly, tis problem as a boundary layer at x = 1. Te exact solution is given by yx = e 1+x 1/ + e x Table 3 gives te maximum absolute errors for different values of and wit and witout fitting factor. 6. Summary and conclusions Non standard fitted finite difference metod for singular perturbation problems using cubic splines was developed. We ave introduced a fitting factor wic is called artificial viscosity in te spline difference sceme to control te rapid canges in te boundary layer. Convergence analysis of te metod is discussed and it sows tat our metod is second order convergent. We ave presented maximum absolute errors for te standard test examples cosen from literature wit te proposed metod wit and witout fitting factor. From te numerical results, it sows te importance of te fitting factor introduced in te spline difference sceme. References 1. Bellman,R.: Perturbation Tecniques in Matematics, Pysics and Engineering, Holt, Rineart & Winston, New York, Bender, C.M., Orszag, S.A.: Advanced Matematical Metods for Scientists and Engineers, McGraw-Hill, New York, Gartland, Jr., E.C.: Graded-mes difference sceme for singular perturbed two point boundary value problems, Mat. Comp Kadalbajoo, M.K., Reddy Y.N.: A non asymptotic metod for general singular perturbation problems, Journal of Optimization Teory and Applications, Kadalbajoo, M.K., Devendra Kumar: Initial value tecnique for singularly perturbed two point boundary value problems using an exponentially fitted finite difference sceme, Computers and Matematics wit Applications, Kevorkian, J., Cole, J.D.: Perturbation Metods in Applied Matematics, Springer-Verlag, New York, Kreiss, B. Kreiss, H.O.: Numerical metods for singular perturbation problems, SIAM Journal on Numerical Analysis

9 NON STANDARD FDM FOR SPP USING CUBIC SPLINE 9 Table 1. Maximum absolute errors in te solution of Example 1 δ\ Wit fitting factor Witout fitting factor Table 2. Maximum absolute errors in te solution of Example 2 δ\ Wit fitting factor Witout fitting factor Table 3. Maximum absolute errors in te solution of Example 3 δ\ Wit fitting factor Witout fitting factor

10 10 K. PHANEENDRA AND E. SIVA PRASAD 8. Lin, P., Vancouver: A Numerical solution of quasilinear singularly perturbed ordinary differential equation witout turning points, Applied Matematics and Mecanics, Miller,J.J.H., O Riordan, E. and Siskin, G.I.: Fitted Numerical Metods for Singular Perturbation Problems, World Scientific, River Edge, NJ Nayfe A.H.: Introduction to Perturbation Tecniques, Wiley, New York, O Malley, R.E.: Introduction to Singular Perturbations, Academic Press, New York, Reinardt, H.J.: Singular Perturbations of difference metods for linear ordinary differential equations, Applicable Anal., Reddy Y.N., Awoke A.: An Exponentially Fitted Special second order finite difference metod for singular perturbation problems, Applied Matematics and Computation, Rasidinia, J., Moammadi, R. and Gasemi, M.: Cubic spline solution of singularly perturbed boundary value problems wit significant first derivatives, Applied Matematics and Computation Siskins, G.I.: A difference sceme for a singular perturbed equation of parabolic type wit a discontinuous initial condition, Sov. Mat. Dokl Varga, R.S.: Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, Young, D.M.: Iterative Solutions of Large Linear Systems, Academic press, New York, K. Paneendra: Department of Matematics, University College of Science, Saifabad, Osmania University, Hyderabad, address: kollojupaneendra@yaoo.co.in URL: ttp:// jondoe optional E.Siva Prasad: Department of Matematics, Kavikulguru Institute of Tecnology & Science, Ramtek, Maarastra, address: emineni@yaoo.co.in 10