Solution of a Fourth Order Singularly Perturbed Boundary Value Problem Using Quintic Spline

Transcription

1 International Mathematical Forum, Vol. 7, 202, no. 44, Solution of a Fourth Order Singularly Perturbed Boundary Value Problem Using Quintic Spline Ghazala Akram and Nadia Amin Department of Mathematics University of the Punjab Lahore 54590, Pakistan toghazala2003@yahoo.com nadiyamen@hotmail.com Abstract Singularly perturbed boundary value problem can be solved using various techniques. The solution of the following fourth order self adjoint singularly perturbed boundary value problem is approximated using quintic spline Ly = ɛy (4) + p(x)y = f(x), p(x) p>0, y(a) = α 0, y(b) =α, y () (a) =α 2, y () (b) =α 3, } or Ly = ɛy (4) + p(x)y = f(x), p(x) p>0, y(a) = α 0, y(b) =α, y (2) (a) =α 4, y (2) (b) =α 5, } Convergence analysis of the method confirms second order convergence. The numerical description of the method is shown by two examples. Keywords: Singularly perturbed boundary value problems; Quintic spline; Self adjoint; Transition Layer; Uniform convergence Introduction The solution of singularly perturbed boundary value problem exhibits a multi scale character. Since there is a thin transition layer, where the solution varies rapidly, while away from the layer the solution behaves regularly and varies gradually, therefore many complications may be faced in solving singularly perturbed boundary value problems using standard numerical methods. In recent years, a large number of special purpose methods have been established to provide accurate results.

2 280 Ghazala Akram and Nadia Amin Consider the self adjoint singularly perturbed boundary value problem of the form: Ly = ɛy (4) } + p(x)y = f(x), p(x) p>0, y(a) = α 0, y(b) =α,y () (a) =α 2,y () (.) (b) =α 3, or Ly = ɛy (4) + p(x)y = f(x), p(x) p>0, y(a) = α 0, y(b) =α,y (2) (a) =α 4,y (2) (b) =α 5, } (.2) where α 0, α, α 2, α 3, α 4 and α 5 are constants and ɛ is a small positive parameter (0 <ɛ ), also f(x) and p(x) are smooth functions. In this problem p(x) =p = constant. Singularly perturbed problems are very famous in the field of science and engineering e.g., fluid dynamics, quantum mechanics, optimal control, convection -diffusion processes and chemical reactor theory etc. Singularly perturbed boundary value problems have been solved numerically using three basic techniques named as finite difference method, finite element method and spline approximation method. Second order singularly perturbed boundary value problem with boundary layer at one end is replaced by three first order initial value problems(ivps) and these problems have been solved using numerical patching method in []. Second order singularly perturbed boundary value problem has been converted into IVP for system of two first order ODEs which are solved using two numerical schemes namely, classical and exponentially fitted difference scheme and the adaptive single-step exponential fitting scheme in [5]. Various types of uniformly convergent mesh and finite difference schemes for the solution of second order singularly perturbed boundary value problem with numerical experiments have been discussed in [5]. A new approach has been defined to solve second order singularly perturbed boundary value problem in which the inner and outer region of the problem is considered as two point boundary layer correction and initial value problem respectively in [7]. A numerical method has been proposed to define second order semi linear singularly perturbed boundary value problem with violating stability condition described uniform convergence in [6]. Second order singularly perturbed boundary value problem has been solved using spline approach where singular and non singular cases are discussed in [0]. Second order singularly perturbed boundary value problem using difference scheme based on quintic spline has been considered, in which the domain of definition is converted into three non overlapping sub domains and this scheme shows fourth order accuracy in [4]. Second order singularly perturbed boundary value problem has been solved using fourth order method based on quintic spline in [2]. Sextic spline has been used to solve the second order singularly perturbed boundary value problem and the method

3 Solution of a fourth order singularly perturbed BVP 28 is proved to be fifth order accurate in [6]. A summary of linear, non linear second order two point singular boundary value problems, two point singularly perturbed BVPs in ordinary differential equations and singularly elliptic BVPs in partial differential equation has been discussed using various computational techniques in [8]. Third order singularly perturbed boundary value problem using quartic spline has been solved and the method is proved to be second order accurate []. Solution of third order singularly perturbed boundary value problem has been developed using sextic spline and the convergence analysis is shown to have third order convergence in [4]. Second and fourth order method have been developed using quintic non polynomial spline for the solution of fourth order two point boundary value problem in [9]. Third and fourth order singularly perturbed boundary value problem having discontinuous source term has been solved using asymptotic finite element method, where these problems are transformed into weakly coupled system according to the type of boundary conditions in [3]. Fourth order singularly perturbed boundary value problem has been converted into two linear and nonlinear ODEs in [2] with the suitable boundary conditions and the domain of definition is divided into two non overlapping sub intervals where the non linear equation has been solved using Newton s method of quasilinearization. Fourth order singularly perturbed two point boundary value problem has been reduced into two ODEs in [3] w.r.t suitable boundary conditions where the domain of definition is transformed into three non overlapping subintervals and Newton s method of quasilinearization is used for the solution of nonlinear equation. The paper is organized in four sections. In section 2, the consistency relation and end conditions required for the solution of BVP (.) and (.2) are determined. In section 3, the convergence analysis of the quintic spline method is discussed. Finally in fourth section, numerical results are illustrated. 2 Consistency Relations To develop the consistency relations the following fifth degree spline is considered: S i (x) =a i (x x i ) 5 +b i (x x i ) 4 +c i (x x i ) 3 +d i (x x i ) 2 +e i (x x i )+f i (2.) defined on [a, b], where x [x i,x i+ ] with equally spaced knots, x i = a + ih, i =0,,..., N, and h = (b a). N Using the following notations S i (x i ) = y i, S i (x i+ ) = y i+, S () i (x i ) = m i, S (2) i (x i ) = k i, S (4) i (x i ) = N i, S (4) i (x i+ ) = N i+.

4 282 Ghazala Akram and Nadia Amin the coefficients in (2.) can be determined, as a i = (N i N i+ ), 20h N i b i = 24, c i = (60h2 k i + 20hm i +4h 4 N i + h 4 N i+ + 20y i 20y i+ ), 20h 3 k i d i = 2, e i = m i, f i = y i. Applying the first, second and third derivative continuities at knots i.e S (µ) (x i ), for μ =, 2 and 3, the following relations can be obtained as S (µ) i i (x i )= 2k i k i + ( 360hm i +8h 4 N i +7h 4 N i 360y i + 360y i ) 60h 2 = 0, 20h 3 ( 60h2 k i +60h 2 k i 20hm i + 20hm i +6h 4 N i +3h 4 N i +h 4 N i+ 20y i + 240y i 20y i+ ) = 0, 2 hk i 2m i m i + 40 h3 N i + h3 N i 60 3y i h + 3y i h = 0, which leads the following consistency relation in terms of N i and y i h 4 N i 2 +26h 4 N i +66h 4 N i +26h 4 N i+ + h 4 N i+2 = 20y i 2 480y i + 720y i 480y i+ + 20y i+2, i =2, 3,..., N (2.2) Using Eq. (.), the Eq. (2.2) can be written as (ph 4 20ɛ)y i 2 + (26ph ɛ)y i + (66ph 4 720ɛ)y i + (26ph ɛ)y i+ +(ph 4 20ɛ)y i+2 = h 4 (f i 2 +26f i +66f i +26f i+ + f i+2 ), i =2,..., N 2.(2.3) Since above system consists of (N 3) equations with (N ) unknowns, so two more equations are required. The following relations describing truncation errors are used in this regard T = h 4 (a 0 N 0 + a N + a 2 N 2 + a 3 N 3 + a 4 N 4 + a 5 N 5 ) (a 6 y 0 + a 7 y +a 8 y 2 + a 9 y 3 + a 0 hy () 0 ), (2.4) T N = h 4 (b 0 N N + b N N + b 2 N N 2 + b 3 N N 3 + b 4 N N 4 ) +b 5 N N 5 (b 6 y N + b 7 y N + b 8 y N 2 + b 9 y N 3 + b 0 hy () N ). (2.5)

5 Solution of a fourth order singularly perturbed BVP 283 Using Taylor s series for the right hand side of Eq. (2.4) along with the coefficients of h 4 y 0,h 3 y () 0,h 2 y (2) 0,h y (3) 0,y (4) 0,hy (5) 0,h 2 y (6) 0,h 3 y (7) 0,h 4 y (8) 0,h 5 y (8) 0, the value of a i s can be calculated, as a 0 =, a = , a 2 = , a 3 = , a 4 = , a 5 = , a 6 = , a 7 = , a 8 = , a 9 = , a 0 = Using the values of a i s in Eq. (2.4), the required end conditions for i = and i = N have been determined, as (8240ph ɛ)y + (5990ph ɛ)y 2 + (40ph ɛ)y 3 35ph 4 y 4 +28ph 4 y 5 h 4 (937f f +5990f f 3 35f 4 +28f 5 ) + (937ph ɛ)α hɛα 2 + O(h 6 )=0. (2.6) (8240ph ɛ)y N + (5990ph ɛ)y N 2 + (40ph ɛ)y N 3 35ph 4 y N 4 +28ph 4 y N 5 h 4 (937f N f N +5990f N f N 3 35f N 4 +28f N 5 ) + (937ph ɛ)α hɛα 3 + O(h 6 )=0. (2.7) Similarly, the end conditions for the system (.2) can be derived as (43274ph ɛ)y + (5662ph ɛ)y 2 + (3432ph ɛ)y 3 39ph 4 y ph 4 y 5 h 4 (4233f f f f 3 39f f 5 ) + (4233ph ɛ)α h 2 ɛα 4 + O(h 6 )=0, (2.8) (43274ph ɛ)y N + (5662ph ɛ)y N 2 + (3432ph ɛ)y N 3 39ph 4 y N ph 4 y N 5 h 4 (4233f N f N +5662f N f N 3 39f N f N 5 ) + (4233ph ɛ)α 60480h 2 ɛα 5 + O(h 6 )=0. (2.9)

6 284 Ghazala Akram and Nadia Amin 3 Convergence of the Method The system of Eqns. (2.6), (2.3) and (2.7) provides the required quintic spline solution of BVP (.), which can be written in the following matrix form AY h 4 DF = C, (3.) where A =[A A 2 ] A = A 2 = 8240ph ɛ 5990ph ɛ 40ph ɛ 26ph ɛ 66ph 4 720ɛ 26ph ɛ ph 4 20ɛ 26ph ɛ 66ph 4 720ɛ ph 4 20ɛ 26ph ɛ ph 4 20ɛ 28ph 4 35ph 4 D = 35ph 4 28ph 4 ph 4 20ɛ 26ph ɛ ph 4 20ɛ ph 4 720ɛ 26ph ɛ ph 4 20ɛ 26ph ɛ 66ph 4 720ɛ 26ph ɛ 40ph ɛ 5990ph ɛ 8240ph ɛ C =(c,c 2,..., c N 2,c N ) T, Y =(y,y 2,..., y N 2,y N ) T and F =(f,f 2,..., f N 2,f N ) T. Also c = (937ph ɛ)α h 4 f hɛα 2, c 2 = (ph 4 20ɛ)α 0 + h 4 f 0, c i = 0, i =3, 4,..., N 3, c N 2 = (ph 4 20ɛ)α + h 4 f N, and c N = (937ph ɛ)α + 937h 4 f N 00800hɛα 3.,,,

7 Solution of a fourth order singularly perturbed BVP 285 If Ȳ =[y(x ),y(x 2 ),...,y(x N )] T denotes the exact solution then from Eq. (3.), it can be written as AȲ h4 DF = T + C, (3.2) where T =[t,t 2,...,t N ] T denotes the truncation error and calculated, as t = ɛh6 y (0) (ζ ), x 0 <ζ <x 2, t i =0ɛh 6 y (6) (ζ i ), x i <ζ i <x i+, i =2, 3,..., N 2, (3.3) and t N = ɛh6 y (0) (ζ N ), x N 2 <ζ N <x N. Moreover, A(Ȳ Y ) = AE = T, (3.4) E = Ȳ Y = (e,e 2,..., e N ) T. (3.5) To determine the error bound the row sums S,S 2,..., S N of matrix A are calculated, as S = j a,j = 24263ph ɛ, S 2 = j a 2,j = 9ph ɛ, S i = j a i,j = 20ph 4, i =3, 4,..., N 3, S N 2 = j a N 2,j = 9ph 4 (3.6) + 20ɛ, and S N = j a N,j = 24263ph ɛ. Since matrix A has been observed to be irreducible and monotone, therefore A exists and its elements are non negative. Hence following results can be obtained from Eq. (3.4) Also, from the theory of matrices it can be written as where A = E = A T. (3.7) A A = I (N ) (N ), (3.8) a, a,2 a,n a 2, a 2,2 a 2,n.... a n, a n,2 a n,n, (3.9) A = a, a,2 a,n a 2, a 2,2 a 2,n. a n,... a n,2 a n,n (3.0)

8 286 Ghazala Akram and Nadia Amin and I =..... (3.) 0 0 Since each row sum of matrix I (N ) (N ) = and A A = I (N ) (N ), therefore each row sum of A A equals to. i.e. a, (a, + a,2 + + a,n )+a,2 (a 2, + a 2,2 + + a 2,n ) + + a,n (a n, + a n,2 + + a n,n ) = a,s + a,2s a,n S N = which can be written in compact form as N i= a k,i S i =, k =, 2,..., N. (3.2) If S j = mins i, then from Eq. (3.2), it can be written as It follows that where, N i= S j (a k, + a k, a k,n ). a k,i min S i = (h 4 B io ), (3.3) B io =( h 4 )min S i > 0, i 0 N. From Eq. (3.4), it can be written as e j = N i= a j,i T i, j =, 2,..., N. (3.4) From Eq. (3.3) and Eq. (3.4), it can be proved that e j Kh2, j =, 2,..., N, B io where K is constant and independent of h. It follows that, E = O(h 2 ).

9 Solution of a fourth order singularly perturbed BVP 287 Similarly, the method developed for the system (2.3), (2.8) and (2.9) is also second order convergent. The result can be summarized in the following theorem Theorem Let Ȳ (x) be the exact solution of the system (.) or (.2) and let y i, i = 0,,..., N be the exact solution of (3.) then E = O(h 2 ). 4 Numerical Results Example : The following boundary value problem is considered, as ɛy (4) + py = ɛx 4 (32ɛ 2 x( 6(7 55x 4 +70x 8 )+ɛ 2 (x 2 3x 6 +2x 0 )) cosɛx +(x 4 (x 4 ) 2 ɛ 5 x 4 (x 4 ) 2 x [0, ] +48ɛ 3 x 2 (7 33x 4 +30x 8 ) 240ɛ(7 99x x 8 ))sinɛx), y() = 0, y () () = 0, y( ) = 0, y () ( ) = 0. (4.) The analytical solution of the problem (4.) is, y(x) =ɛx 8 (x 4 ) 2 sin ɛx. The observed maximum errors associate with y i s for Example, corresponding to different values of ɛ are summarized in Table. It is noted from the Table that if h is reduced by factor, then E is reduced by factor, which 2 4 indicates that the present method gives second order results. Table. ɛ h = h = h = h = e e e e e e e e e e e e Example 2: The following boundary value problem is considered, as ɛy (4) + py = ɛ(2x 4 + cosx ɛ(48 + cosx)), x [, ], y( ) = ɛ(2 + cos), y (2) ( ) = ɛ(24 cos), y() = ɛ(2 + cos), y (2) () = ɛ(24 cos). (4.2)

10 288 Ghazala Akram and Nadia Amin The analytical solution of the problem (4.2) is, y(x) =ɛ(2x 4 + cosx). The observed maximum errors associate with y i s for Example 2, corresponding to different values of ɛ are summarized in Table 2. Table 2. ɛ h = 6 h = 32 h = 64 h = e e e e e e e e e e e e It is confirmed from the Table 2 that if h is reduced by factor, then E is 2 reduced by factor, which indicates that the present method gives second order 4 results. 5 Conclusion Quintic spline method is developed for the approximate solution of fourth order singularly perturbed boundary value problem. In addition to the boundary conditions corresponding to the st derivatives, the boundary conditions corresponding to the 2nd derivatives are also considered. The method has been proved to be second order convergent. Two examples are considered for numerical illustration of the method. It is also observed that the results of these examples preserve O(h 2 ). References [] Ghazala Akram, Quartic spline solution of third order singularly perturbed boundary value problem, Submitted in ANZIAM. [2] T. Aziz and A. Khan, Quintic spline approach to the solution of a singularly-perturbed boundary-value problem, Optimization Theory and Applications 2 (2002), [3] A. Ramesh Babu and N. Ramanujam, An asymptotic finite element method for singularly perturbed third and fourth order ordinary differential equations with discontinuous source term, Applied Mathematics and Computation 9 (2007), [4] R. K. Bawa and S. Natesan, A computational method for self-adjoint singular perturbation problems using quintic spline, Computers and Mathematics with Applications 50 (2005),

11 Solution of a fourth order singularly perturbed BVP 289 [5] Dragoslav Herceg, Katarina Surla, Ivana Radeka, and Helena Malicic, Numerical experiments with different schemes for a singularly pertured problem, Advances in Engineering Software 3 (200), [6] Arshad Khan, Islam Khan, and Tariq Aziz, Sextic spline solution of a singularly perturbed boundary value problems, Applied Mathematics and Computation 98 (2008), [7] Manoj Kumar, Hradyesh Kumar Mishra, and Peetam Singh, A boundary value approach for a class of linear singularly perturbed boundary value problems, Advances in Engineering Software 40 (2009), [8] Manoj Kumar and Neelima Singh, A collection of computational techniques for solving singular boundary-value problems, Advances in Engineering Software 40 (2009), [9] M. A. Ramadan, I. F. Lashien, and W. K. Zahra, Quintic nonpolynomial spline solutions for fourth order two point boundary value problems, Communications in Nonlinear Science and Numerical Simulation 4 (2009), [0] J. Rashidinia, M. Ghasemi, and Z. Mahamoodi, Spline approach to the solution of a singularly-perturbed boundary-value problems, Applied Mathematics and Computation 89 (2007), [] Y. N. Reddy and P. Pramod Chakravarthy, An initial-value approach for solving singularly perturbed two-point boundary value problems, Applied Mathematics and Computation 55 (2004), [2] V. Shanthi and Ramanujam, A boundary value technique for boundary value problem for singularly perturbed fourth-order ordinary diffrential equations, Computers and Mathematics with Applications 47 (2004), [3] V. Shanthi and N. Ramanujam, A numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations, Applied Mathematics and Computation 29 (2002), [4] Shahid S. Siddiqi and Ali Tabraiz, Sextic spline solution of a third order singularly perturbed boundary value problem, Submitted in Advances in Numerical Analysis. [5] J. Vigo-Aguiar and S. Natesan, An efficient numerical method for singular perturbation problems, Computational and Applied Mathematics 92 (2006), 32 4.

12 290 Ghazala Akram and Nadia Amin [6] Relja Vulanovic, A uniform convergence result for semilinear singular perturbation problems violating the standard stability condition, Applied Numerical Mathematics 6 (995), Received: April, 202