New Fourth Order Quartic Spline Method for Solving Second Order Boundary Value Problems

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1 MATEMATIKA, 2015, Volume 31, Number 2, c UTM Centre for Industrial Applied Matematics New Fourt Order Quartic Spline Metod for Solving Second Order Boundary Value Problems 1 Osama Ala yed, 2 Te Yuan Ying 3 Azizan Saaban 1,2,3 Scool of Quantitative Sciences, UUM College of Arts Sciences, Universiti Utara Malaysia, UUM Sintok, Keda Darul Aman, Malaysia 1 alayedo@yaoo.com, 2 yuanying@uum.edu.my, 3 azizan.s@uum.edu.my Abstract In tis article, a fourt order quartic spline metod as been developed to obtain te numerical solution of second order boundary value problem wit Diriclet boundary conditions. Te development of te quartic spline metod convergence analysis ave been presented. Tree test problems ave been used for numerical experimentations purposes. Numerical experimentations sowed tat te quartic spline metod generates more accurate numerical results compared wit an existing cubic spline metod in solving second order boundary value problems. Keywords Boundary value problem; spline interpolation; quartic spline metod; sooting metod Matematics Subject Classification 65L10, 65L20, 65D07, 41A15, 41A25. 1 Introduction Boundary value problems (BVPs are arising frequently in various fields of sciences engineering. Generally, it is very difficult to solve tese types of problems analytically. Hence, numerous numerical metods ave been developed to find te approximate solutions for tese problems. One of tese metods is called te spline metod. Te use of spline metod for solving BVPs was first discussed by Bickley in 1968, [1]. Following is work, many researcers started using spline metods to approximate BVPs. For instance, Caglar et al. [2], Ramadan et al. [3], Rasidinia et al. [4], Al-Said et al. [5], Hamid et al. [1,6] Fauzi Sulaiman [7] ave used different degrees of splines to approximate second order BVPs. Most of tese researcers used teir spline metods to approximate special cases of BVPs suc as, linear BVPs BVPs witout te presence of te first derivative. In our work, owever, we are considering te general second order BVPs of te form subject to te boundary conditions u = f (x, u, u, a x b, (1 u (a = α, u (b = β. (2 Keller [8] sown tat problem (1 togeter wit te boundary conditions in (2, as a unique solution if f (x, u, u satisfies te following conditions: (i f (x, u, u is continuous on a domain Ω, were te domain Ω is defined as Ω = {(x, u, u a x b, < u <, < u < }; (ii f f exist continuous for all (x, u, u Ω; u u

2 150 Osama Ala yed, Te Yuan Ying Azizan Saaban (iii f u > 0 f u W, for some positive constant W. However, in te rest of tis discussion, we ave to assume tat u C 5 [a, b]. Te main objective of our researc is to introduce a new quartic spline metod to approximate te second order BVPs as in (1. Tis paper is organized as follows. In Section 2, a quartic spline metod is constructed. Section 3 discussed te convergence of te proposed metod. To sow te performance of te proposed metod for comparison purposes, some numerical examples are given in Section 4. Finally, te conclusion is given in Section 5. 2 Quartic Spline Let P be te partition for te interval [a, b] suc tat P : a = x 0, x 1,..., x n = b, were x i = a +i = b a. We assumed tat our quartic spline function as to satisfy n te following conditions: (i S (x = s i (x, x [x i, x i+1 ], i = 0, 1, 2,..., n 1; (ii S (a = u (a, S (b = u (b; (iii s (r i (x i+1 = s (r i+1 (x i+1, r = 0, 1, 2, 3. We let u (x be te exact solution of problem (1 s i be te approximate solution to u i = u (x i obtained by te quartic spline s i (x on te interval [x i, x i+1 ]. Since our spline is of degree four, te tird derivative is a linear polynomial, wic can be written as follows s i (x = Z i+1 (x x i (x i+1 x + Z i, (3 were Z i = s i (x, x [x i, x i+1 ]. On integrating equation (3 tree times, we obtain s i (x = Z i+1 (x x i 4 24 Z i (x i+1 x A i (x x i 2 + B i (x i+1 x + C i (x x i, (4 were A i, B i C i, i = 0, 1, 2,..., n 1, are coefficients wic need to be determined in terms of u i, u i+1, µ i Z i. In order to derive explicit expressions for te tree coefficients of equation (4, we define te following relations: u i = s i (x i, (5 u i+1 = s i (x i+1, (6 µ i = s i (x i. (7 From equations (5, (6 (7, by using straigtforward calculation, we obtain te following equations: A i = µ i Z i, (8

3 New Fourt Order Quartic Spline Metod 151 C i = u i+1 B i = u i Z i, (9 2 4 Z i 2 24 Z i+1 2 µ i. (10 Now, we impose te first second continuity conditions of quartic spline s i (x at te point x i+1 i.e. s (r i (x i+1 = s (r i+1 (x i+1, r = 1, 2, te following relations are obtained Z i Z i Z i µ i + 2 µ i+1 = u i 2u i+1 + u i+2, (11 2 Z i + 2 Z i+1 + µ i µ i+1 = 0. (12 Ten, we eliminate µ i+1 from equations (11 (12 to obtain µ i = u i 2u i+1 + u i Z i Z i+1 24 Z i+2. (13 On substituting equation (13 into equation (11, we obtain te following main recurrence relation given by Z i + 11Z i Z i+2 + Z i+3 = 24 3 ( u i + 3u i+1 3u i+2 + u i+3,i = 0, 1, 2,..., n 3. (14 Equation (14 forms a system of n 2 equations wit n + 1 unknowns, wic are te Z i, i = 0, 1, 2,..., n. To solve tis system uniquely, we ave to add tree more conditions at te end points i.e. x 0 x n. Hence, we coose Z 0 = Z n = µ 0 = 0. To obtain te last equation, we substitute Z 0 = 0 µ 0 = 0 in equation (13 to obtain 12Z 1 + Z 2 = 24 2 (u 0 2u 1 + u 2. (15 Equations (14 (15 form a system of n 1 equations wit n 1 unknowns. Tese unknowns can be solved using te MATHEMATICA software. Finally, to construct an algoritm for te quartic spline metod, we can use te following steps: Step 1: Divide te interval [a, b] into n 1 subintervals by taking x i = a + i, were = 1/n i = 0, 1, 2,..., n. Step 2: Apply sooting metod wit te fourt order explicit Runge-Kutta metod to problem (1, to obtain te approximate solution u i at te grid points. Step 3: Use equations (14 (15 to form a system of linear equations, ten solve for te values of A i, B i C i for i = 0, 1, 2,..., n 1. Step 4: Use te values of A i, B i, C i, Z i u i obtained from Step 2 Step 3 to construct te quartic spline solution s i (x in equation (4, to approximate te solution of problem (1.

4 152 Osama Ala yed, Te Yuan Ying Azizan Saaban 3 Convergence Analysis Let s i (x given by equation (4, denotes te quartic spline using te exact values u i, µ i Z i. Also, let s i (x denotes te quartic spline constructed using ũ i, µ i Z i, were ũ i is te approximate solution of problem (1 obtained by te sooting metod wit fourt order explicit Runge-Kutta metod; wile µ i Z i are te second te tird derivative of te function s i (x at te point (x i, ũ i, respectively. Ten, s i (x is given by were s i (x = Z i+1 (x x i 4 24 Z i (x i+1 x 4 24 C i = ũi+1 + Ãi (x x i 2 + B i (x i+1 x + C i (x x i, Ã i = µ i Z i, B i = ũi Z i, 2 4 Z i 2 24 Z i+1 2 µ i, for x [x i, x i+1 ]. Assume tat e(x defines te error between te exact solution u (x te spline function s i (x for problem (1 given by (16 e(x = u (x S (x,x [a, b]. (17 It is easy to verify tat we can rewrite te error function e(x as follows [ ] e(x = [u (x S (x] + S (x S (x e(x = e I (x + e D (x, (18 were e I (x is te error caused by spline interpolation e D (x is te error caused by te discretization of problem (1. Now, to estimate e(x, we ave to estimate e I (x e D (x separately. Since our spline is a polynomial of degree four, ten we can write e I (x over te interval [x i, x i+1 ] as u (x s i (x = u(5 (ζ i 5! (x x i 2 (x x i 1 (x x i (x x i+1 (x x i+2, (19 for some ζ i [x i, x i+1 ]. We recalled tat every subinterval as lengt of, if we let t = x x i, ten equation (19 can be rewritten as u (x s i (x = u(5 (ζ i 5! (2 + t( + t(t ( t(2 t. (20 Calculation on te expression (2 + t( + t(t( t(2 t in equation (20 sows tat 15+ it as maximum value at t = 145, it is equal to Ten, u (x s 10 i (x is bounded by u (x s i (x u (5 (ζ i. (21

5 New Fourt Order Quartic Spline Metod 153 Let W 5 = max x [a,b] u (5 (x. Terefore, it is easy to conclude tat e I (x W 5 5. (22 In order to estimate te error function e D (x, we can subtract equation (16 from equation (4 to obtain s i (x s i (x = (Z i+1 Z (x xi 4 i+1 24 (Z i Z ( (xi+1 x 4 i 24 + A i Ãi (x x i 2 + (B i B i (x i+1 x + (C i C i (x x i, (23 for x [x i, x i+1 ]. Let X = (x 0, x 1,..., x n T, U = (u 0, u 1,..., u n T, Ũ = (ũ 0, ũ 1,..., ũ n T, µ = (µ 0, µ 1,..., µ n T, µ = ( µ 0, µ 1,..., µ n T, Z = (Z 0, Z 1,..., Z n T Z = ( Z 0, Z 1,..., Z n T. From equation (23, it is easy to see tat e D (x U Ũ + 2 µ µ + 3 Z 2 Z. (24 We first estimate µ µ. We use equation (13 to obtain µ i µ i = ui ũi 2 ui+1 ũi ui+2+ũi (Z i Z i (Z i+1 Z i+1 24 (Z i+2 Z i+2. Terefore, from equation (25, we obtain µ µ 3 U Ũ 2 + Z Z. (26 2 On substituting equation (26 into equation (24, we obtain e D (x 4 U Ũ + 3 Z Z. (27 Next, to estimate Z Z, we let Q = (q i,j to denote a matrix wit q 1,1 = 12, q 1,2 = 1, q i,i = 15, i = 2, 3,..., n 1, q i,i+1 = q i,i 2 = 1, i = 2, 3,..., n 2, q i,i 1 = 11, i = 2, 3,..., n 1. We also let J = (j m,l to denote a matrix wit j 1,1 = 2, j 1,2 = 1, j m,m = 3, m = 2, 3,..., n 1, j m,m+1 = 1, m = 2, 3,..., n 2, j m,m 1 = 3, m = 2, 3,..., n 1, j m,m 2 = 1, m = 2, 3,..., n 2. (25

6 154 Osama Ala yed, Te Yuan Ying Azizan Saaban Let ψ = 24 (u 3 0, u 0, 0,..., 0, u n T, ten te system (14 can be rewritten in a matrix form as QZ = 24 3JU + ψ. (28 From equation (28, we obtain Q Z = 24 3JŨ + ψ + τ (, (29 were τ ( = (τ 0 (, τ 1 (,..., τ n ( T is te error in te tird derivative due to te discretization. On subtracting equation (29 from equation (28, we obtain ( Q Z Z = 24 ( 3J U Ũ τ (. (30 Since u 0 = ũ 0 u n = ũ n, ten it is not difficult to sow tat τ 0 ( = τ n ( = 0, (31 τ i ( = 3 2 u(4 (ζ i, (32 for ζ i (x i, x i+1. From equations (31 (32, it follows tat τ ( 3c 1 2, (33 were c 1 = max u (4 (ζ. Since Q is strictly diagonally dominant matrix, ten Q 1 a ζ b exists, Q 1 1 2, Q = 28 J = 8. Togeter wit equations (30 (33, we obtain Z Z 96 U 3 Ũ c 1. (34 From equations (27 (34, we obtain e D (x 100 U Ũ c 1 4. (35 In order to estimate U Ũ, we may assume te following result proved by Cawla Subramanian [9]. Teorem 1 Assume tat u (x is sufficiently smoot. Ten tere exist a constant c independent of suc tat U Ũ c 4. Terefore, from equation (34 Teorem 1, we arrived at e D (x c 2 4, (36 were c 2 = 100c+ 3c1 4. Finally, from equation (18 togeter wit our findings from equations (22 (36, we obtain te following result.

7 New Fourt Order Quartic Spline Metod 155 Teorem 2 Wit te assumptions of Teorem 1, our proposed quartic spline metod S (x as described in Section 2, provides order 4 uniformly convergent approximations for te solution u (x of problem (1, tat is were c 3 = (b aw5 5 + c 2. 4 Numerical Experiments e(x e I (x + e D (x c 3 4, In tis section, we implemented te proposed metod on tree examples of te second order BVPs. We denote max u ( ( x i+1/2 si xi+1/2 as te maximum absolute errors 0<i<n 1 between te nodal points tat are tabulated in Table 1 for step size equal to 0.1. We also compared our results wit tose obtained by te cubic spline metod developed by Cawla Subramanian [9]. Problem 1 [10] Consider te following linear second order BVPs u (x = 2 x u (x + 2 sin (lnx x2u (x + x 2, u (1 = 1, u (2 = 2, 1 x 2. Te exact solution for Problem 1 is given by u (x = c 1 x + c2 were c 2 = 1 70 (8 12 sin(ln 2 4 cos(ln2 c 1 = c 2. Problem 2 [10] Consider te following nonlinear second order BVPs x u (x = 2u (x 3, u ( 1 = 1 2, u (0 = 1, 1 x 0. 3 Te exact solution for Problem 2 is given by u (x = 1 x+3. Problem 3 [11] Consider te following second order Bratu type equation u (x + 2e u(x = 0, u (0 = u (1 = 0, 0 x 1. Te exact solution for Problem 3 is given by ( cos ( (x 0.5 u (x = 2 ln. cos ( sin (lnx 10 cos (ln x, From Table 1, we observed tat our proposed metod te existing cubic spline metod by Cawla Subramanian [9] are found to ave comparable accuracy in solving Problem 1 Problem 3. However, our proposed metod is more accurate tan te existing cubic spline metod in solving Problem 2.

8 156 Osama Ala yed, Te Yuan Ying Azizan Saaban Table 1: Maximum Absolute Errors for Problem 1, Problem 2 Problem 3 Problem Metods Maximum absolute errors 1 Cawla Subramanian [9] Te proposed metod Cawla Subramanian [9] Te proposed metod Cawla Subramanian [9] Te proposed metod Conclusion In tis article, we ave presented a new quartic spline metod for te numerical solution of second order BVPs in (1. An algoritm to apply te new metod is presented as well. Convergence analysis sowed tat te order of convergence of te new metod is 4. We ave cosen tree test problems to evaluate te effectiveness of te proposed metod; compared wit an existing metod by Cawla Subramanian [9] in terms of numerical accuracy. Numerical experimentations seemed to indicate tat te new proposed metod is reliable may generate more accurate numerical results in solving second order BVPs in (1. References [1] Hamid, N. N. A., Majid, A. A. Ismail, A. I. M. Extended cubic B-spline metod for linear two-point boundary value problems. Sains Malaysiana (11: [2] Caglar, H., Caglar, N. Elfaituri, K. B-spline interpolation compared wit finite difference, finite element finite volume metods wic applied to two-point boundary value problems. Applied Matematics Computation (1: [3] Ramadan, M. A., Lasien, I. F. Zara, W. K. Polynomial nonpolynomial spline approaces to te numerical solution of second order boundary value problems. Applied Matematics Computation (2: [4] Rasidinia, J., Moammadi, R. Jalilian, R. Cubic spline metod for two-point boundary value problems. International Journal of Engineering Science : [5] Al-Said, E. A., Noor, M. A., Almualim, A. H., Kokkinis, B. Coletsos, J. Quartic spline metod for solving second-order boundary value problems. International Journal of Pysical Sciences (17: [6] Hamid, N. N. A., Majid, A. A. Ismail, A. I. M. Quartic B-spline interpolation metod for linear two-point boundary value problem. World Applied Sciences Journal :

9 New Fourt Order Quartic Spline Metod 157 [7] Fauzi, N. I. M. Sulaiman, J. Half-sweep modified successive overrelaxtion metod for solving second order two-point boundary value problems using cubic spline. International Journal of Contemporary Matematical Sciences (32: [8] Keller, H. B. Numerical Metods for Two-Point Boundary-Value Problems. Massacusetts: Blaisdell Publising Company [9] Cawla, M. M. Subramanian, R. A new fourt-order cubic spline metod for second-order nonlinear two-point boundary-value problems. Journal of Computational Applied Matematics (1: [10] Burden, R. L. Faires, J. D. Numerical Analysis. 7t Edition. Boston: Prindle Weber Scmidt [11] Kuri, S. A. A new approac to Bratu s problem. Applied Matematics Computation (1: