QUINTIC SPLINE SOLUTIONS OF FOURTH ORDER BOUNDARY-VALUE PROBLEMS

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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 5, Number, Pages 0 c 2008 Institute for Scientific Computing Information QUINTIC SPLINE SOLUTIONS OF FOURTH ORDER BOUNDARY-VALUE PROBLEMS SHAHID S. SIDDIQI AND GHAZALA AKRAM Dedicated to the memory of Dr. M. Rafique Abstract. In this paper quintic spline is used for the numerical solutions of the fourth order linear special case boundary value problems. End conditions for the definition of spline are derived, consistent with the fourth order boundary value problem. The algorithm developed approximates the solutions, their higher order derivatives. It has also been proved that the method is a second order convergent. Numerical illustrations are tabulated to demonstrate the practical usefulness of method. Key Words. Quintic spline; Boundary value problem; End conditions.. Introduction Spline functions are used in many areas such as interpolation, data fitting, numerical solution of ordinary partial differential equations. Spline functions are also used in curve surface designing. Usmani [7], considered the fourth order boundary value problem to be the problem of bending a rectangular clamped beam of length l resting on an elastic foundation. The vertical deflection w of the beam satisfies the system (. ) [ L + ( K D )] w = D q(x), L d4 dx 4, (. 2) w(0) = w(l) = w (0) = w (l) = 0, D is the flexural rigidity of the beam, is the spring constant of the elastic foundation the load q(x) acts vertically downwards per unit length of the beam. Mathematically, the system (.) belongs to a general class of boundary value problems of the form (. 3) ( d 4 dx 4 ) + f(x) y(x) = g(x), x [a, b], y(a) = α 0, y(b) = α, y (a) = γ 0, y (b) = γ, α i, γ i ; i= 0, are finite real constants the functions f(x) g(x) are continuous on [a,b]. The analytic solution of (.3) for special choices of f(x) g(x) are easily obtained, but for arbitrary choices, the analytic solution cannot be determined. Numerical methods for obtaining an approximation to y(x) are introduced. Usmani [7] derived numerical techniques of order 2, 4 6 for solution of a fourth Received by the editors July 4, 2004, in revised form December 0, Mathematics Subject Classification. 65L0. 0

2 02 SHAHID S. SIDDIQI AND GHAZALA AKRAM order linear boundary value problem. Usmani [8] derived cubic, quartic, quintic sextic spline solution of nonlinear boundary value problems. Usmani Saai [9] developed a quartic spline for the approximation of the solution of third order linear (special case) two point boundary value problems involving third order linear differential equation. Papamichael Worsey [4] derived end conditions for cubic spline interpolation at equally spaced nots. Papamichael Worsey [5] have developed a cubic spline method, similar to that proposed by Daniel Swartz [3] for second order problems. Siddiqi Twizell [0-3] presented the solutions of 6, 8, 0 2 order linear boundary value problems, using the sixth, eighth, tenth twelfth degree splines. In this paper a quintic spline method is described for the solution of (.3). The end conditions for quintic spline interpolation, at equally spaced nots are derived, which is discussed in the next section. 2. Quintic Spline Let Q be a quintic spline defined on [a, b] with equally spaced nots (2. 4) x i = a + ih, i = 0,, 2,...,, (2. 5) h = b a. Moreover, for i = 0,, 2,...,, taing (2. 6) Q(x i ) = y i, Q () (x i ) = m i, (2. 7) Q (2) (x i ) = M i, Q (3) (x i ) = n i (2. 8) Q (4) (x i ) = N i. Also, let y(x) be the exact solution of the system (.3) y i be an approximation to y(x i ), obtained by the quintic spline Q(x i ). It may be noted that the Q i (x), i =, 2, 3,..., can be defined on the interval [x i, x i ], integrating (2. 9) Q 4 i (x) = h [N i (x i x) + N i (x x i )], four times w.r.t. x, gives (2. 0) Q i (x) = 20h [N i (x i x) 5 + N i (x x i ) 5 ] + Ax3 6 + Bx2 2 + Cx + D. To calculate the constants of integrations, the following conditions are used: (2. ) Q i (x i ) = y i, Q (2) i(x i ) = M i, Q i (x i ) = y i, Q (2) i(x i ) = M i. The identities of quintic splines for the solution of (.3) can be written as m i m i + 66m i + 26m i+ + m i+2 = 5 h [ y i 2 0y i + 0y i+ + y i+2 ] ; (2. 2) i = 2, 3,..., 2,

3 QUINTIC SPLINE SOLUTIONS OF FOURTH ORDER BOUNDARY-VALUE PROBLEMS 03 (2. 3) (2. 4) M i M i + 66M i + 26M i+ + M i+2 = 20 h 2 [y i 2 + 2y i 6y i + 2y i+ + y i+2 ]; n i n i + 66n i + 26n i+ + n i+2 = h 3 [ y i 2 + 2y i 2y i+ + y i+2 ]; N i N i + 66N i + 26N i+ + N i+2 (2. 5) = 20 h 4 [y i 2 4y i + 6y i 4y i+ + y i+2 ]; for i = 2, 3,..., 2, (2. 6) (2. 7) N i + 4N i + N i+ 6 h 2 [M i 2M i + M i+ ] = 0; h{m i + 2m i + m i+ } h 3 {3n i + 4n i + 3n i+ } = 20{ y i + y i+ }; (2. 8) for i =, 2, 3,...,, 8h{m i+ m i } h 2 {M i 6M i + M i+ } = 20{y i 2y i + y i+ }; m i = h 6 {2M i + M i } h3 3 {8N i + 7N i } + h {y i y i }; (2. 9) i =, 2, 3,...,, m i = h 6 {2M i + M i+ } + h3 3 8N i + 7N i+ + h {y i+ y i }; (2. 20) i = 0,,,...,, m i = h2 20 {n i + 8n i + n i+ } + 2h {y i+ y i }; (2. 2) i =, 2, 3,...,, M i = h2 20 {N i + 8N i + N i+ } + h 2 {y i 2y i + y i+ }; (2. 22) i =, 2, 3,...,, M i = 32h {m i m i 32m i+ m i+2 } h 2 {y i 2 + 6y i 34y i + 6y i+ + y i+2 }; (2. 23) i = 2, 3,..., 2 N i = 3 2h 2 {M i + 8M i + M i+ } + 30 h 4 {y i 2y i + y i+ }; (2. 24) i =, 2, 3,...,. The relations (2.2) (2.24) can be derived from the results of Albasiny Hosins [] Ahlberg, Nilson Walsh [2] discussed by Papamichael Behforooz [6]. The uniqueness of Q can be established showing that any of the four (+) (+) linear systems, obtained, using one of the relations (2.2), (2.3), (2.4) (2.5) together with the four end conditions of Q, is non singular. In this paper the linear system corresponding to (2.5) has been chosen has a unique solution

4 04 SHAHID S. SIDDIQI AND GHAZALA AKRAM for N i s, i = 0,,...,. Equation (2.6) (2.22) give the parameters M i s, i = 0,,...,. Consider the system (2.5) (2. 25) N i N i + 66N i + 26N i+ + N i+2 = 20 h 4 [y i 2 4y i + 6y i 4y i+ + y i+2 ]; i = 2, 3,..., 2, (2. 26) N i = f i y i + g i, i = 0,,...,. The above system gives ( 3) linear algebraic equations in the ( ) unnowns (y i, i =, 2,..., ). Two more equations are needed to have complete solution of y i s, which are derived in section 3. It may be noted that Papamichael Worsey [5] needed two consistency systems for the solution of boundary value problem (.3) but the method developed in this paper, needs only one such system. Papamichael Behforooz [6] proved the following lemma to determine the end conditions in terms of first derivative of interpolatory quintic spline. Lemma Let λ i = m i y () i. If y C 7 [a, b] then (2. 27) λ i λ i + 66λ i + 26λ i+ + λ i+2 = β i, i = 2, 3,..., 2, β i 2 h6 y (7), i = 2, 3,..., 2. Following the above lemma along with equation (2.5) with Taylor series expansion about the point x i, the following lemma can easily be proved, to determine the end conditions for the solution of boundary value problem (.3). Lemma 2 Let (2. 28) λ i = N i y (4) i, i = 0,,...,, then for y C 6 [a, b], (2. 29) λ i λ i + 66λ i + 26λ i+ + λ i+2 = β i, i = 2, 3,..., 2, while β i 58 3 h2 y (6), i = 2, 3,..., 2. Finally the required end conditions are derived in the following section. 3. End Conditions Consider the end conditions of the form (3. 30) N 0 + αn + βn 2 + γn 3 + N 4 = h 4 [ 3 i=0 a i y i + bhy () 0 + h 4 cy (4) 0 ]

5 QUINTIC SPLINE SOLUTIONS OF FOURTH ORDER BOUNDARY-VALUE PROBLEMS 05 (3. 3) N + αn + βn 2 + γn 3 + N 4 = h 4 [ 3 i=0 Recalling that y C 6 [a, b] λ i = N i y (4) i, i = 0,,...,, the equations (3.30), (2.5) (3.3) give (3. 32) λ 0 + αλ + βλ 2 + γλ 3 + λ 4 = β, a i y i bhy () + h 4 cy (4) ]. (3. 33) λ i λ i + 66λ i + 26λ i+ + λ i+2 = β i ; i = 2, 3,..., 2, (3. 34) λ + αλ + βλ 2 + γλ 3 + λ 4 = β, β = (3. 35) h 4 [ 3 i=0 ( a i y i + bhy () 0 + h 4 cy (4) 0 h 4 y (4) 0 + αy (4) + βy (4) 2 )] +γy (4) 3 + y (4) 4 β = h 4 [ 3 i=0 (3. 36) Moreover, from lemma 2 a i y i bhy () + h 4 cy (4) h 4 ( y (4) + αy (4) + βy(4) 2 )] +γy (4) 3 + y(4) 4. (3. 37) β i 58 3 h2 y (6), i = 2, 3,..., 2. The scalars b, c a i, i = 0,, 2, 3 are determined in terms of α, β γ, such that β β are bounded by O(h 2 ). Moreover, α, β γ are chosen such that the system (3.32) (3.34) has unique solution. Following Papamichael Behforooz [6], the required end conditions may be written as (3. 38) (3. 39) N 0 + N 4 = h 4 [ h4 y (4) 0 N + N 4 = h 4 [ y y 20y y 3 40 ] 3 hy() 0 y + 40 y 20 y y 3 ] hy() 4 3 h4 y (4) The quintic spline solution of the system (.3) is defined in the next section.

6 06 SHAHID S. SIDDIQI AND GHAZALA AKRAM 4. Quintic Spline Solution Since the interpolatory quintic spline Q alongwith end conditions (3.38) (3.39) satisfies Q (r) y (r) = O(h 6 r ), r = 0,,..., 5, therefore, it follows from (.3) that (4. 40) N i + f i y i = g i + O(h 2 ); i = 0,,...,, y 0 = α 0, y = α, m 0 = γ 0, m = γ The system (4.40) leads naturally to the method of collocation, a quintic spline Q approximating the solution of (.3) is attained from (4.40) by simply dropping the O(h 2 ) terms. That is, the quintic spline Q is defined by the following + 5 linear equations (4. 4) Ñ i + f i ỹ i = g i ; i = 0,,...,, ỹ 0 = α 0, ỹ = α, m 0 = γ 0, m = γ The quintic spline solution Q is based on the linear equations (3.38), (2.5) (3.39) together with (4.40). The linear system obtained can, thus, be written in matrix form as (4. 42) (A + h 4 BF)Y = C + e, e i = O(h 6 ), i =, 2,...,. The parameters ỹ i of the approximating spline Q satisfy the linear system (4. 43) (A + h 4 BF)Ỹ = C, Ỹ = [ỹ, ỹ 2,..., y ] T C = [c, c 2,..., c ] T. Also A = (a m,n ), m, n =, 2,...,, a, = a, = 9, a,2 = a, 2 = 9/2; otherwise, 6, m = n, 4, m n =, (4. 44) a m,n =, m n = 2, 0, m n > 2, B =

7 QUINTIC SPLINE SOLUTIONS OF FOURTH ORDER BOUNDARY-VALUE PROBLEMS 07 F = diag(f i ); i =, 2,...,. Moreover, c = ( 2 2 ỹ0 + 3h m 0 + h 4 40 (f 0 ỹ 0 g 0 ) + 9 ) (4. 45) 40 g 4, (4. 46) (4. 47) (4. 48) (4. 49) c 2 = h4 20 ( g 0 f 0 ỹ g + 66 g g 3 + g 4 ) ỹ 0, c i = h4 20 ( g i g i + 66 g i + 26 g i+ + g i+2 ); i = 3, 4,..., 3, c 2 = h 4 20 ( g f ỹ + 26 g + 66 g g 3 + g 4 ) ỹ c = ( 2 2 ỹ 3h m + h 4 40 (f ỹ g ) + 9 ) 40 g 4. Extending the method for the solution of sixth, eighth higher order boundary value problems, is in process. Convergence analysis of the method is discussed in the next section. 5. Convergence Analysis Using eq.(4.42) eq.(4.43), it can be written as That is (A + h 4 BF)(Y-Ỹ ) = e. (5. 50) A(I + h 4 A BF)(Y-Ỹ ) = e. To determine the bound on Y -Ỹ, the following lemma is needed [7]. Lemma 5. If G is a matrix of order N G <, then (I + G) exists (I + G) < G. Equation (5.50) can be expressed as (5. 5) (Y-Ỹ ) = A (I + h 4 A BF) e Using lemma 5., (5. 52) Y-Ỹ provided A e h 4 A B F, (5. 53) h 4 A B F <. Now (5. 54) (5. 55) B =, F f = max x [a,b] f(x).

8 08 SHAHID S. SIDDIQI AND GHAZALA AKRAM From Usmani[7], (5. 56) A (b a)4 384h4 so ) ( + 8h3 (b a) 3, (5. 57) h 4 A B F h 4 (b a)4 384h4 which shows that eq.(5.53) holds only if, (5. 58) f < (b a) (b a) 4 + 8h 3 (b a). ( ) (b a)4 + 8h3 (b a) f 384h4 3 ) ( + 8h3 (b a) 3 f, Considering the restriction (5.58) on f, Y -Ỹ can be determined as Y-Ỹ ( + 8h3 (b a) 3 (5. 59) = ( f G) Gh 2, (5. ) G = (b a)4 384 ) ( + 8h3 (b a) 3 ) h 6 which shows that the method developed for the solution of fourth order boundary value problem (.3), is second order convergent. To implement the method for the quintic spline solution of the boundary value problem (.3), three examples are discussed in the following section. 6. Numerical Examples Example Consider the following boundary value problem y (iv) + 4y =, x, (6. 6) y( ) = y() = 0, y ( ) = y () = The analytic solution of the given problem is sinh(2) sin(2) 4(cosh(2) + cos(2)) y(x) = 0.25[ 2[sin() sinh() sin(x) sinh(x) + cos() cosh() cos(x) cosh(x)]/(cos(2) + cosh(2))]. The observed maximum errors (in absolute values) associated with y (µ) i, µ = 0,, 2, 3, 4, for the system (6.6) are summarized in Table, which confirms the method to be second order convergent. Example 2 Consider the following boundary value problem (6. 62) y (iv) + xy = (8 + 7x + x 3 )e x, 0 x, y(0) = y() = 0, y (0) =, y () = e The analytic solution of the above differential system is y(x) = x( x) e x. }

9 QUINTIC SPLINE SOLUTIONS OF FOURTH ORDER BOUNDARY-VALUE PROBLEMS 09 Table. Maximum absolute errors for problem (6.6) in y (µ) i, µ = 0,, 2, 3, 4. h y i y () i y (2) i y (3) i y (4) i The observed maximum errors (in absolute values) associated with y (µ) i, µ = 0,, 2, 3, 4, for the system (6.62) are summarized in Table 2. Table 2. Maximum absolute errors for problem (6.62) in y (µ) i, µ = 0,, 2, 3, 4. h y i y () i y (2) i y (3) i y (4) i

10 0 SHAHID S. SIDDIQI AND GHAZALA AKRAM Example 3 Consider the differential system (6. 63) y (iv) y = 4(2x cos(x) + 3 sin(x)), x, y( ) = y() = 0, y ( ) = y () = 2 sin() The analytic solution of the above system is y(x) = (x 2 ) sin(x). The observed maximum errors (in absolute values) associated with y (µ) i, µ = 0,, 2, 3, 4, for the system (6.63) are summarized in Table 3. } Table 3. Maximum absolute errors for problem (6.63) in y (µ) i, µ = 0,, 2, 3, 4. h y i y () i y (2) i y (3) i y (4) i References [] E.L.Albasiny, W.D.Hosins, Cubic Spline Solutions to Two-Point Boundary Value Problems, Computer J. Vol. 2 No 2, (969), pp [2] J.H.Ahlberg, E.N.Nilson J.L.Walsh, The Theory of Splines their Applications, Academic Press, New Yor, 967. [3] J. W. Daniel B. K. Swartz, Extrapolated Collocation For Two-Point Boundary-Value Problems Using Cubic Splines, J. Inst. Maths Applics., Vol. 6, (975), pp [4] N. Papamichael A. J. Worsey, End Conditions For Improved Cubic Spline Derivative Approximations, J. Comptut. Appl. Math. Vol. 7, (98), pp [5] N. Papamichael A. J. Worsey, A Cubic Spline Method For The Solution Of A Linear Fourth-Order Two-Point Boundary Value Problem, J. Comptut. Appl. Math. Vol. 7, (98), pp [6] N. Papamichael G. H. Behforooz, End Conditions For Interpolatory Quintic Splines, IMA J. Numer. Anal. Vol., (98), pp [7] Riaz A. Usmani, Discrete Variable Methods For A Boundary Value Problem With Engineering Applications, Mathematics of Computation, Vol. 32 No. 44, (978), pp

11 QUINTIC SPLINE SOLUTIONS OF FOURTH ORDER BOUNDARY-VALUE PROBLEMS [8] Riaz A. Usmani, Spline Solutions For Nonlinear Two Point Boundary Value Problems, Internat. J. Math Math. Sci. Vol. 3 No., (980), pp [9] Riaz A. Usmani Manabu Saai, Quartic Spline Solutions For Two-Point Boundary Value Problems Involving Third Order Differential Equations, Journal Mathematical Phy. Sci. Vol. 8, No. 4, (984), India. [0] S.S. Siddiqi E.H. Twizell, Spline Solution of linear sixth-order boundary value problems, Intern. J. Computer Math. Vol. (3), (996), pp [] S.S. Siddiqi E.H. Twizell, Spline Solution of linear eighth-order boundary value problems, Comp. Meth. Appl. Mech. Eng. Vol. 3, (996), pp [2] S.S. Siddiqi E.H. Twizell, Spline Solution of linear tenth-order boundary value problems, Intern. J. Computer Math. Vol. 68(3), (998), pp [3] S.S. Siddiqi E.H. Twizell, Spline Solution of linear twelfth-order boundary value problems, J. Comp. Appl. Maths. Vol. 78, (997), pp Department of Mathematics, Punjab University, Lahore, Postcode Paistan. shahidsiddiqiprof@yahoo.co.u Department of Mathematics, Punjab University, Lahore, Postcode Paistan toghazala2003@yahoo.com