Non-Polynomial Spline Solution of Fourth-Order Obstacle Boundary-Value Problems

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1 Non-Polynomial Spline Solution of Fourth-Order Obstacle Boundary-Value Problems Jalil Rashidinia Reza Jalilian Abstract In this paper we use quintic non-polynomial spline functions to develop numerical methods for approximation to the solution of a system of fourth-order boundaryvalue problems associated with obstacle unilateral contact problems. The convergence analysis of the methods has been discussed shown that the given approximations are better than collocation finite difference methods. Numerical examples are presented to illustrate the applications of these methods to compare the computed results with other known methods. Keywords Quintic non-polynomial spline Boundary formula Convergence Obstacle problems. I. INTRODUCTION In this paper we apply non-polynomial spline functions to develop numerical methods for obtaining smooth approximations to the solution of a system of fourth-order boundary-value problem of the form: u (4) = f(x) a x c g(x)u(x) + f(x) + r c x d f(x) d x b (1) subected to the boundary continuity conditions u(a) = u(b) = α 1 u (a) = u (b) = α u(c) = u(d) = β 1 u (c) = u () (d) = β where f(x) g(x) are continuous functions on [a b] [c d] respectively. The parameters r α i β i (i = 1 ) are real constants. Such type of system arise in the study of obstacle unilateral moving free boundary-value problems has important applications in other branches of pure applied science [ ]. In general it is not possible to obtain the analytical solution J. Rashidinia is with the School of Mathematics Iran University of Science & Technology Narmak Tehran Iran rashidinia@iust.ac.ir. R. Jalilian is with the Department of Mathematics Ilam University PO Box Ilam Iran rezaalilian@iust.ac.ir of (1) for arbitrary choices of f(x) g(x). A special form of problem (1) have been considered by the numbers of authors [ ] they used finite difference collocation spline methods. In the present paper we apply non-polynomial spline functions [16170] that have a polynomial trigonometric part to develop numerical methods for obtaining smooth approximation to the solutions of such system. These methods are based on a nonpolynomial spline space. The spline functions we propose in this paper have the form a sin(kx) + b cos(kx) + cx 3 + dx + ex + f. We develop the class of various methods. Our method perform better than the other collocation finite difference spline methods of same order. This approach has the advantage over finite difference methods that it provides continuous approximations to not only for u(x) but also for u (i) (x) i = 1 3 at every point of the range of integration. Also the c -differentiability of the trigonometric part of non-polynomial spline compensates for the loss smoothness inherited by polynomial spline. The spline function we propose in this paper has the form Span1 x x x 3 sin( k x) cos( k x)} where k is the frequency of trigonometric part of the spline functionwhen k 0 our spline reduce to the form: Span1 x x x 3 x 4 x 5 } (when ). The above fact is evident when correlation between polynomial non-polynomial splines basis is investigated in the following manner T 5 = span1 x x x 3 sin(kx) cos(kx)} = span1 x x x 3 4 (kx) k4(cos(kx) 1 + ) 10 (kx)3 (sin(kx) (kx) + )}. k5 6 00

2 From the above equation it follows that lim k 0 T 5 = 1 x x x 3 x 4 x 5 } so that the Usmani s method [18] based on quintic splines is a special case (k = 0) of our approach. II. CLASS OF METHODS For simplicity we first develop the quintic nonpolynomial spline for solving the fourth-order boundary value problem d 4 u = g(x)u + f(x) for x [c d] dx4 u(c) = u(d) = β 1 u (c) = u (d) = β. (3) For this purpose we divide the interval [cd] into n equal subintervals using the grid points. Let u(x) be the exact solution of the boundary-value problem (3) u be an approximation to u(x ) in order to develop the numerical method for approximating solution of differential equations (3) we introduce the set x } so that x = c + h h = d c = n n the non-polynomial quintic spline p (x) in subinterval x x x +1 has the form p (x) = a sin k(x x ) + b cosk(x x ) +c (x x ) 3 + d (x x ) +e (x x ) + l = n(4) e = u +1 u h + h3 [(6 θ )S (6 + θ )S +1 ] 6θ 4 h(m +1 + m ) (6) 6 θ = kh. Using the continuity of the first third derivatives at (x u ) we get the following relation for = 1... n 1: m m + m = 6(u +1 u + u ) h + 6(S +1 S cosθ + S ) hk 3 sin θ (6 + θ )S +1 (1 θ )S k θ 3(S + S ) k + (6 θ )S k θ (7) m +1 m + m = S S S +1 k + h(s +1 S cosθ + S ). (8) k sin θ Using equations (7) (8) we get the following scheme: where a b c d e l are constants k is free parameter. If k 0 then p (x) reduces to quintic spline in [cd]. By using continuity conditions at the common nodes (x u ) to derive expression for the coefficients of (4) in terms of u u +1 m m +1 S S +1 we have: p (x ) = u p (x +1 ) = u +1 p () (x ) = m p () (x +1 ) = m +1 p (4) (x ) = S p (4) (x +1 ) = S +1. (5) Using the (5) we get the following expressions: b = h4 S θ l 4 = u h4 S θ 4 a = S +1 S cosθ d k 4 = k m + S sin θ k c = h(s +1 S ) + θk(m +1 m ) 6θ u + 4u u 4u + u = h 4 [α(s + + S ) +β(s +1 + S ) + γs ] (9) where = 3... n α = θ3 6(θ sin θ) 6θ 4 sin θ β = 1θ(1 + cosθ) θ3 (cosθ ) 4 sinθ 6θ 4 sin θ γ = 36 sinθ 1θ(1 + cosθ) θ3 (4 cosθ 1). 6θ 4 sin θ If θ 0 then (α β γ) ( Using u (4) = g u + f + r f f(x ) u u(x ) g g(x ) at nodal points x by Taylor expansion the local truncation errors t = 10 ). 01

3 3...n associated with our scheme is: t = (1 (α + β) γ)h 4 u (4) +( 1 6 (4α + β))h6 u (6) III. DEVELOPMENT OF THE BOUNDARY FORMULAS For discretization of boundary conditions we define: +( 1 80 (16 1 α β))h8 u (8) 17 +( (64α + β))h10 u (10) 31 +( ( 4α β 0160 ))h1 u (1) (ζ ) +O(h 13 ). For different choices of parameters α β γ we get the class of methods such as: (i) Second-Order Method For α = 1 β = 6 γ = 1 α β gives: δ 4 u = h 4 u (4) 1 1 h6 M 6 +O(h 7 ) = 3...n. (10) (ii)second-order Method For α = 6 β = 7 γ = 1 α β gives: δ 4 u = h 4 u (4) h6 M 6 + O(h 7 ) (11) = 3...n. (iii) Fourth-Order Method For α = 0 β = 1 4α γ = 1 α β gives: 6 δ 4 u = h 4 [u (4) u (4) + u (4) ] 1 70 h8 M 8 +O(h 9 ) = 3...n. (1) (iv) Sixth-Order Method For α = β = 3 16α γ = 1 α β 70 0 gives: δ 4 u = h4 70 [ (u(4) + + u (4) ) + 14(u (4) +1 + u (4) ) +474(u (4) )] h10 M 10 +O(h 11 ) = 3...n (13) where M 6 = max c x d u (6) (x) M 8 = max c x d u (8) (x) M 10 = max c x d u (10) (x). Each of the above recurrence relations gives n linear equations in n unknowns we need two more equations at each end of the rang of integration. (i) b ku k + c h u 0 + h 4 d ku (4) k + t 1 = 0 (ii) b ku n k + c h u n + h 4 d ku (4) n k +t n = 0 (14) where b k c d k are arbitrary parameters to be determined. In order to obtain the second-order method we find that: (b 0 b 1 b b 3 c ) = ( ) (d 0 d 1 d d 3) = ( 1 0 0). 1 We obtain the second order boundary formulas as follows: (5 h 4 g 1 )u 1 4u + u 3 = ( 1 1 h4 g 0 )β 1 h β h 4 ( 1 1 (f 0 + r) (f 1 + r)) h6 u (6) (x 1 ) + O(h 7 ) u n 3 4u n + (5 h 4 g n )u n = ( 1 1 h4 g n )β 1 h β h 4 ( (f n + r) (f n + r)) h6 u (6) (x n ) + O(h 7 ). (15) For four-order method we find that: (b 0 b 1 b b 3 c ) = ( ) (d 0 d 1 d d 3) = ( ) 360 ( h4 g 1 )u 1 + ( h4 g )u +(1 h4 360 g 3)u 3 = ( h4 g 0 )β 1 h β + h4 360 (8(f 0 + r) + 45(f 1 + r) + 56(f + r) +(f 3 + r) h8 u (8) (ζ 1 ) + O(h 9 ) ( h4 g n 3 )u n 3 + ( h4 g n )u n +( h4 g n )u n = ( h4 g n )β 1 h β + h4 360 (8(f n + r) + 45(f n + r) +56(g n + r) + (g n 3 + r)) 0

4 h8 u (8) (ζ n ) + O(h 9 ). (16) For discretization of boundary conditions for sixth-order method we define: 5 (i) b ku k = c h u 0 + h 4 (ii) b ku n k = c h u n 5 +h 4 d ku (4) k + t 1 d 5 ku (4) n 5+k + t n. (17) In order to obtain the truncation errors of t 1 t we find that: (b 0 b 1 b b 3 c c ) = ( ) (d 0 d 1 d d 3 d 4 d 5) = ( ) (5 d 1h 4 g 1 )u 1 + ( 4 d h 4 g )u +(1 d 3h 4 g 3 )u 3 (d 4h 4 g 4 )u 4 (d 5h 4 g 5 )u 5 = ( d 0h 4 g 0 )β 1 h β + h 4 (d 0(f 0 + r) +d 1(f 1 + r) + d (f + r) + d 3(f 3 + r) +d 4(f 4 + r) + d 5(f 5 + r)) h10 u (10) (ζ 1 ) + O(h 11 ) (5 d 1h 4 g n )u n + ( 4 d h 4 g n )u n +(1 d 3h 4 g n 3 )u n 3 (d 4h 4 g n 4 )u n 4 (d 5h 4 g n 5 )u n 5 = ( d 0h 4 g n )β 1 + h β +h 4 (d 0(f n + r) + d 1(f n + r) + d (f n + r) +d 3(f n 3 + r) + d 4(f n 4 + r) + d 5(f n 5 + r)) h10 u (10) (ζ n ) + O(h 11 ). (18) IV. CONVERGENCE ANALYSIS Here we prove the convergence of the methods. Let us write the error equation of the methods as follows: AE = T (19) where E = (e ) is the (n-1)-dimensional column vector with e the error of discretization defined by e = u(x ) u. In other words e is the amount by which computed solution u deviates from the actual solution u(x ) at x = x A is nine-b matrix which can be described as A = M +BG G = h 4 diag(g ) = 1... n (0) here M = P where P = (p i ) is a tridigonal monotone matrix defined by: p i = B = i = = 1...N 1 i = 1 0 otherwise d 1 d d 3 d 4 d 5 β γ β α α β γ β α α β γ β α... α β γ β α α β γ β d 5 d 4 d 3 d d 1 E = A T = [M + BG] T (1) () E [I + M BG] M T. (3) By using (I + A) (1 A ) Usmani et. al. [19] we obtain M 5(d c)4 + 4(d c) h 384h 4 (4) M T 1 M B G. (5) Provided that M BG < 1. Also we can obtain B G h4 M g M g = max c x d g(x). (6) For second order we obtain T 59h6 360 M 6 M 6 = max c ζ d u (6) (ζ) using (4)-(6) we obtain 118ωh 6 M h 4 74ω G O(h ) (7) for fourth order we get T 41h M 8 M 8 = max c ζ d u (8) (ζ) 03

5 using (4)-(6 we obtain 1735ωh 8 M h ω G O(h4 ) for sixth order we get T 167h M 10 M 10 = max c ζ d u (10) (ζ). By using (4)-(6we get (8) 104ωh 10 M h ω G O(h6 ) (9) where ω = 5(d c) 4 + 4(d c) h G = max g(x) c x d provided G < 6910h4 181ω. It follows E 0 as h 0. Therefore the convergence of the methods have been established. V. NUMERICAL RESULTS We consider the system of differential equations [ ] 1 x u (4) = 1 x 1 4u x 1 (30) with the boundary conditions: u() = u( ) = u(1 ) = u(1) = 0 u () = u ( ) = u ( 1 ) = u (1) = 0 (31) the conditions of continuity of u u atx = 1. The analytical solution for this boundary value problem is Γ 1 (x) x u(x) = Γ (x) x 1 (3) 1 Γ 3 (x) x 1 where Γ 1 (x) = 1 4 x x x + 3 x Γ (x) = ϕ 1 [ϕ sin x sinh x+ϕ 3 cosxcosh x] Γ 3 (x) = 1 4 x4 1 8 x x 3 x ϕ 1 = cos(1) + cosh(1) ϕ = sin( 1) sinh( 1) ϕ 3 = cos( 1) cosh( 1). We solved this example over the whole interval [-11] by using the Quintic non-polynomial spline methods with step lengths h = m m = The maximum absolute errors in solution for our various methods are listed in tables 1 also the maximum absolute errors in the solution at middle points of interval are tabulated in table. To compare our computed results obtained by second fourth order methods with the results obtained by other known methods in [ ] the maximum absolute errors in the solution of example 1 are listed in tables Spline approach has the advantage over finite difference method that it provides continuous approximations to u (i) (x) i = 1 3 at every point of the range of integration beside approximation to u(x). Following [0] to obtain the necessary formula for computing values of first second third derivatives of solution of example 1 by using equation (7) (8) solving the resulting identity for m = 1...n we have m = (u +1 u +u ) + h (S +1 S cos θ+s ) h 3 sin θ h (S S ) h [(6+θ )S +1 (1 θ )S +(6 θ )S ] θ 6θ 4 (S S S +1 ) 6θ h (S +1 S cos θ+s ) 6θ sin θ with m 0 = m n+1 = β being known from the boundary conditions. Having computed u m S = 0...n + 1 it is possible to evaluate the coefficient of the spline function (4) as given by (6). Since y = p (x ) = 0...n y n+1 = p n(x n+1 ) it follows that y a k + e = 0...n Ψ n + 3c n h + d n h + e n = n + 1 (33) where Ψ n = a n k cosθ b n k sin θ. Similarly from y = p (x ) = 0...n y n+1 = p n (x n+1 ) we can obtain y a k 3 + 6c = 0...n a n k 3 cosθ + b n k 3 sin θ + 6c n = n + 1. (34) The values of u (i) (x) i = 1 3 have been computed by our second order method (i). To compare with the method in [1] the maximum absolute errors are listed in tables 6. 04

6 Example : We consider the system of differential equation solved by Al-Said Noor [1]. 0 x u (4) = 1 x 1 1 4u x 1 (35) with the boundary conditions: u() = u( ) = u(1 ) = u(1) = 0 u () = u ( ) = u ( 1 ) = u (1) = ǫ (36) where ǫ 0. The analytical solution for this boundary value problem is Λ 1 (x) for 1 x u(x) = Λ (x) for x 1 (37) 1 Λ 3 (x) = for x 1 where Λ 1 (x) = ( 3 x3 3 x 13x 1)ǫ 1 4 Λ (x) = ϕ 1 [ϕ sin x sinh x+ϕ 3 cosxcosh x] Λ 3 (x) = ( 3 x3 + 3 x 13x + 1)ǫ 1 4 ϕ 1 = cos(1) + cosh(1) ϕ = sin( 1) sinh( 1) ϕ 3 = cos( 1) cosh( 1). We solved this example over the whole interval [-11] by using our second order methods (i)(ii) with step lengths h = m m = The maximum absolute error in solution are listed in table 7 8 our results compared with the results obtained in [111]. The results shows superiority of our second orders methods. TABLE I m O(h )(i) O(h )(ii) O(h 4 ) O(h 6 ) TABLE II IN MIDDLE POINTS m O(h )(i) O(h )(ii) O(h 4 ) O(h 6 ) TABLE III m Our fourth-order Fourth-order[4] Finite difference[14] TABLE IV h Second-order(i) Second-order(ii) Second-order[1] 1/ / / VI. CONCLUSION We have developed a new non-polynomial quintic spline for solving a system of fourth-order boundary-value problems. This approach has the advantages over finite difference methods that it provides continuous approximations to not only for u(x) but also for u (i) (x) i = 1 3 at every point of the range of integration. Our numerical results are better than those produced by collocation finite difference methods for solution of equation(1). REFERENCES [1] AL-SAID E.A. NOOR M.A. Quartic Spline Method for Solving Fourth-Order Obstacle Boundary Value Problems Journal of Computational Applied Mathematics Vol.143pp [] AL-SAIDE.A. NOORM.A. Computational Methods for Fourth- Order Obstacle Boundary Value Problems Comm. Appl. Nonlinear. Anal. Vol.pp [3] AL-SAID E.A. NOORM.A. RASSIAST.M. Cubic Splines Method for Solving Fourth-Order Obstacle Problems Appl. Math. Comput.Vol.174 pp [4] AL-SAIDE.A. NOORM.A. KAYAD. Al-KHALEDK. Finite difference method for solving fourth-order obstacle problems Int. J. Comput. Math. Vol.81pp [5] BAIOCCHIC.C. CALEOA. Variational quasivariational Inequalities John Wiley Sons New York [6] CHAWLAM.M. SUBRAMANIAN High accuracy quintic spline solution of fourth-order two-point boundary value problems Int. J. Computer. Math. Vol.31pp [7] HENRICIP. Discrete variable method in ordinary differential equations John Wiley New York [8] JAINM.K. Numerical solution of differential equations Second EditionsWiley Eastern Limited [9] JAINM.K. IYANGERS.R.K. SOLDHANHAJ.S.V. Numerical solution of a fourth-order ordinary differential equation J. Engg. Math. Vol.11pp [10] KIKUCHIN. ODENJ.T. Contact problem in elasticity SIAM Publishing Co. Philadelphia

7 TABLE V h Second-order In [] Second-order[3] 1/ / / TABLE VI MAXIMUM ABSOLUTE ERRORS OF u u AND u FOR EXAMPLE 1 m u (x i ) u i u (x i ) u i u (x i ) u i Our O(h )(i) O(h ) in[1] TABLE VII MAXIMUM ABSOLUTE ERRORS FOR EXAMPLE WHEN ǫ = 10 6 m Second-order(i) Second-order(ii) Second-order In [1] TABLE VIII MAXIMUM ABSOLUTE ERRORS FOR EXAMPLE WHEN ǫ = 10 6 m Second-order In [11] Second-order In [] Jalil Rashidinia Associate professor School of Mathematics Iran University of Science & Technology Narmak Tehran Iran rashidinia@iust.ac.ir Nationality: Iranian Research Interests: Applied mathematics computational sciences Numerical solution of ODE PDE IE Spline approximations Numerical linear algebra [11] KHALIFAA.K. NOORM.A. Quintic spline solutions of a class of contact problems Math. Comput. Modlelling Vol.13pp [1] LEWYH. STAMPACCHIAG. On the regularity of the solution of the variational inequalities Comm. pure Appl. Math. Vol.pp [13] NOORM.A. Al-SAIDE.A. Fourth order obstacle problems.in: Th.M.Rassias H.M.Srivastava(Eds) Analytic geometric inequalities applications kluwer Academic Publishers Dordrecht Netherlspp [14] NOORM.A. AL-SAIDE.A. Numerical solution of fourth order variational inequalities Inter. J. Comput. Math. Vol.75pp [15] PAPAMICHELN. WORSEYE.A. A cubic spline method for the solution of linear fourth-order two-point boundary value problem J. Comput. Appl. Math. Vol.7pp [16] RASHIDINIA J. Applications of splines to the numerical solution of differential equations Ph.D. thesis Aligarh Muslim University Aligarh India [17] SIRAJ-UL-ISLAM TIRMIZIS.I.A. SAADAT ASHRAF A class of method based on non-polynomial spline function for the solution of a special fourth-order boundary-value problems with engineering applications Appl. Math. Comput. Vol.174pp [18] USMANIR.A. WARSIS.A. Smooth spline solutions for boundary value problems in plate deflection thoery Comput. Maths. with Appls. Vol.6 pp [19] USMANIR.A. Discrete variable method for a boundary value problem with engineering applications Math. Comput. Vol [0] Van DAELEM. VANDEN BERGHEG. De MEYERH. A smooth approximation for the solution of a fourth-order boundary value problem based on non-polynomial splines J. Comput. Appl. Math. Vol.pp Reza Jalilian Assistant professor Department of Mathematics Ilam University PO Box Ilam Iran rezaalilian@iust.ac.ir Research Interests: Spline approximations 06