Solution of Fourth Order Obstacle Problems Using Quintic B-Splines

Transcription

1 Applied Mathematical Sciences, Vol. 6, 202, no. 94, Solution of Fourth Order Obstacle Problems Using Quintic B-Splines Shahid S. Siddiqi, Ghazala Akram and Kalsoom Arshad Department of Mathematics University of the Punjab, Lahore, Pakistan Abstract In this paper, a numerical method is developed for solving a system of fourth order boundary value problem associated with obstacle, contact and unilateral problems. It is shown that the Quintic B-spline method is very effective tool and yields better results. Numerical examples are presented to illustrate the applicability of the new method. Keywords: Quintic B-Spline; Boundary Value Problems; Variational Inequalities; Obstacle Problems; Unilateral Problems Introduction For studying the contact, unilateral, obstacle and equilibrium problems arising in different branches of pure and applied sciences, variational inequality theory has become an effective and powerful tool. Variational inequality theory has proved to be immensely useful in the study of many branches of mathematical and engineering sciences. Penalty methods and projection methods have been developed for the solution of general variational inequalities(see[-4, 6, 8-0, 2, 4, 9]). The projection methods are not supposed to be suitable as in projection methods, the projection is needed, which is difficult to be obtained. The penalty methods are inefficient as in these methods, instability is created. However, the general variational inequalities can be characterized by a system of differential equations using the penalty function technique, if the obstacle function is known. This technique was used by Lewy and Stampacchia [8] to study the regularity of the solution of variational inequalities. The main advantage of this technique is its simple applicability in solving obstacle and

2 4652 S. S. Siddiqi, Gh. Akram and K. Arshad unilateral problems. This technique has been used for solving fourth order system of differential equations associated with obstacle and unilateral problems by finite difference and quintic spline methods [, 2, 9]. Noor and Tirmizi [0] solved the system of second order boundary value problems using pade approximation. Al-Said [4] developed the method for the solution of system of second order boundary value problems using cubic spline. It is claimed that the method can be considered as an improvement for the cubic spline method developed in [3]. Gao and Chi [6] solved a system of third-order boundary value problems associated with third-order obstacle problems using the quartic B-splines and the method is claimed to be of second order. Usmani [5], developed the method of the solution of fourth order boundary value problem, considering it 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), (. ) w(0) = w(l) =w (0) = w (l) =0, where L d4 dx 4, D is the flexural rigidity of the beam, K is the spring constant of the elastic foundation and the load q(x) acts vertically downwards per unit length of the beam. Usmani [6], developed the discrete methods for the solution of fourth order linear special case boundary value problem, similar to the problem (.) with the change in boundary conditions in terms of second order derivatives instead of first order derivatives. Twizell and Tirmizi [5] developed and analysed a sixth-order method for the numerical solution of the linear fourth-order boundary value problem for which the boundary conditions are given in terms of functional values and (i) first-order derivatives (the clamped-clamped beam problem) or (ii) second-order derivatives (the simple-simple beam problem). Papamichael and Worsey [3] developed the cubic spline method for the solution of problem (.). Daele et al. [] developed second order method for the solution of fourth-order boundary value problem using non polynomial spline which is mixture of mixed spline function consisting of cubic polynomial function and a trigonometric function. Siddiqi and Ghazala [7, 8] developed the non polynomial spline technique for the solution of linear special case fifth and eighth order boundary value problems, respectively. Siddiqi et al. [20] developed the method for the solution of linear special case fifth order boundary value problems using the non polynomial sextic spline. The method is proved to be fifth order convergent. Siddiqi et al. [2] presented the method for the solution of sixth order boundary value problems using quintic spline. Ghazala and Siddiqi [7] solved the sixth order linear special case boundary value prob-

3 Fourth order obstacle problems 4653 lems using non polynomial spline. The following system of fourth order boundary value problems, is considered f(x), a x c, y (4) (x) = f(x)+y(x)g(x)+r, c x d, f(x), d x b, (.2 ) along with the boundary conditions y(a) =y(b) =α 0, y (a) =y (b) = α, y(c) =y(d) =α 2, y (c) =y (d) = α 3, } (.3 ) where r and α i, i =0,, 2, 3 are finite real constants and the functions f(x) and g(x) are continuous on [a, b] and [c, d], respectively. Such type of systems arise in connection with contact, obstacle and unilateral problems. In this paper, quintic B-spline function is used to develop a technique for the solution of system (.2). In Section 2, a class of contact problems in elasticity is considered and is formulated in terms of variational inequalities. The quintic B-spline method for the solution of system (.2) is derived in Section 3. In Section 4, two examples are considered for the usefulness of the method developed. 2 Formulation Khan et al. [] considered the linear fourth order boundary value problem describing the equilibrium configuration of an elastic beam, pulled at the ends and lying over an elastic obstacle of constant height /4 and unit rigidity of the type y (4) f(x), y ψ(x), (y (4) f(x)) (y(x) ψ(x)) = 0, y(0) = y() = y (0) = y () = 0, on Ω = [0, ], (2. ) where f is a given force acting on the beam string and ψ(x) is the elastic obstacle. To study the problem (2.) via the variational inequality formulation, the set K can be defined as K = {v : v H 2 0(Ω) : v ψ,on Ω}, (2.2 ) which is a closed convex set in H 2 0 (Ω), where H2 0 (Ω) is a Sobolev space [9, 2, 4], which is basically a Hilbert space.

4 4654 S. S. Siddiqi, Gh. Akram and K. Arshad The Kikuchi and Oden technique [2], shows that the energy functional associated with the obstacle problem (2.) is where I[v] = = 0 0 {d 4 v/dx 4 2f(x)} v(x)dx, v H 2 0(Ω), (d 2 v/dx 2 ) 2 dx 2 0 f(x)v(x)dx, = a(v, v) 2 <f,v>, (2.3 ) a(u, v) = and <f,v> = 0 0 ( d 2 u/dx 2)( d 2 v/dx 2) dx (2.4 ) f(x)v(x)dx. (2.5 ) It can be proved that the form a(u, v) defined by (2.4) is bilinear, positive and symmetric. Moreover, the functional f defined by (2.5) is linear and continuous. The minimum y of the functional I[v] defined by (2.3) on the closed convex set K in H0 2 (Ω) can be characterized by the following variational inequality [9, 2, 4] a(y, v y) <f,v y>, v K. (2.6 ) The obstacle problem (2.) is, thus, equivalent to solving the variational inequality problem (2.6). The equivalence has been used to study the existence of a unique solution of (2.), see for example [9, 4]. Following Lewy and Stampacchia [8], the problem (2.6) can be written as y (4) + μ(y ψ)(y ψ) =f, 0 <x<, (2.7 ) y(0) = y() = 0, y (0) = y () = ɛ, (2.8 ) where ɛ is a small positive constant, ψ is the obstacle function and μ(t) is the penalty function defined by μ(t) = { 4, t 0, 0, t < 0. (2.9 ) Equation (2.6) describes the equilibrium configuration of an elastic beam, pulled at the ends and lying over an elastic obstacle of constant height /4. Since the obstacle function ψ is known, it is possible to find the exact solution of the problem in the interval /4 x 3/4. Assuming that the obstacle function ψ is defined by { /4, 0 x /4, 3/4 x, ψ(x) = /4, /4 x 3/4. (2.0 )

5 Fourth order obstacle problems 4655 From equation (2.6) (2.9), the following system of equations can be obtained as { f, 0 x /4, y (4) = 4y + f, /4 x 3/4, 3/4 x, (2. ) with the following boundary conditions y(0) = y(/4) = y(3/4) = y() = 0, y (0) = y (/4) = y (3/4) = y () = ɛ, (2.2 ) such that y and y are continuous at x =/4 and 3/4, (see []). 3 Quintic B-Spline Method Extending the technique developed by Gao and Chi [6], the boundary value problem (2.) is transformed into the following initial value problems { y (4) 0, 0 x /4, 3/4 x, = 4y, /4 x 3/4, (3. ) with initial values y (0) = 0, y () (0) = 0, y (2) (0) = 0, y (3) (0) = and { y (4) f, 0 x /4, 3/4 x, 2 = 4y 2 + f, /4 x 3/4, with initial values (3.2 ) y 2 (0) = 0, y () 2 (0) = 0, y (2) 2 (0) = 0, y (3) 2 (0) = 0. It may be noted that the exact solution of the problem (2.) is y(x) =y 2 (x) (y 2 ()/y ()) y (x). (3.3 ) To determine the quintic B-spline solution, the interval [0, ] is divided into n equal subintervals with knots x i = ih, i =0,, 2,..., n, h =/n, where n is a multiple of 4. Let Ω n = {x 5,x 4,x 3,x 2,x,x 0,x..., x n } (3.4 )

6 4656 S. S. Siddiqi, Gh. Akram and K. Arshad be the set of extended equally spaced knots with x i = ih. It may be mentioned that the quintic B-spline s(x) = n i= 5 γ i B i,5 (x) (3.5 ) is a polynomial of degree five in each subinterval [x i, x i+ ]. It may also be noted that the quintic B-spline s, by definition, C 4 [0, ]. Let s (x) = n i= 5 α ib i,5 (x) and s 2 (x) = n i= 5 β ib i,5 (x) be the approximate solution to y (x) and y 2 (x), respectively. Moreover, s (x) and s 2 (x) are determined such that they must satisfy the initial problems (3.) and (3.2) respectively. i.e. s (4) (x i )= 0, 0 x i /4, 4s (x i ), /4 x i 3/4, 0, 3/4 x i, alongwith the conditions s (0)=0, s () (0)=0, s (2) (0)=0, s (3) (0) =, and s (4) 2 (x i )= f(x i ), 0 x i /4, f(x i ) 4s 2 (x i )+, /4 x i 3/4, f(x i ), 3/4 x i, (3.6 ) (3.7 ) along with the conditions s 2 (0)=0, s () 2 (0)=0, s (2) 2 (0)=0, s (3) 2 (0) = 0. The first five coefficient α 5, α 4, α 3,α 2, α and β 5, β 4, β 3, β 2, β of s (x) and s 2 (x), x [x 0, xn ] can be determined by applying the initial 4 conditions which leads to the following s (4) (x i )=/h 4 [α i 5 3α i 4 +6α i 3 3α i 2 + α i ], (3.8 ) and s (x i )=/20[α i 5 +26α i 4 +66α i 3 +26α i 2 + α i ], (3.9 ) s (4) 2 (x i )=/h 4 [β i 5 3β i 4 +6β i 3 3β i 2 + β i ] (3.0 ) s 2 (x i )=/20[β i 5 +26β i 4 +66β i 3 +26β i 2 + β i ]. (3. ) From Eqns. (3.5) (3.0), it can be written as α i = α i 5 +4α i 4 6α i 3 +4α i 2, α i 5 ( 4/h 4 +26/30)/(/h 4 +/30)α i 4 (6/h 4 +66/30)/(/h 4 +/30)α i 3 ( 4/h 4 +26/30)/(/h 4 +/30)α i 2, i n/4, (3.2 ) n/4 i 3n/4, α i 5 +4α i 4 6α i 3 +4α i 2, 3n/4 x i n,

7 Fourth order obstacle problems 4657 β i = β i 5 +4β i 4 6β i 3 +4β i 2 + h 4 f(x i ), x i n/4, β i 5 ( 4/h 4 +26/30)/(/h 4 +/30)β i 4 (6/h 4 +66/30)/(/h 4 +/30)β i 3 ( 4/h 4 +26/30)/(/h 4 (3.3 ) +/30)β i 2 +(f(x i )+)/(/h 4 +/30), n/4 x i 3n/4, β i 5 +4β i 4 6β i 3 +4β i 2 + h 4 f(x i ), 3n/4 x i n, 4 Error Bound Considering the quintic splines s (x) and s 2 (x) to be the approximate solutions to y (x) and y 2 (x) respectively. Since s (x) and s 2 (x) are defined piecewise, on [x 0, x n ], therefore error bound of s (x) over any of the three subintervals [x 0,x n/4 ], [x n/4,x 3n/4 ] and [x 3n/4,x n ] will help in calculating the error bound over [x 0, x n ]. The error bound of s (x) over [x 0, x n/4 ] can be calculated as under: Consider the following initial value problem y (x) =f, x [x 0,x n/4 ] (4. ) along with the following conditions s (0) = y (0), s () (0) = y () (0), s (2) (0) = y (2) (0), s (3) (0) = y (3) (0), s (4) (0) = y (4) (0), (4.2 ) let the error term corresponding to each knot can be denoted by then e (x i )=s (x i ) y (x i ), i =0,,..., n 4, (4.3 ) e (x 0 )=s (x 0 ) y (x 0 )=0, e () (x 0 )=s () (x 0 ) y () e (2) (x 0 )=s (2) (x 0 ) y (2) (x 0 )=0, e (3) (x 0 )=s (3) (x 0 ) y (3) e (4) (x 0 )=s (4) (x 0 ) y (4) Expanding s (x) and y (x) as (x 0 )=0. (x 0 )=0, (x 0 )=0, (4.4 ) s (x ) = s(x 0 )+hs () (x 0 )+s (2) (x 0 )h 2 /2+s (3) (x 0 )h 3 /6+s (4) (x 0 )h 4 /4! + O(h 5 ) y (x ) = y(x 0 )+hy () (x 0 )+y (2) (x 0 )h 2 /2+y (3) (x 0 )h 3 /6+s (4) (x 0 )h 4 /4! + O(h 5 )

8 4658 S. S. Siddiqi, Gh. Akram and K. Arshad The order of error terms can be expressed as e (x ) = s(x ) y(x )=O(h 5 ) e () (x ) = s () (x ) y () (x )=O(h 4 ) e (2) (x ) = s (2) (x ) y (2) (x )=O(h 3 ) e (3) (x ) = s (3) (x ) y (3) (x )=O(h 2 ) and e (4) (x ) = s (4) (x ) y (4) (x )=O(h). Similarly, expanding s (x) and y (x) using the Taylor s series, give s (x 2 ) = s(x )+hs () (x )+s (2) (x )h 2 /2+s (3) (x )h 3 /6+s (4) (x )h 4 /4! + O(h 5 ) y (x 2 ) = y(x )+hy () (x )+y (2) (x )h 2 /2+y (3) (x )h 3 /6+s (4) (x )h 4 /4! + O(h 5 ) The order of error terms can be expressed as In general, it can be written as e (x 2 ) = e(x )+O(h 5 ), e () (x 2 ) = e () (x )+O(h 4 ), e (2) (x 2 ) = e (2) (x )+O(h 3 ), e (3) (x 2 ) = e (3) (x )+O(h 2 ), and e (4) (x 2 ) = e (4) (x )+O(h). which shows that e (x i+ ) = e (x i )+O(h 5 ), (4.5 ) and e () (x i+ ) = e () (x i )+O(h 4 ). (4.6 ) e (x n ) = n 4 4 O(h5 )=O(h 4 ) and e () (x n ) = n 4 4 O(h4 )=O(h 3 ). (4.7 ) Similarly, the error term of s 2 can be determined to be of O(h 4 ), which shows that the error bounds of s and s 2 over the remaining two subintervals are also of O(h 4 ). Thus from Eq. (3.3) it can be concluded that quintic B-spline solution of BVP (2.) is second order convergent method.

9 Fourth order obstacle problems Numerical Results In this section, two examples are considered to illustrate the quintic B-spline method. Example When f = 0, the following problem can be considered as 0, 0 x /4, y (4) (x) = 4y(x), /4 x 3/4, (5. ) 0, 3/4 x, with the conditions y(0) = 0, y () (0) = 0, y() = 0, y () () = 0. The analytic solution of the problem (5.) is y(x) = 0, 0 x /4, exp(x)( cos(x) sin(x) + exp( x)( cos(x) sin(x), /4 x 3/4, 0, 3/4 x. (5.2 ) Table : Maximum absolute errors for the problem (5.) in y i. h Maximum absolute errors Example 2 For f =, the following BV P is considered as, 0 x /4, y (4) (x) = 4y(x)+2, /4 x 3/4,, 3/4 x, (5.3 ) with the conditions y(0) = 0, y () (0) = 0, y() = 0, y () () = 0.

10 4660 S. S. Siddiqi, Gh. Akram and K. Arshad The analytic solution of the problem (5.2) is x 4 /24 x 3 /48 + x 2 /384, 0 x /4, exp(x)( cos(x) sin(x) y(x) = (5.4 ) + exp( x)( cos(x) sin(x), /4 x 3/4, x 4 /24 7x 3 / x 2 /384 7/64x +3/28, 3/4 x. Conclusion. A new B-spline method for solving a system of fourth order Table 2: Maximum absolute errors for the problem (5.2) in y i. h Maximum absolute errors problems is developed. The present method enables us to approximate the solution as well as its first, second, third, derivative at every point of range of integration. It has been observed that the results obtained from the method developed in the paper is acceptable. References [] A. Khan, M. A. Noor and T. Aziz, Parametric Quintic-Spline Approach to the Solution of a System of Fourth-Order Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol 22, No. 2, pp , [2] E.A. Al-Said and M. A. Noor, Computational Methods for Fourth-Order Obstacle Boundary-Value Problems, Communications in Applied Nonlinear Analysis, Vol 2, pp , 995. [3] E.A. Al-Said, Spline Methods for Solving System of Second-Order Boundary-Value Problems, Int. J. Comput. Math. Vol 70, pp , 999. [4] E.A. Al-Said, The Use of Cubic Splines in the Numerical Solution of a System of Second-Order Boundary Value Problems, Computers and Mathematics with Applications, Vol 42, pp , 200.

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12 4662 S. S. Siddiqi, Gh. Akram and K. Arshad [7] Shahid S. Siddiqi and Ghazala Akram, Solution of Fifth-Order Boundary- Value Problems using Non Polynomial Spline Technique, Applied Mathematics and Computation, Vol. 75, No. 2, pp , [8] Shahid S. Siddiqi and Ghazala Akram, Solution of Eighth-Order Boundary-Value Problems using Non Polynomial Spline Technique, International Journal of Computer Mathematics, Vol. 84,No. 3, pp , [9] Shahid S. Siddiqi and Ghazala Akram, Solution of tenth order boundary value problems using non polynomial spline technique, Applied Mathematics and Computation, Vol. 90, pp , [20] Shahid S. Siddiqi, Ghazala Akram and Salman Amin Malik, Non Polynomial Sextic Spline method for the Solution along with the convergence of linear Special Case Fifth Order Two-point boundary value problems, Applied Mathematics and Computation, Vol. 90, pp , [2] Shahid S. Siddiqi, Ghazala Akram and Saima Nazeer, Quintic Spline Solutions of Linear Sixth Order Boundary Value Problems, Applied Mathematics and Computation, Vol. 89, pp , Received: April, 202