Solving Integral Equations by Petrov-Galerkin Method and Using Hermite-type 3 1 Elements

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1 5 1 July 24, Antalya, Turkey Dynamical Systems and Applications, Proceedings, pp Solving Integral Equations by Petrov-Galerkin Method and Using Hermite-type 3 1 Elements M. Karami Department of Mathematics, Technical Faculty, Islamic Azad University (South Unit), Ahang Blvd., Tehran, Iran m_karami@azad.ac.ir Abstract In this paper, we use the Petrov-Galerkin method for solving Fredholm integral equations of the second kind on [, 1] that the trial space is piecewise Hermite-type cubic polynomials and the test space is piecewise linear polynomials, and for showing the efficiency of method, we use numerical examples. Key words: The Petrov-Galerkin method, Piecewise polynomials, Regular pair, Trial space, Test space. 1 Introduction In this paper, we solve Fredholm integral equations of the second kind given in the form u(t) (Ku)(t) =f(t), t [, 1], (1) (Ku)(t) = k(t, s)u(s) ds, the function f L 2 [, 1], thekernelk L 2 ([, 1] [, 1]) are given and u L 2 [, 1] is the unknown function to be determined. Numerical methods including least square, collocation and Galerkin methods for equation (1) are used and their analysis may be found in [1, 2, 3]. The Petrov- Galerkin method is established in [4] for equation (1). In [4] it has been shown that the Petrov-Galerkin method includes the Galerkin, collocation and least square methods. One of the advantages of the Petrov-Galerkin method is that it allows us to achieve the same order of convergence as the Galerkin method with much less computational cost by choosing the test spaces to be spaces of piecewise polynomials of lower degree. In [5] we used continuous and discontinuous Lagrange-type k elements for solving equation (1) on [, 1]. 436

2 Solving integral equations by Petrov-Galerkin method 437 This paper is organized as follows: In Section 2, we review the Petrov-Galerkin method for equation (1). In Section 3 we use Hermite-type 3 1 elements for solving equation (1) on [, 1]. 2 The Petrov-Galerkin method In this section we follow the paper [4] with a brief review of the Petrov-Galerkin method. Let be a Banach space and be its dual space of continuous linear functionals. For each positive integer n, we assume that n, Y n and n, Y n are finite dimensional vector spaces with dim n =dimy n, n =1, 2,... (2) Also n, Y n satisfy condition (H) :For each x and y,thereexistx n n and y n Y n such that kx n xk and ky n yk as n. Define, for x, anelementp n x n called the generalized best approximation from n to x with respect to Y n, by the equation (x P n x, y n )= for all y n Y n. (3) It is proved in [4] that, for each x, the generalized best approximation from n to x with respect to Y n exists uniquely if and only if Y n n = {}. (4) Under this condition, P n is a projection, i.e., P 2 n = P n. Assume that, for each n, there is a linear operator Π n : n Y n with Π n n = Y n satisfying the following two conditions: (H-1) for all x n n, kx n k C 1 (x n, Π n x n ) 1/2, (H-2) for all x n n, kπ n x n k C 2 kx n k. If a pair of sequences of spaces { n } and {Y n } satisfy (H-1) and (H-2), we call { n,y n } a regular pair. It is proved in [4] that, if a regular pair { n,y n } satisfies dim n =dimy n and condition (H), then the corresponding generalized projection P n satisfies: (1) for all x, kp n x xk as n, (2) there is a constant C> such that kp n k <C, n =1, 2,...,

3 438 M. Karami (3) for some constant C> independent of n, kp n x xk CkQ n x xk, Q n x is the best approximation from n to x. The Petrov-Galerkin method for equation (1) is a numerical method for finding u n n such that (u n Ku n,y n )=(f,y n ) for all y n Y n. (5) If { n,y n } is a regular pair with a linear operator Π n : n Y n,thenequation (5) may be rewritten as (u n Ku n, Π n x n )=(f,π n x n ) for all x n n. (6) Furthermore, equation (5) is equivalent to u n P n Ku n = P n f. (7) Equation (7) can also be derived from the fact that P n x =for an x if and only if (x, y n )=for all y n Y n. Now, assume u n n and {b i } n i=1 is a basis for n and {b j }n is a basis for Y n. Therefore the Petrov-Galerkin method on [, 1] for equation (1) is (u n Ku n,b j)=(f,b j), j =1,...,n. (8) P Let u n (t) = n a i b i (t). Then equation (1) leads to determining {a 1,a 2,...,a n } as i=1 the solution of the linear system n ½ a i b i (t)b j(t) dt i=1 = ¾ K(s, t)b i (s)b j(t) ds dt f(t)b j(t) dt, j =1,...,n. (9) The Petrov-Galerkin methods using regular pairs { n,y n } of piecewise polynomial spaces are called Petrov-Galerkin elements. If we use piecewise polynomials of degree k and k for the spaces n and Y n respectively, we call the corresponding Petrov-Galerkin elements k k elements. In Section 3 we solve the equation (1) using Hermite-type 3 1 elements.

4 Solving integral equations by Petrov-Galerkin method Hermite-type 3-1 elements We subdivide the interval [, 1] into n subintervals by a sequence of points =t < t 1 < <t n =1.DenoteI i =[t i 1,t i ] and h i = t i t i 1 and let n be the space of piecewise Hermite-type cubic polynomials, that is: n = {x n C 1 [, 1] : x n Ii is a cubic polynomial determined by x l n(t i 1 ),x l n(t i ), l =, 1, i =1,...,n} = span{b 1 (t),b 2 (t),...,b 2n+2 (t)}. Using Hermite interpolation, for each x n n it holds that b j (t) = b 2i+j (t) = b 2n+j (t) = and x n (t) = {x n (t j 1 )b 2j 1 (t)+x n(t j 1 )b 2j (t)}, φ j (τ)(h 1 ) j 1, τ = t t h 1,t I 1, j =1, 2,, t 6 I 1, φ j+2 (τ)(h i ) j 1, τ = t t i 1 h i, t I i, ( φ j (τ)(h i+1 ) j 1, τ = t t i =1,...,n 1, i h i+1,t I i+1, j =1, 2, S, t 6 I i Ii+1, φ j+2 (τ)(h n ) j 1, τ = t t n 1 h n,t I n, j =1, 2,, t 6 I n, φ 1 (τ) = (1 τ) 2 (2τ +1), φ 2 (τ) = τ(1 τ) 2, φ 3 (τ) = τ 2 (3 2τ), φ 4 (τ) = (τ 1)τ 2. Now, let Y n be the space of piecewise linear polynomials, that is, Y n =span{b 1(t),b 2(t),...,b 2n+2(t)},

5 44 M. Karami b 2i+1(t) = b 2i+2(t) = 1, t [t i h i 2,t i + h i+1 2 ], i =, 1,...,n,, otherwise, t t i, t [t i h i 2,t i + h i+1 2 ], i =, 1,...,n,, otherwise, h = h =. Then dim n =dimy n =2n +2 and in [4] it is proved that { n,y n } form a regular pair. Let B be a (2n +2) (2n +2)matrix with entries A direct computation gives B =. b ij =(b i (t),b j(t)), i,j =1,...,2n +2. L N 1 M 1 L 1 N 2 M 2 L 2 N M n 2 L n 2 N n 1 M n 1 L n 1 N n M n L n, L = L i = M i = h h h h h i h i h2 i h2 i h i h2 i h2 i 3 32 h3 i, L n = h n h2 n h2 i h2 i h3 i h3 i+1,,n i = 3 32 h i h2 i h2 n 1 6 h3 n,,i=1, 2,...,n 1, h2 i 3 32 h3 i,i=1,...,n.

6 Solving integral equations by Petrov-Galerkin method 441 For solving equation (1), we seek a function u n n,anditcanbewrittenas u n (t) = 2n+2 c j b j (t). (1) From (5), the Petrov-Galerkin method for equation (1) is µ u n (t) k(t, s)u n (s) ds, b i (t) = f(t),b i (t), (11) i =1,...,2n +2. If we substitute (1) in (11), we have 2n+2 µ ¾ c j ½(b j (t),b i (t)) k(t, s)b j (t) ds, b i (t) = f(t),b i (t), (12) Now, we approximate k(t, s) and f(t) in n : k(t, s) = f(t) = i =1,...,2n +2. {k(t p 1,s)b 2p 1 (t)+k t (t p 1,s)b 2p (t)}, p=1 {f(t q 1 )b 2q 1 (t)+f (t q 1 )b 2q (t)}, q=1 and then substitute in (12). Therefore, this leads to determining {c 1,c 2,,c 2n+2 } as the solution of the linear system 2n+2 Example 1 c j {b ji p=1 d pj b 2p 1,i + e pj b 2p,i } = f(t q 1 )b 2q 1,i + f (t q 1 )b 2q,i, d pj = R 1 k(t p 1,s)b j (s) ds, e pj = R 1 k t(t p 1,s)b j (s) ds, u(t) with exact solution u(t) =t. q=1 i =1, 2,...,2n +2, j =1,...,2n +2, p =1,...,. (t + s)u(s) ds =(t/2) (1/3), t 1,

7 442 M. Karami Example 2 u(t) with exact solution u(t) =e t. Example 3 u(t) (t 2 e s(t 1) )u(s) ds =(1 t)e t + t, t 1, ( 1 3 e2t 5s/3 )u(s) ds = e 2t+1/3, t 1, with exact solution u(t) =e 2t. In the following table we computed ku n (t i ) u(t i )k 2 for n =1, 2, 4, 1 with equally spaced points: n Example 1 Example 2 Example References [1] Atkinson K. E., The Numerical Solution of Integral Equation of the Second Kind, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, [2] Kress R., Linear Integral Equations, Springer, New York, [3]DelvesL.M.andMohamedJ.L.,Computational Methods for Integral Equations, Cambridge University Press, [4] Chen Z. and u Y., The Petrov-Galerkin and iterated Petrov-Galerkin methods for second-kind integral equations, SIAMJ. Num. Anal., 35 (1998), No. 1, [5] Maleknejad K. and Karami M., The Petrov - Galerkin method for solving Fredholm integral equations of the second kind, 34th Iranian Mathematics Confrence, Shahrood University, Shahrood, Iran, Aug. 3 Sept. 2, 23.