Wavelet Galerkin Solution for Boundary Value Problems

Transcription

1 Internationa Journa of Engineering Research and Deveopment e-issn: X, p-issn: X, Voume, Issue 5 (May 24), PP.2-3 Waveet Gaerkin Soution for Boundary Vaue Probems D. Pate, M.K. Abeyratna 2, M.H.M.R. Shyamai Dihani 3* Center for Industria Mathematics, Facuty of Technoogy & Engineering, The M S University of Baroda, Gujarat, India. 2 Department of Mathematics, Facuty of Science, University of Ruhuna, Matara, Sri Lanka. 3 Department of Interdiscipinary Studies, Facuty of Engineering, University of Ruhuna, Gae, Sri Lanka. Abstract:- Waveet-Gaerkin technique has very important advantages over cassica finite difference and finite eement method. In this paper, we have made an attempt to deveop a technique for Waveet-Gaerkin soution of Neumann and mixed boundary vaue probem in one dimension in parae to the work of J. Besora [5]. The Tayor s approach has been used to incude Neumann condition in Waveet-Gaerkin setup. The test exampes are given to vaidate techniques with exact soutions. Keywords:- Boundary Vaue Probems, Waveets, Scaing Function, Connection Coefficients, Waveet Coefficient,Waveet- Gaerkin Method. I. INTRODUCTION Waveet is an important area of mathematics and now a days it becomes an important too for appications in many areas of science and engineering. The orthonorma bases of compacty supported waveets for the space of square-integra function L 2 R was constructed by Daubecheis in 988 (see [7]). Those waveets are bases in a Gaerkin method to sove Dirichet as we as Neumann mixed boundary vaue probem. The approximations of partia differentia equations with waveet basis are more attentive, because the reaity of orthogonaity of compacty supported waveets. The concepts of Mutiresoution Anaysis based Fast waveet transform agorithm have given great momentum to make waveet best approximations of ODE s and PDE s. Waveet-Gaerkin technique is frequenty used now a days, and its numerica soution of partia differentia equations have been deveoped by severa researchers. Many contributions have been done in the area of waveets and differentia equations by authors ike Beykin et a.[4], Jawerth et a. [9], Qian et a.[4], Qian et a. [5], Wiiams et a. [6], Wiiams et a. [9]. Severa researchers now a day s uses Daubechies waveets as bases in a Gaerkin method to sove boundary vaue probem. The contribution in this area is due to the remarkabe work by Latto et a. [], Xu et a. [2], [22], Wiiams et a. [6], [7], [8], Mishra et a [3], Jordi Besora [5] and Amartunga et a. [], [2], [3]. It is known that the probems with periodic boundary conditions have been handed successfuy. Fictious boundary approach with Dirichet boundary conditions have been appied by Dianfeng et a. [8] in anayzing SH wave equation. Latto et a. [] presents a connection coefficients for j= and N=6. The probem with Distinct boundary conditions is deveoped by Jordi Besora [5]. In this paper, we have proposed waveet-gaerkin approximation to the Neumann and mixed boundary vaue probem, using compacty supported waveets as basis functions introduced by Daubechies [6], [7]. We have used Tayors approach to dea with Neumann and mixed Boundary conditions. II. WAVELETS An osciatory function ψ(x) L 2 R is a waveet if it has foowing properties: ψ(x) is n times differentiabe and its derivatives are continuous. ψ(x)is we ocaized both in time and frequency domains, i.e. ψ(x) and its derivatives must decay very rapidy. For frequency ocaization ψ ω must decay sufficienty fast as ω and that ψ ω becomes fat in the neighbourhood of ω =. The fatness is associated with number of vanishing moments of ψ(x), i.e. or equivaenty x k ψ x dx = 2

2 for k =,, n. d k ψ ω = dωk in the sense that the arger number of vanishing moments more is the fatness when ω is sma. ψ ω 2 dω < ω Daubechies waveets are compacty supported functions. Since they have non zero vaues within a finite interva and zero vaues esewhere in some intervas and they can be used for representation of soution of BVP. In order to have mutiresoution properties, the scaing function is defined as φ(x), and it must satisfy, φ x = L k= a k φ(2x k) or N N φ x = 2 k= C k φ 2x k ; when a k = 2C k with the property that k= a k = 2. where L denotes the genus of the Daubechies waveet. The functions generated with these coefficients wi have supp φ =, L and ( L ) vanishing waveet moments. 2 () Fig.: Daubechies Scaing function with L=6 and j= The basic waveet is defined in terms of scaing function by, ψ x = k= L ( ) k a k φ k (2x k) (2) A famiy of scaing functions is generated by transation from φ k x = φ x k (3) Simiar a famiy of waveets is denoted by transation and scaing from ψ j,k x = ψ 2 j x k (4) Since φ x dx = ; the scaing function satisfies the three conditions.. L k= a k = 2 (5) 2. The orthonormaity condition of the scaing function φ x k φ x m dx = δ k,m Impies that L a k a k 2m = δ,m (6) k= 22

3 where δ is a kroneker deta function. 3. The moment of the smooth waveet function to be zero. Waveet Gaerkin Soution for Boundary Vaue probems x m ψ x dx = impies that L k= ( ) k k m a k = (7) for m =,, L 2 The reation between scaing function and waveet function can be expressed as ; V j + = V j W j It impies that, for any integer m, φ x ψ x m dx = (8) Where denotes the direct sum and V j and W j be the subspaces generated, respectivey as the L 2 - cosure of the inear spans of φ j,k x = 2 j φ 2 j x k and ψ j,k x = 2 j ψ 2 j x k, k Z Using condition (8) we can write V V V j V j + And V j + = V W W W 2 W j where j is the diation parameter as scae. The support of the scae function φ 2 j x k for a certain vaue of j and L is given as supp φ 2 j x k = k L+k 2j, 2 j Each functionf(x) L 2 R, can be written in the form f x = k 2 j c k φ 2 j x k (9) and this shoud be satisfied the convergence property where f c k φ 2 j x k 2 jp C f p k c k = f(x)φ 2 j x k dx and C and p are constants. III. MULTIRESOLUTION ANALYSIS Construction of smooth waveets with compact support is a major task in todays research scenario. Meyer[6] and Maat [] found conditions which we caed mutiresoution anaysis which is use by Daubchies to construct waveet of compact support having arbitrary smoothness. A MRA of L 2 R is defined as a sequence of cosed subspace of V j j Z of L 2 R and a scaing function φ satisfy the foowing conditions.. The space V j are nested i.e. V V V, 2. The space L 2 R is a cosure of the union of a V j and the intersection of a V j is empty. i.e. j V j = L 2 R and j V j = 3. f x V j f 2x V j + ; j Z 4. f x V f x k V ; k Z 5. {φ(x k)} is an orthonorma basis for the space V. IV. WAVELET-GALERKIN METHOD Gaerkin Method was introduced by V.I Gaerkin. We can discuss this using a one-dimensiona differentia equation. Lu x = f x ; x ; () where L = d a x du + b(x)u x dx dx with boundary conditions, u = and u =. 23

4 Where a, b and f are given rea vaued continuous function on [,]. We aso assume that L is a eiptic differentia operator. Consider, v j is a compete orthonorma basis of L 2 [,] and every v j C 2 (, ) such that, v j = v j = We can seect the finite set Λ of indices j and then consider the subspace S, S = span{v j ; j Λ}. Approximate soution u s can be written in the form, u s = k Λ x k v k S () where each x k is scaar. We may determine x k by seeing the behaviour of u s as it ook ike a true soution on S. i.e. L u s, v j = f, v j j Λ, (2) such that the boundary conditions u s = and u s = are satisfied. Substituting u s vaues into the equation (2), k Λ L v k, v j x k = f, v j j Λ (3) Then this equation can be reduced in to the inear system of equation of the form a jk x k = y i (4) or AX=Y where A = a jk j,k Λ and a jk = L v k, v j, x denotes the vector x k k Λ and y denotes the vector y k k Λ. In the Gaerkin method, for each subset Λ, we obtain an approximation u s S by soving inear system (4). If u s converges to u then we can find the actua soution. Our main concern is the method of inear system (4) by choosing a waveet Gaerkin method. The matrix A shoud have a sma condition number to obtain stabiity of soution and A shoud sparse to perform cacuation fast. Simiary we can do the same thing in above set up. Let ψ j,k x = 2 j ψ 2 j x k (5) is a basis for L 2 [,] with boundary conditions ψ j,k = ψ j,k = j, k Λ and ψ j,k is C 2. We can repace equations (2) and (3) by u s = j,k Λ x j,k ψ j,k and j,k Λ L ψ j,k, ψ,m x j,k = f, ψ,m, m Λ So that AX=Y. Where A = a,m ;j,k,m,(j,k) Λ ; X = x j,k (j,k) Λ Y = y,m (,m ) Λ Then a,m ;j,k = L ψ j,k, ψ,m y,m = f, ψ,m Where, m and j, k represent respectivey row and coumn of A. This is an accurate method to find the soution of partia differentia equation. V. CONNECTION COEFFICIENT To find the soution of differentia equation using the Waveet Gaerkin technique we have to find the connection coefficients which is aso expored in Latto et a.([]), d d 2 d = d x 2 (x)dx (6) Ω 2 Φ Φ 2 Taking derivatives of the scaing function d times, we get φ d x = 2 d L k= a k φ d k (2x k) (7) d We can simpify equation (6) then for a Ω d 2 2 gives a system of inear equation with unknown vector Ω d d 2 TΩ d d 2 = 2 d Ωd d 2 (8) where d = d + d 2 and T = i a i a q 2+i. These are so caed scaing equation. But this is the homogeneous equation and does not have a unique nonzero soution. In order to make the system inhomogeneous, one equation is added and it derived from the moment equation of the scaing function φ. This is the normaization equation, d! = d 24 d,d M Ω Connection coefficient Ω,d can be obtained very easiy using Ω d d 2,

5 Ω,d = φ d φ d 2 dx = [φ d d φ 2 ] φ d φ d2+ dx As a resut of compact support waveet basis functions exhibit, the above equation becomes Ω d d 2 = φ d d φ 2 + dx After d integration, d Ω d 2 = d φ d d φ 2 +d 3 dx = d,d Ω k The moments M i of φ i are defined as With M = Latto et a derives a formua as M i k = m t= x k φ i (x)dx m t t t M m m t i = i 2(2 m ) = i= a i i t (2) Where a i s are the Daubechies waveet coefficients. Finay, the system wi be T 2d I (2) Ω d d = M d d! Matab software is used to compute the connection coefficient and moments at different scaes. Latto et a [] computed the coefficients at j= and L=6 ony. The computation of connection coefficients at different scaes have been done by using the program given in Jordi Besora [5]. The scaing function at j= and L=6 connection coefficient prepared by Latto et a [] is given in Tabe and Tabe 2 respectivey. Tabe I: Scaing function at j= and L=6 have been provided by Latto et a. [] x Phi(x) x Phi(x) E E E-7 L E-2 (9) 25

6 Tabe II: Connection coefficients at j= and L=6 have been provided by Latto et a. [3] using Ω n k=φ x kφ(x n)dx Ω[ 3] e-3 Ω[ 2] e- Ω[ ] e- Ω[] e+ Ω[] e+ Ω[2] e+ Ω[2] e- Ω[3] e- Ω[4] e-3 Consider VI. TEST PROBLEMS d 2 u(x) + βu x = f (22) dx 2 Now we use Waveet-Gaerkin method soution Here, we consider L = 6 and j = We can write the soution of the differentia equation (22) is, 2 j u x = c k 2 j 2Φ 2 j x k, x [,] k=l where c k are the unknown constant co-efficient Substitute (23) in (22) we get d 2 = k= 5 c k Φ x k, x [,] (23) dx 2 c kφ x k + β c k Φ x k = k= 5 k= 5 Taking inner product with φ x k We have k= 5 c k L +2 j 2 j L 26 k= 5 c k φ x k + β c k φ x k = k= 5 L +2 j 2 j L φ x k φ(x n) dx + β k= 5 c k φ x k φ(x n) dx = 2 j 2 j c k Ω n k + β k= 5 c k δ n,k = n = L, 2 L, 2 j i.e; n = 5, 4,, where By using Neumann Boundary conditions u = ; u = k= 5 (24) Ω n k = φ x k φ(x n)dx δ n,k = φ x k φ(x n)dx Considering eft and right boundary conditions we can write u = k= 5 c k φ k = (25) u = k= 5 c k φ k = (26) We use Tayors method to approximate derivative on the right side

7 u x = u a + hu a + h2 u a + 2! which gives the approximation of derivative at the boundary using forward, backward or centra differences. Equation (25) and (26) represent the reation of the coefficient c k. We can repace first and ast equations of (24) using (25) and (26) respectivey. Then we can get the foowing matrix with L=6 TC=B φ(4) φ(3) φ(2) φ() Ω[] Ω Ω[ ] Ω[ 2] Ω[ 3] Ω[ 4] Ω[ 5] Ω[2] Ω[] Ω Ω[ ] Ω[ 2] Ω[ 3] Ω[ 4] T = Ω[3] Ω[2] Ω Ω Ω[ ] Ω[ 2] Ω[ 3] Ω[4] Ω[3] Ω[2] Ω Ω Ω[ ] Ω[ 2] Ω[5] Ω[4] Ω[3] Ω[2] Ω Ω Ω[ ] p p 2 p 3 p 4 p = φ 4 φ 4 h p 2 = φ 3 φ 3 h p 3 = φ 2 φ 2 h p 4 = φ φ h C = c 5 c 4 c 3 c 2 c c c and B =. Suppose given Boundary Vaue Probem is, u xx = 2 (27) with boundary conditions u = and u () = The exact soution is, u x = x 2 + 2x + c 5 = c 4 = c 3 = c 2 = c = c = c =

8 Fig.2: Waveet-Gaerkin Soution for u xx = 2, u = and u () = 2. The given boundary vaue Probem is, u + u = ; u = and u = The exact soution is, u x = cos x + tan sin x c 5 = c 4 = c 3 = c 2 = c = c = c = Fig.3: Waveet-Gaerkin Soution for u + u = ; u = and u = 3. The given boundary vaue Probem is, u + u = ; u = and u + u() = The exact soution is, u x = cos x + (tan 2 sec 2) sin x 28

9 c 5 = c 4 = c 3 = c 2 = c = c = c = Fig.4: Waveet-Gaerkin Soution for u + u = ; u = and u + u() = VII. CONCLUSIONS Waveet method has shown a very powerfu numerica technique for the stabe and accurate soution of given boundary vaue probem. The comparison shows that the exact soution correates with numerica soution, using Daubechies waveets. REFERENCES []. K. Amaratunga, J.R. Wiiam, S. Qian and J. Weiss, waveet Gaerkin soution for one dimensiona partia differentia equations, Int. J. numerica methods eng. 37, (994). [2]. K. Amaratunga, J.R. Wiiams, Qian S. and J. Weiss, Waveet-Gaerkin soutions for one Dimensiona Partia Differentia Equations, IESL Technica Report No. 92-5, Inteigent Engineering Systems Laboratory, M. I. T., 992. [3]. K. Amaratunga and J.R. Wiiam, Waveet-Gaerkin Soutions for One dimensiona Partia Differentia Equations, Inter. J. Num. Meth. Eng. 37(994), [4]. J. Besora, Gaerkin Waveet Method for Goba Waves in D, Master Thesis,Roya Inst. of Tech. Sweden, 24. [5]. G. Beykin, R. Coifman, and V. Rokhin, 99,Fast Waveet Transfermation and Numerica Agorithm, Comm. Pure Appied Math.,44,4-83. [6]. I. Daubechis, 992, Ten Lectures on Waveet, Capita City Press, Vermont. [7]. I. Daubechies, 988, Orthonorma bases of compacty supported waveets, Commun. Pure App. Math., 4, [8]. L.U. Dianafeng, Tadashi Ohyoshi and Lin ZHU, Treatment of Boundary condition in the Appication of Waveet-Gaerkin Method to a SH Wave Probem, 996, Akita Uni. (Japan) 29

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