Wavelet-Galerkin Solutions of One and Two Dimensional Partial Differential Equations

Transcription

1 VOL 3, NO0 Oct, 202 ISSN CIS Journal All rights reserved Wavelet-Galerkin Solutions of One and Two Dimensional Partial Differential Equations Sabina, 2 Vinod Mishra, 2 Department of Mathematics, Sant Longowal Inst of Engg And Tech, Longowal (Pb), India sabinajindal8@gmailcom, 2 vinodmishra20@rediffmailcom ABSTRACT In recent years wavelets have received much attention because of its comprehensive mathematical power and good application potential in many interesting branches of science and technology The advantage of wavelet techniques over finite difference or element method is well known This paper offers wavelet based Galerkin methods for solving partial differential equations and provides some examples as test problems INTRODUCTION Since the contribution of orthogonal bases of compactly supported wavelet by Daubechies (988) and multiresolution analysis based fast wavelet transform algorithm by Beylkin (99), wavelet based approximation of ordinary and partial differential equations gained momentum in attractive way Wavelets have the capability of representing the solutions at different levels of resolutions, which make them particularly useful for developing hierarchical solutions to engineering problems Among the approximations, wavelet-galerkin technique is the most frequently used scheme these days Daubechies wavelets as bases in a Galerkin method to solve differential equations require a computational domain of simple shape This has become possible due to the remarkable work by Amartunga et al (993, 994 & 996) [-3], Latto et al (992 )[5], Xu et al (994) [0] and Williams et al (993 & 994) [8-9] Yet there is difficulty in dealing with boundary conditions So far problems with periodic boundary conditions or periodic distribution have been dealt successfully Advantage of wavelet-galerkin method over finite difference or element method has lead to tremendous applications in science and engineering To a certain extent, the wavelet technique is a strong competitor to the finite element method Although the wavelet method provided an efficient alternative technique for solving partial differential equations [PDEs] numerically, it is not as easy to implement as the traditional finite-difference method The reason is that the use of the wavelet-galerkin method to solve PDEs leads to the problem of computing integrals whose integrands involves products of compactly supported wavelets and their derivatives These integrals are evaluated using what is known as the connection coefficient method Notice that with increasing resolution (for Daub6) accuracy deteriorates, since condition number increases But for higher order wavelets, condition number is consistently lower Moreover, the condition number also depends on the order of derivatives, increases with increase in derivatives Wavelet-Galerkin methods such as Amaratunga et al method, fictitious boundary approach, capacitance matrix method, difference wavelet-galerkin difference method and wavelet Taylor Galerkin approach for PDEs are well known due to their own advantages For wavelet-galerkin technique for ordinary differential equations refer to Mishra et al [6-7] 2 2D MULTIRESOLUTION ANALYSIS 2D Multiresolution Analysis [MRA] (Christov [4]) can be constructed by taking tensor product of D ones Let be the space of two-dimensional square integrable function Given an MRA with scaling function with wavelet and corresponding wavelet space, define Will be spanned by and each one of new wavelets will span its corresponding wavelet space, and Furthermore For any function, projection of onto scaling space at resolution may be defined by 373

2 VOL 3, NO0 Oct, 202 ISSN CIS Journal All rights reserved 3 2D WAVELET-GALERKIN TECHNIQUE Consider the following problem: On the region () For some will capture all details of the original function Select Substituting () and forcing the condition (2), we find (3) With boundary conditions boundary of on the (4) Assume that can be represented accurately by a set of analytic function such that is so chosen as to satisfy the initial conditions For approximation to be good, the residual of Clearly this is D problem and cannot be applied to 2D Let us assume function of and for each, we can solve the system as for D problem Ultimately we obtain (5) Solving (5), we find for each the coefficients thus the solution to () must reduce to minimum, for that we use the simplest Galerkin Method, namely Ritz-Raleigh This method minimizes residual to the effect that which in turn gives For details refer to Christov [4] 4 WAVELET METHODS FOR PDES 4 Wavelet-Galerkin Solution of the Periodic Problem [3] Consider the two dimensional Poission s equation (6) (2) Where of period are periodic in Next step is to find equation formation () form the matrix be Let the approximate solution at scale,,, Let be an MRA with scaling function acts as orthonormal basis for At each approximation level, orthogonal projection of onto is taken in the manner (fit fixed) Put Where are periodic wavelet coefficients of so that (7) 374

3 VOL 3, NO0 Oct, 202 ISSN CIS Journal All rights reserved periods and also are periodic in with at all dyadic points Let us discretize Inverse FT gives the solution 42 Capacitance Matrix Method and the Boundary Conditions [3] The matrix representation is (8) (9) Consider the problem (6), ie are periodic with period with Dirichlet s boundary conditions on the boundary of region If is not periodic, it can be made periodic making it zero or extending smoothly outside Let be the solution in with periodic boundary conditions The solution to the differential equation with Dirichlet boundary conditions is obtained by adding another function such that Since are the convolution kernels, ie the first column of the scaling function matrices and is the wavelet coefficient matrix Similarly, RHS of (6) can be expressed for in, must satisfy However, on or outside, may take such values as to make satisfy the given boundary conditions The desired effect may be achieved by placing sources (or delta function) along a closed boundary which encompasses the region In other words, is given by as in takes as (0) (0) () and is the delta function at So the solution ; (2) Substitute the expansion of and into the given differential equation (6) and then take inner product on both sides with,, Use We obtain, we find and Taking Fourier Transforms of (9), (0) and (), Discretize (2) at the points on stands for number of points on (3) Similarly considering the mesh points on The discretized solution (3) takes the form Solving the matrix formation, 375

4 VOL 3, NO0 Oct, 202 ISSN CIS Journal All rights reserved yields which when substituted in (3) gives 43 Wavelet-Galerkin Fictitious Boundary Approach Consider the equation Let approximate solution be Or (5) That is,, Similarly are the connection coefficients Solving (5) will give the coefficients and hence the solution 5 SOLUTIONS OF PDES: TEST PROBLEMS Substituting in the given equation and letting inner product with Problem : Consider the D wave equation gives with initial condition condition and the boundary Using 2-term connection coefficients and following the method as in Section43, the comparable solution with exact one with at N = 7 376

5 VOL 3, NO0 Oct, 202 ISSN CIS Journal All rights reserved u(x,t) WG Solution Exact solution x 3 x error x Fig : Wavelet solution of D wave equation (Problem ) with error estimation for N = 7, Problem 2: Consider 2D problem with u(x,y,t) Fig 2: Solution 2D Problem (Problem 2) with error estimation for N = 0, 377

6 VOL 3, NO0 Oct, 202 ISSN CIS Journal All rights reserved Above is the 3D graph of solution obtained through wavelet-galerkin technique and error in comparison to finite difference solution 6 CONCLUSION Despite some disadvantages wavelet-galerkin technique provides efficient solutions for PDEs The graph of wavelet-galerkin solutions for D and 2D are shown with error estimates in comparison to known/finite difference solutions Fairly good solutions are obtained with wavelet-galerkin technique REFERENCES [] Amaratunga, Kevin, Hierarchical Wavelet-Based Numerical Models for Digital Data, PhD Thesis, MIT (USA), 996 [2] Amaratunga, Kevin, JR Williams, Sam Qian and John Weiss, Wavelet-Galerkin Solutions for One Dimensional Partial Differential Equations, Int J Numer Meth Engg 37(994), [3] Amartunga, Kevin and JR Williams, Wavelet Based Green s Function Approach to 2D PDEs, MIT (USA), Engg Comput 0 (993), [5] Latto, A, HL Resnikoff and E Tenenbaum, The Evaluation of Connection Coefficients of Compactly Supported Wavelets, in: Proceedings of the French USA Workshop on Wavelets and Turbulence (ed Y Maday), Princeton University, New York, Springer- Verlag, 992 [6] Mishra, Vinod and Sabina, Wavelet Galerkin Solutions of Ordinary Differential Equations, International Journal of Mathematical Analysis 5(20), [7] Mishra, Vinod and Sabina, Wavelet Galerkin Finite Difference Solutions of ODEs, Advanced Modeling and Optimization 3(20), [8] Williams, JR and Kevin Amaratunga, High Order Wavelet Extrapolation Schemes for Initial Problems and Boundary Value Problems, July 994, IESL Tech Rep, No 94-07, Intelligent Engineering Systems Laboratory, MIT(USA) [9] Williams, JR and Kevin Amaratunga, Simulation Based Design using Wavelets, Intelligent Engineering Systems Laboratory, MIT (USA), pp- 2 [4] Christov, Ivan, Wavelet-Galerkin Method for Operator Equations, Acta Numerica 6 (997), [0] Xu, J-C And W-C Shann, Galerkin-Wavelet Methods for Two-point Boundary Value Problems, Numer Math 63(992),