1. Wavelets As Galerkin Basis Aleksandr Yakovlev Department of Computational Physics St Petersburg State University

2. Contents 1.Introduction 2.Galerkin Method 3.Multiresolution analysis 4.Multiresolution analysis and Galerkin Method 5.Solution for an example equations 5.1 Finite differences method 5.2 Wavelet Galerkin solution 5.3 Incorporation of boundary conditions 5.4 Offsetting boundary conditions to control error 5.5 Comparison of results

3. Introduction 1.Differential equations ODE PDE constant coefficients variable coefficients 2.Domain and boundary conditions Dirichlet Neuman Cyclic 3.Method Galerkin method 4.Improvements Use Wavelet basis Increase resolution Increase order

4. The Galerkin Method

5. Multiresolution Analysis

6. Daubechies D6 scaling function

7. Multiresolution Analysis and Galerkin method

8. Finite difference solution to DE

9. Finite difference solution to DE (1)

10. Wavelet Galerkin Method

11. Wavelet Galerkin Method (1)

12. Wavelet Galerkin Method (2)

13. Wavelet Galerkin Method (3)

14. Wavelet Galerkin Method (4)

15. Incorporation of boundary conditions

17. Offsetting boundary conditions to control error

18. D12 wavelet coefficients of the delta function

19. Error in wavelet solution to boundary value problem

20. Offsetting of boundary sources to control error

21. Error in wavelet solution to boundary value problem with offset sources

22. Comparsion of results

23. Decay in error of wavelet and finite difference solutions with increasing sample size

24. Variation of computation time with increasing sample size

25. Acknowlegements Prof. S.Yu. Slavyanov, St Petersburg State Univercity Prof. A.V. Tsiganov, St Petersburg State Univercity Prof. S.L. Yakovlev, St Petersburg State Univercity And other colleagues of mine from the Department of Computational Physics