Siraj –ul – Islam Laboratory for Multiphase Processes University of Nova Gorica, Slovenia

1. Siraj –ul – Islam Laboratory for Multiphase Processes University of Nova Gorica, Slovenia

2. Siraj –ul – Islam Laboratory for Multiphase Processes University of Nova Gorica, Slovenia

3. Some Applications of Wavelets Siraj –ul – Islam Laboratory for Multiphase Processes University of Nova Gorica, Slovenia

4. Some Applications of Wavelets Siraj –ul – Islam Laboratory for Multiphase Processes University of Nova Gorica, Slovenia

5. Khyber Pass"Khyber is a Hebrew word meaning a fort" Some Applications of Wavelets Siraj –ul – Islam Laboratory for Multiphase Processes University of Nova Gorica, Slovenia

6. Khyber Pass"Khyber is a Hebrew word meaning a fort" • Alexander the Great and his army marched through the Khyber to reach the plains of India ( around 326 BC) • In the A.D. 900s, Persian, Mongol, and Tartar armies forced their way through the Khyber • Mahmud of Ghaznawi, marched through with his army as many as seventeen times between 1001-1030 AD • Shahabuddin Muhammad Ghaur, a renowned ruler of Ghauri dynasty, crossed the Khyber Pass in 1175 AD to consolidate the gains of the Muslims in India • In 1398 AD Amir Timur, the firebrand from Central Asia, invaded India through the Khyber Pass and his descendant Zahiruddin Babur made use of this pass first in 1505 and then in 1526 to establish a mighty Mughal empire • January 1842, in which about 16,000 British and Indian troops were killed

7. Some Applications of Wavelets Siraj –ul – Islam Laboratory for Multiphase Processes University of Nova Gorica, Slovenia

8. Some Applications of Wavelets Siraj –ul – Islam Laboratory for Multiphase Processes University of Nova Gorica, Slovenia

9. Some Applications of Wavelets Siraj –ul – Islam Laboratory for Multiphase Processes University of Nova Gorica, Slovenia

10. What are Wavelets? A wavelet is a function which • maps from the real line to the real line • has an average value of zero • has values zero except over a bounded domain

11. What are Wavelets? The word wavelet refers to the function h(t) that generates a basis for the orthogonal complement of V0 in V1 • A small wave • Extends to finite interval Wavelets analysis is a procedure through which we can decompose a given function into a set of elementary waveforms called wavelets

12. Types Of Wavelets ICCES 2010 Las Vegas, March 28 - April 1, 2010

13. The Haar Scaling Functions and Haar Wavelets • Haar scaling function (Father function) • Haar Wavelet function (Mother wavelets)

14. The Haar Scaling Function and

15. The Haar Wavelets

16. The Haar Wavelets and its Integrals with the collocation points The repeated integral of Haar wavelet is given by

17. The Haar Wavelets and its Integrals

18. Some applications of wavelets Numerical Analysis Ordinary and Partial Differential Equations Signal Analysis Image processing and Video Compression (FBI adopting a wavelet-based algorithm as a the national standard for digitized finger prints) Control Systems Seismology

19. Highly Oscillating function

20. Multi-Resolution Analysis

21. Multi-Resolution Analysis

22. Multi-Resolution Analysis

23. Multi-Resolution Analysis

24. Multi-Resolution Analysis Scaling function (Father wavelet) basis in V Wavelet function (Mother wavelets) basis in W

25. Gaussian Quadrature

26. Gaussian Quadrature

27. Gaussian Quadrature

28. Problems with Gaussian Quadrature • Solution 2n by 2n system • Search for better nodal values • Finding optimized values for the unknown weights

29. Numerical Integration based on Haar wavelets Inter. J. Computer Math. 2010

30. Numerical Integration based on Haar wavelets

31. Numerical Integration based on Haar wavelets

32. Numerical Integration based on Haar wavelets

33. Numerical integration for double and triple integrals

34. Numerical integration for double and triple integrals

35. Numerical double integration with variable limits To extend the present idea to numerical integration with variable limits and make it more efficient, we use an iterative approach instead of using two and three dimensional wavelets

36. Numerical triple integration with variable limits

37. Numerical results

38. Numerical results

39. Numerical results

40. Numerical results

41. Numerical results

42. Numerical results Symmetric Gauss Legendre Symmetric Gauss Legendre

43. Convergence of the method

44. Numerical Solution of Ordinary Diff. Eqs. Existing Methods • Runge-Kutta family of Methods (Need shooting like to convert BVP into IVP, Stability limits) • Finite difference Methods (Low accuracy and large matrix inversion) • Asymptotic Methods (Series solution convergence problem)

45. Convergence can be problematic Use the same algorithms used for IVP Shooting method • Idea: transform the BVP in an initial value problem (IVP), by guessing some of the initial conditions and using the B.C. to refine the guess, until convergence is reached Target Too high: reduce the initial velocity! Too low: increase the initial velocity!

46. Shooting Method for Boundary Value Problem ODEs Definition: a time stepping algorithm along with a root finding method for choosing the appropriate initial conditions which solve the boundary value problem. Second-order Boundary-Value Problem y(a)=A and y(b)=B

47. Computational Algorithm Based on Haar Wavelets Computer Math. Model. 2010 • Contrary to the existing methods, the new method based on wavelets can be used directly for the numerical solution of both boundary and initial value problems 2. Stability in time integration is overcome. • Variety of boundary condition can be implemented with equal ease 4. Simple applicability along with guaranteed convergence.

48. Haar Wavelets for Boundary Value Problem in ODEs Consider the following coupled nonlinear ODEs Along with boundary conditions

49. Haar Wavelets for Boundary Value Problem in ODEs Wavelets approximation for and can be given by,

50. Haar Wavelets for Boundary Value Problem in ODEs