Siraj ul Islam Laboratory for Multiphase Processes University

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

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

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

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. 900 s, 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

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

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 10

What are Wavelets? The word wavelet refers to the function h(t) that generates a basis for the orthogonal complement of V 0 in V 1 • 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 11

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

The Haar Scaling Functions and Haar Wavelets a) Haar scaling function (Father function) b) Haar Wavelet function (Mother wavelets) 13

The Haar Scaling Function and 14

The Haar Wavelets 15

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

The Haar Wavelets and its Integrals 17

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

Highly Oscillating function 19

Multi-Resolution Analysis 20

Multi-Resolution Analysis 21

Multi-Resolution Analysis 22

Multi-Resolution Analysis 23

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

Gaussian Quadrature 25

Gaussian Quadrature 26

Gaussian Quadrature 27

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

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

Numerical Integration based on Haar wavelets 30

Numerical Integration based on Haar wavelets 31

Numerical Integration based on Haar wavelets 32

Numerical integration for double and triple integrals 33

Numerical integration for double and triple integrals 34

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 35

Numerical triple integration with variable limits 36

Numerical results 37

Numerical results 38

Numerical results 39

Numerical results 40

Numerical results 41

Numerical results Symmetric Gauss Legendre 42

Convergence of the method 43

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) 44

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! Convergence can be problematic Use the same algorithms used for IVP

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

Computational Algorithm Based on Haar Wavelets Computer Math. Model. 2010 1. 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. 3. Variety of boundary condition can be implemented with equal ease 4. Simple applicability along with guaranteed convergence.

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

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

Haar Wavelets for Boundary Value Problem in ODEs

Adaptivity Through Non-uniform Haar Wavelets Inter. J. Comput. Method Eng Science & Mechanics (2010)

Adaptivity Through Non-uniform Haar Wavelets Where

Nodes Generations Through Cubic Spline

Nodes Generations initial temperature initial shape solve temperature of the slice at the new position calculate deformation of the slice at the new position final velocity in rolling direction initial velocity In rolling direction initial shape initial nodes final shape final nodes renoding

Nodes Generations Nodal points are generated through the following procedures: Transfinite Interpolation Elliptic Grid Generation

Nodes Generations TRANSFINITE INTERPOLATION Through this technique we can generate initial grid which is confirming to the geometry we encounter in different stages of plate and shape rolling. We suppose that there exists a transformation which maps the unit square, in the computational domain onto the interior of the region ABCD in the physical domain such that the edges map to the boundaries AB, CD and the edges are mapped to the boundaries AC, BD. The transformation is defined as Where respectively represents the values at the bottom, top, left and right edges

Nodes Generations An example of transformation from computational domain to physical domain.

Nodes Generations ELLIPTIC GRID GENERATION The mapping procedure defined above form the physical domain to the computational domain is described by are continuously differentiable maps of all order. The grid generated through transfinite interpolation can be made more conformal to the geometry by using the following elliptic grid generators where is the Jacobean of the transformation.