**Wavelet Based Parallel Solution**

**To Two-Point Boundary Value Problems**

**With Non-Periodic Boundary Conditions**

H. Narang (Tuskegee Univ.)

Abstract Number: **hn_125**

**Introduction**: Wavelets have been used for over a decade now for
signal processing, image processing, and data compression and produced
impressive results. Recently researchers have made attempts to find the
solution of ordinary and partial differential equations arising in
various fields of science and engineering trying to exploit the
multi-resolution character of wavelets. If successful, this will pave
the way for solving many science and engineering problems of a practical
nature where numerical solutions are either extremely difficult or too
time-consuming particularly near singularities and discontinuities. The
multi-resolution character of wavelets could be fruitfully exploited to
employ course grids in smooth regions and fine grids near singularities
and discontinuities in automatic and natural fashion to complete the
solutions.

**Methods and Materials**: Although some attempts have been recently
made in the solution of partial differential equations with Galerkin's
method, all their attempts have used periodic boundary conditions, which
seem artificial and non-practical in many real-world problems of science
and engineering. This may have been done to just establish the wavelet
technique in the area of applied mathematics and engineering but does
not provide the solution to realistic science and engineering problems,
which have mostly non-periodic boundary conditions. We have successfully
applied Galerkin's wavelet method to solve a variety of initial boundary
value problems ranging from diffusion, diffusion with chemical
reactions, diffusion with advection to wave propagation. We also solved
various non-linear problems, problems with discontinuous initial
conditions, and singular boundary value problems. This has resulted in a
system of differential algebraic equations which have been solved by the
back-ward difference method.

**Results**: Our research shows a close match with available exact
solutions to time dependent initial boundary value problems in science
and engineering with non-periodic boundary conditions. Future research
will concentrate on wavelet Galerkin's method technique to
two-dimensional initial boundary value problems in various fields of
science and engineering.

**References**:

1.  H. Narang and R. Zarrieff, "Wavelet Based Parallel Solution of
    Two-Point Boundary Value Problems," (invited paper), [Fifth
    International Conference of the Forum for Interdisciplinary
    Mathematics, International Conference on Combinatorics, Statistics,
    Pattern Recognition and Related Areas]{.underline}, Mysore
    University, Mysore, Karnataka, India, December 28-30, 1998.

2.  C. H. Romine and B. W. Peyton, "Computing Connection Coefficients of
    Compactly Supported

Wavelets on Bounded Intervals," Argonne National Laboratory, Argonne,
IL, April 1997.

3\. H. Narang, "Wavelet based Solution to Time Dependent Two-Point
Initial Boundary Value Problems," (invited

paper), [2001 SIAM Annual Conference on Linear Algebra in Signals,
Systems and Controls]{.underline}, Boston, MA,

2001\.

4\. R. Nekkanti, "Wavelet based Solution to Time Dependent Two-Point
Initial Boundary Value Problems",

M.S. Thesis, Tuskegee University, USA, 2001.