The discrete orthogonal wavelet-Galerkin method is illustrated as an effective method for solving partial differential equations (PDE's) with spatially varying parameters on a bounded interval. Daubechies scaling functions provide a concise but adaptable set of basis functions and allow for implementation of varied loading and boundary conditions. These basis functions can also effectively describe C0 continuous parameter spatial dependence on bounded domains. Doing so allows the PDE to be discretized as a set of linear equations composed of known inner products which can be stored for efficient parametric analyses. Solution schemes for both free and forced PDE's are developed; natural frequencies, mode shapes, and frequency response functions for an Euler–Bernoulli beam with piecewise varying thickness are calculated. The wavelet-Galerkin approach is shown to converge to the first four natural frequencies at a rate greater than that of the linear finite element approach; mode shapes and frequency response functions converge similarly.

References

References
1.
Gockenbach
,
M.
,
2011
,
Partial Differential Equations: Analytical and Numerical Methods
,
SIAM USA
,
Philadelphia
.
2.
Fletcher
,
C.
,
1984
,
Computational Galerkin Methods
,
Springer-Verlag
,
New York
.
3.
Lapidus
,
L.
, and
Pinder
,
G.
,
1982
,
Numerical Solution of Partial Differential Equations in Science and Engineering
,
Wiley
,
New York
.
4.
Mallat
,
S.
,
2009
,
A Wavelet Tour of Signal Processing
,
Academic
,
New York
.
5.
Strang
,
G.
, and
Nguyen
,
T.
,
1996
,
Wavelets and Filter Banks
,
Wellesley-Cambridge Press
,
Wellesley, MA
.
6.
Cohen
,
A.
,
Daubechies
, I
.
, and
Vial
,
P.
,
1993
. “
Wavelets on the Interval and Fast Wavelet Transforms
,”
Appl. Comput. Harmon. Anal.
,
1
, pp.
54
81
.10.1006/acha.1993.1005
7.
Daubechies
,
I.
,
1992
,
Ten Lecturers on Wavelets
,
SIAM
,
Philadelphia
.
8.
Williams
,
J.
, and
Amaratunga
,
K.
,
1994
, “
Introduction to Wavelets in Engineering
,”
Int. J. Numer. Methods Eng.
,
37
, pp.
2365
2388
.10.1002/nme.1620371403
9.
Amaratunga
,
K.
, and
Williams
,
J.
,
1997
, “
Wavelet-Galerkin Solution of Boundary Value Problems
,”
Arch. Comput. Methods Eng.
,
4
(
3
), pp.
243
285
.10.1007/BF02913819
10.
Ala
,
G.
,
Silvestre
,
M. L. D.
,
Francomano
,
E.
, and
Tortorici
,
A.
,
2003
, “
An Advanced Numerical Model in Solving Thin-Wire Integral Equations by Using Semi-Orthogonal Compactly Supported Spline Wavelets
,”
IEEE Trans. Electron.
,
45
, pp.
218
228
.
11.
Joly
,
P.
,
Maday
,
Y.
, and
Perrier
, V
.
,
1994
, “
Towards a Method for Solving Partial Differential Equations by Using Wavelet Packet Bases
,”
Comput. Methods Appl. Mech. Eng.
,
116
, pp.
301
307
.10.1016/S0045-7825(94)80036-7
12.
Beylkin
,
G.
, and
Keiser
,
J.
,
1997
, “
On the Adaptive Numerical Solution of Nonlinear Partial Differential Equations in Wavelet Bases
,”
J. Comput. Phys.
,
132
(
2
), pp.
233
259
. 10.1006/jcph.1996.5562
13.
Chen
,
M.
,
Hwang
,
C.
, and
Shih
,
Y.
,
1996
, “
The Computation of Wavelet-Galerkin Approximation on a Bounded Interval
,”
Int. J. Numer. Methods Eng.
,
39
, pp.
2921
2944
.10.1002/(SICI)1097-0207(19960915)39:17<2921::AID-NME983>3.0.CO;2-D
14.
Lakestani
,
M.
,
Rassaghi
,
M.
, and
Dehghan
,
M.
,
2006
, “
Numerical Solution of the Controlled Duffing Oscillator by Semi-Orthogonal Spline Wavelets
,”
Phys. Scr.
,
74
, pp.
362
366
.10.1088/0031-8949/74/3/010
15.
Lakestani
,
M.
, and
Dehghan
,
M.
,
2006
, “
The Solution of a Second-Order Nonlinear Differential Equation With Neumann Boundary Conditions Using Semi-Orthogonal B-Spline Wavelets
,”
Int. J. Comput. Math.
,
83
, pp.
685
694
.10.1080/00207160601025656
16.
Restrepo
,
J. M.
, and
Leaf
,
G. K.
,
1993
, “
Inner Product Computations Using Periodized Daubechies Wavelets
,”
Int. J. Numer. Methods Eng.
,
40
(
19
), pp.
3557
3578
.10.1002/(SICI)1097-0207(19971015)
17.
Pernot
,
S.
, and
Lamarque
,
C.-H.
,
2001
, “
A Wavelet-Galerkin Procedure to Investigate Time-Periodic Systems: Transient Vibration and Stability Analysis
,”
J. Sound Vib.
,
245
, pp.
845
875
.10.1006/jsvi.2001.3610
18.
Beylkin
,
G.
,
1992
, “
On the Representation of Operators in Bases of Compactly Supported Wavelets
,”
SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
,
6
(
6
), pp.
1716
-
1740
.10.1137/0729097
19.
Romine
,
C. H.
, and
Peyton
,
B. W.
,
1997
, “
Computing Connection Coefficients of Compactly Supported Wavelets on Bounded Intervals
,”
U.S. Department of Energy
, Report No. ORNL/TM-13413.
20.
Plonka
,
G.
,
Selig
,
K.
, and
Tasche
,
M.
,
1995
, “
On the Construction of Wavelets on a Bounded Interval
,”
Adv. Comput. Math.
,
4
, pp.
357
388
.10.1007/BF02123481
21.
Hein
,
H.
, and
Feklistova
,
L.
,
2011
, “
Free Vibrations of Non-Uniform and Axially Functionally Graded Beams Using Haar Wavelets
,”
Eng. Sci.
,
33
(
12
), pp.
3696
3701
.10.1016/j.engstruct.2011.08.006
22.
Boyd
,
J. P.
,
2001
,
Chebyshev and Fourier Spectral Methods
,
Dover
,
New York
.
23.
Welstead
,
S.
,
1999
,
Fractal and Wavelet Image Compression Techniques
,
SPIE
,
Bellingham, WA
.
24.
Latto
,
A.
,
Resnikoff
,
H.
, and
Tenenbaum
,
E.
,
1992
, “
The Evaluation of Connection Coefficients of Compactly Supported Wavelets
,” Proceedings of the French–USA Workshop on Wavelets and Turbulence, Princeton University, Princeton, NJ, June,
Springer-Verlag
,
New York
.
25.
Doyle
,
J.
,
1997
,
Wave Propagation in Structures
,
Springer-Verlag
,
New York
.
26.
Zhang
,
T.
,
Tian
,
Y.-C.
,
Tadé
,
M.
, and
Utomo
,
J.
,
2007
, “
Comments on ‘The Computation of Wavelet-Galerkin Approximation on a Bounded Interval
,”
Int. J. Numer. Methods Eng.
,
72
, pp.
244
251
.10.1002/nme.2022
27.
Humar
,
J.
,
1990
,
Dynamics of Structures
,
Prentice-Hall
,
Upper Saddle River, NJ
.
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