## Table of contents for An introduction to scientific computing : twelve computational projects solved with MATLAB / Ionut Danaila ... [et al.].

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```1 Numerical Approximatin of Model Partial Differential
E quations..........   ...             ..... ... ............... . .   I
1.1 Discrete Integration Methods for Ordinary Differential
E q uations  .......... .... .. .......... ......... . ...   1
1.1   C,onstnlction of Nu merlcal Integration Schemnes .   2
1.1.2  Genernl Form of Numierical Schemes . . , ,...      6
1.1.3  Appiication to the Absorption Equation .............  8
1.1.4  Stability of a Numerical Scheme  ................  . 9
1 2 Model Partial Differential Equations . . .                11
12.1.  The  Convection  Equation  ................. ....   11
12.22  The Wave Equation  .......... . . ......  . ... .  14
1.2,3  The  Heat Equation  ... ... ................ ...  17
1.3 Soutions and Programs ,......   .......    . . ...        19
Chapter References   ..................................       30
2   Nonlinear Differential Equations: Application to Chemical
K in etics  .......  .... ....  .................. ........ .....  33
.1 -Physticl  Problew and Mathematical Modeling .......... 33
22 Stability of the System .   ........                       34
2.3 Model for the Maintained Reaction ..............36
2.3.1 iExistence of a Criticai Point and Stability .  . ..... .. 36
2.3.2  Ni merical Solution  ................    ..       37
2.4 Model of Reaction with a Delay Term ........   . . ... . .37
5    SolUtions an   Programs  . .  ............. .... .  ... .  41
Chapter  RPeferences  .  . .  ... . . ...................     48
3   Polynomial Approximation ...          .................. 49
3.1  I ntroduction  .. . ... .. ... ..........   ............  9
3.2  Polynomial Interpolation  . . . . .  ...........       .  50
3.2.1  Lagrange  Interpolation. ............  ..........  51
3.2.2  Hen ite Interpoltion  ........ ......         .   57
3.=  HeH,,e nterola,on                     5
S3.P  est Polynomial App oximation  ...  ...................... .  59
33. 1 Best Uniform Approximation  .......                59
33.2  Be st Hilbertiai Approximation  . ... .......      61
3.3.  Discrete Least Squares Approximation ............. .64
3.4 Piecewise Polynomial Approxiniationr ..........           65
.4. 1 Picewise Con tant Approximation . . .  . .  . ..  . .66
34.2  Piecewise Affine Appoximation .. .... . ........ 67
3.4.3  Piecewise Cubic Approximnatin  .. ....... ....  .. 6. 6.
3.6  Solitons and Programs ..........                         70
SChapir References. .......                                   83
SSolving an Advectionr-Difusion Equation by a Finite
"El  ent  M ethod  .  .. .   .  .  .. .. .  .. ... .........  85
A.   Variationial Formuiation of the Problem  . . . .  .  .  .. . . 85
.2 /I P fI Fi tie Eleuienit MAli hod                         87
. 2  A  P2  Finie I tieient - ithod  ..       .  . ..          7.
"4.3  A  St2 a ilization iMethod  .  .....  . . ..  . ....   .  . .....  93
4 4.1 (omnputatio n of the Solution at the Endpoints of the
Intervals                  ........                93
14 .2 An'alysis of the Stabilized  1 hod . .... . . . .95
4,5 .The Case of a Variable Source Term . . . .   . .  . . . . .97
16  Solutons  and  Prog ams  . .. ...  ............ ...... .  97
(hapter Refeirence .. ....           .           .  . .. ... . 108
5   Soiving a Differential Equation by a Spectral Mlethod .      t.11
5.1 Siome Properties of th L egendre Pol vnomials . . . .......... 12
,2  (Gauss  Legendre  Qu drat e . ... ... . ...  . . .. . .. . . . 113
.3  TLe endu re Expansions .. ....  .  ...  .. .  .. . ..  . 115
S  A   p r al Discretization  ... ... ..        ...    .  . .   11
POssible  Extensions  ...... .   .  .  .  . . ..... ..  .119
-.6  S lonutio s  a mnd  Program s  ....................... ...120
(   pter  Refer nces .. .. ............                     .  25
SSignal Processing: Multiresolution Analysis .        ...        127
6.   1wntroductio i. . ........  ..........                  127
G6   Approximation of . Fun, tion: Theoretic'al Aspect .. . .  . . . 127
.2.1  Pie  ewlise  (Col stant Functions ........ .....  ....... 2:I
6.2.2 Decomposition of the Space S ,j . . . .... . . . . . 129
6.9.3 Decomposition and Reconstirction Algorithms .. . . . 132
6.2.4  inportance of Multiresolution Analysis. ..... .. . ... 133
6.   biltiresolution Analysis: Pa ctical Aspect . ....  .  .34
60   Multiresolution Analysis: Impleunk tation. .. . . ..  ......135
6.5 Introductionu to Wavelet Theory ........ ..       . .   .. .137
6,5.1  Scaling  Funm tions  wa d  Wav  lets  ............. . .. . .137
6,5.2  The Schauder W avelet  ...  . ...... . ....... . .. .. .  139
6.5.3 Implementation of the Schauder Wavelet ........... 41
6.5.4 The Daubechies Wavelet ...........    ..   ....142
6.5.5 Im plemientation of the Daubechies Wavelet D4 . . . .  144
6.6  Generalization: image Processing ..  ... .. .   . . . . . . 146
6.6.1 Image Processing: Implementation. . . .. ......... . . .. 147
6.7  Solutions  and  Programs  ..... .................. .....  148
Chapter  References  ................................ ......  150
7   Elasticity: Elastic Deformation of a Thin Plate .........   . 151
7.1  Introduction  .  . . ....  ........ ..... ...... .. .... .. .  151
7.2 Modeling Elastic Deformations (Linear Problem) ......... 152
7.3 Modeling Electrostatic Forces (Nonlinear Problem) ......... 153
7.4 Numerical Discretization of the Problem. .. .  .............. .154
7.5 Programming Tips....                    ....             M
7.5.2  Program  Validation  .... .. .. . .....  . . ...  158
7.6  Solving  the Linear Problem  .......................... ..   159
7.7 Solving the Nonlinear Problem  ...... ... ............. 159
7.7.1  A  Fixed-Point Algorithm . .. ...  ........ ..... ...... 159
7.7.2  Numerical Solution  ..................... ........160
7.8  Solutions and  Programs  .. ............. .. ..... ....... 162
7.8.1  Further Comments  ......... ....... .......  .... 62
Chapter  References ..............   ............      ..    64
8   Domain Decomposition Using a Schwarz Method .......... [165
8.1  Principle and Application Field of Domain Decomposition .. . 165
8.2 One-Dimensional Finite Difference Solution ..... ..._. ..,.. 166
8.3 Schwarz M ethod in One Dimension . . .. . . . .  . . . . . . . _167
8.3.1.  Discretization  . ...     I; .. .........   .   1, 68
8.4 Extension to the Two-Dimensional Case . . . . . . ... ... . 171.
8.4.1  Finite Difference Solution  . ...... ..... ..  . ...... .171
8.4.2 Domain Decomposition in the Two-Dimensional Case .. 175
8.4.3 Implementation of Realistic Boundary Conditions . . . 1.78
8.4.4  Possible Extensions ...... ................ ...... 180
.5 Solutions and Programs ...................           . ... 181
Chapter  References .......  ...... ....................... ..  190
9   Geometrical Design: Bezier Curves and Surfaces .     . .. ..... 193
9.1  Introduction  ..... . .. ....  .................... ........  193
9.2  B zier Curves .......  .. .. . ......                    193
9.3 Basic Properties of Bezier Cuves ... ....  ...... ...... 195
9.3.1  Convex Hull of the Control Points ..... . ... .  .. . . .. 195
9.3.2 Multiple Control Points .   ............         . _196
9.3.3 Tangent Vector to a Bezier Curve . . .  . ...... . . . 197
9.13.  Junction of Berzier Curves  ...... ..  . ...... .  . 197
9.3.5  Generation of the Point P(t)  .......... 198
9.   Splitting  Bezier  Curves  ..................   .......... ......   201
.6   Intersectionu  of Bezier Curves .....  .  .............203
9.6.1  limple mentation  ..... . .. .   .......... .  . .205
9.7  rezier  Surfaces  ..  .          . .................     20
9.8.1  Convex  Hull  ...........   .   .......... .......  206
9.,.2  Tangent  Vector  ...... ..........  . .... ..... .   207
9.8.3  Junction  of Bzier Patches, . ..        ..........
0.84 Zonstructioon of the Point 'P'f)                ...208
9.9  (on,i_-tru t'ion  oif Ip'7iT" ss f                ' .
-9.  Construction  of B  ei r SurfAces  ............ ...... ...
9.10  Solutions and  - ro urams  .. .  . . .  . . ..... . .... ...  210
C1a: pter  References. .   .............. .... ........      212
10 Gas Dynamics: The Riemann Problem and Discontinuous
Solutions: Application to the Shock Tube Problem. . .   .... . 213
10.1 Physical Description of the Shock Tube Problem  ........... 213
10.2 Euler Equations of Gas Dynamcs .. .......     ........  215
10.21  Dinienslonless Equations  . ....... .. 218
10.2.2  Exact Solution  . ..... ...  .. . . . . . ..........  21
1.0.3  Numr ric a  Solution  .. .....  .  ... .. . . . ..... ......   22
10 3.1 Lax-Wendroff and MacConrnack Centfered Schemes ...2. 2
10.3.2 Upwind Schemes (Roe s Approximate Solver) ......... 227
10.4 Solutions and Progranms .                          . . . . 232
C,apter  References  .  ... . ..... ........      . . .....  233
11 Thermal Engineering: Optimization of an Industrial
.F urn ace  .  . .  .  .  ...... .. ...   .  ... ...... .  . .. ...  .. 235
111  If troduction  ..... ... . . ............ .......... ......  235
11.2 F
Library of Congress subject headings for this publication: MATLAB, Numerical analysis Data processing

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