 اسم المؤلف
P. Dechaumpwwhai, N. Wansophark
التاريخ
1 أبريل 2022
المشاهدات
215
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Numerical Methods in Science and Engineering
Theories with MATLAB, Mathematica, Fortran, C and Python Programs
P. Dechaumpwwhai
N. Wansophark
Contents
Preface v

1. First Step to Numerical Methods 1
1.1 Introduction 1
1.2 What are the Numerical Methods? 3
1.3 Need for Studying Numerical Methods 3
1.4 Computer Hardware and Software 8
1.5 Errors 11
1.6 Closure 13
Exercises 13
2. Root of Equations 19
2.1 Introduction 19
2.2 Graphical Method 20
2.3 Bisection Method 22
2.4 False-Position Method 25
2.5 One-Point Iteration Method 28
2.6 Newton-Raphson Method 32
2.7 Secant Method 37
2.8 MATLAB Functions for Finding Root of Equation 38
2.9 Roots of System of Non-linear Equations 41
2.9.1 Direct iteration method 42
2.9.2 Newton-Raphson iteration method 43
2.10 Closure 46
Exercises 46
3. System of Linear Equations 53
3.1 Introduction 53
3.2 Cramer’s Rule 55
3.3 Gauss Elimination Method 57
3.4 Problems of Gauss Elimination Method 62
3.4.1 Division by zero 62
3.4.2 Round-off error 62
3.4.3 Ill-conditioned system 63viii Contents
3.5 Improved Gauss Elimination Method 63
3.5.1 Pivoting 64
3.5.2 Scaling 65
3.5.3 Tridiagonal system 66
3.6 Gauss-Jordan Method 68
3.7 Matrix Inversion Method 70
3.8 Solving System of Linear Equations by MATLAB 71
3.9 LU Decomposition Method 72
3.10 MATLAB Function for LU Decomposition 77
3.11 Cholesky Decomposition Method 78
3.12 MATLAB Function for Cholesky Decomposition 81
3.13 Jacobi Iteration Method 82
3.14 Gauss-Seidel Iteration Method 85
3.15 Successive Over-relaxation Method 87
3.17 Closure 101
Exercises 101
4. Interpolation and Extrapolation 111
4.1 Introduction 111
4.2 Newton’s Divided Differences 112
4.2.1 Linear interpolation 112
4.2.3 nth-order Polynomial interpolation 115
4.3 Lagrange Interpolating Polynomials 118
4.3.1 Linear interpolation 118
4.3.3 Polynomial interpolation 122
4.4 Spline Interpolations 124
4.4.1 Linear spline 125
4.4.3 Cubic spline 128
4.5 MATLAB Functions for Interpolations 132
4.6 Extrapolation 134
4.7 Closure 135
Exercises 136
5. Least-Squares Regression 141
5.1 Introduction 141
5.2 Linear Regression 142
5.3 Linear Regression for Nonlinear Data 146
5.4 Polynomial Regression 150
5.5 MATLAB Functions for Least-Squares Regression 154Contents ix
5.6 Multiple Regression 156
5.6.1 Linear 156
5.6.2 Polynomial 162
5.7 Closure 164
Exercises 164
6. Numerical Integration and Differentiation 173
6.1 Introduction 173
6.2 Trapezoidal Rule 175
6.3 Composite Trapezoidal Rule 180
6.4 Simpson’s Rule 184
6.5 Composite Simpson’s Rule 186
6.6 Newton-Cotes Formulas 188
6.7 Romberg Integration 192
6.8 Gauss Integration 197
6.9 Multiple Integration 205
6.10 MATLAB Commands for Integration 208
6.11 Differentiation 210
6.12 MATLAB Commands for Differentiation 216
6.13 Closure 217
Exercises 217
7. Ordinary Differential Equations 227
7.1 Introduction 227
7.2 Euler’s Method 230
7.3 Heun’s Method 234
7.4 Modified Euler’s Method 237
7.5 Runge-Kutta Method 239
7.5.1 Second-order 240
7.5.2 Third-order 242
7.5.3 Fourth-order 243
7.6 System of Equations 246
7.7 MATLAB Commands 250
7.8 Multistep Methods 252
7.8.1 Non-self-starting Heun’s method 253
7.9 Closure 260
Exercises 260
8. Partial Differential Equations 269
8.1 Introduction 269x Contents
8.1.1 Definitions 269
8.1.2 Types of equations 270
8.1.3 Boundary and initial conditions 272
8.2 Elliptic Equation 273
8.2.1 Differential equation 273
8.2.2 Computational procedures 274
8.2.3 Example 277
8.3 Parabolic Equation 283
8.3.1 Differential equation 283
8.3.2 Explicit method 284
8.3.3 Implicit method 289
8.3.4 Crank-Nicolson method 292
8.4 Hyperbolic Equation 297
8.4.1 Differential equation 297
8.4.2 Computational procedures 299
8.4.3 Example 301
8.5 Closure 305
Exercises 305
Bibliography 317
Appendix A Matrices 319
A.1 Definitions 319
A.2 Matrix Addition and Subtraction 321
A.3 Matrix Multiplication 321
A.4 Matrix Transpose 322
A.5 Matrix Inverse 322
A.6 Matrix Partitioning 323
A.7 Calculus of Matrices 323
Appendix B MATLAB Fundamentals 325
B.1 Introduction 325
B.2 MATLAB Environment 325
B.2.1 Command Window 326
B.2.2 Command History Window 328
B.2.3 Edit/debug Window 328
B.2.4 Figure Window 329
B.2.5 Workspace Window 330
B.2.6 Help Window 330
B.3 Variables in MATLAB 331
B.3.1 Scalar variable 331
B.3.2 Vector variable 331
B.3.3 Use of colon symbol 333
B.3.4 Displaying data 333
B.3.5 Use of long commands 334Contents xi
B.4 Mathematical Operations 334
B.5 Built-In Functions 336
B.6 Plotting Graphs 338
B.7 Programming 343
B.7.1 Script file 343
B.7.2 Function file 344
B.7.3 Input and output commands 344
B.7.4 Read and write data file 347
B.7.5 Programming commands 349
B.7.5.1 Decision commands 349
B.7.5.2 Iteration commands 352
Appendix C Derivation of Fourth-Order Runge-Kutta Formula 355
Appendix D Mathematica Commands 359
Index 37
dex
Acceleration, 4, 20, 298
formulas, 256
fourth-order, 256
closed formulas, 258
fourth-order, 259
Aerodynamic heating rate, 160
Air density, 111, 125,
Airfoil, 111, 125,
Approximate solution, 7
Approximation
first-order, 32
second-order, 33
third-order, 33
zero-order, 33
Atmospheric pressure, 166
Back substitution, 57, 59, 60, 63, 64
Backward divided differences, 211
Beam bending, 48
Bessel function, 112
Binary system, 12
Bisection method, 20, 22
Bits, 12
Boole’s rule, 191, 206
Boundary condition
Dirichlet, 272
Neumann, 272
Boundary layer thickness, 161
Bracketing method, 24, 28, 39
Buckling, 16, 20, 47
C language, 9
Cable deflection, 50
CAE, 9
Central difference approximation, 285, 289
Central divided differences, 212
Cholesky decomposition method, 55, 78
Commercial software, 3, 22, 46, 208, 243
Complex geometry, 1, 305
Computer
hardware, 8
language, 9
mainframe, 9
personal, 9
software, 1, 9, 13
asymmetric matrix, 78, 97,
bi-conjugate, 100
generalized minimal residual, 100
modified, 94
quasi-minimal residual, 100
square, 100
Conservation of energy, 54, 269, 273, 283
Continuous function, 112
Convergence, 23, 26,
criterion, 23, 26
Coordinate transformation, 202
Corrector, 235, 254
Courant number, 300
Cramer’s rule, 55, 101
Crank-Nicolson method, 292, 305
Dependent variable, 5, 39, 156, 210, 227372 Numerical Methods in Science and Engineering
Derivative
Approximate, 210
exact, 37, 211
Determinant, 55, 63
Differences
backward, 215
central, 215
forward, 215
Differentiation, 174, 210
Direct iteration method, 41
Divergence, 32, 35
Divided differences
central, 212,
first backward, 211, 255
first forward, 211
second forward, 213
Drag coefficient, 4, 228
Elliptic integral
first kind, 174, 218
Error
absolute, 36
approximate, 12
blunder, 11
data, 11
exact, 179, 232
modeling, 11
percentage, 12
propagation, 11
relative, 36, 195
round-off, 11
true, 12
truncation, 11
Euler’s method, 229, 230, 260
Exact
solution, 3, 5
temperature, 54, 277, 294
Experimental data, 111, 142
Explicit method, 284, 297
Exponential
function, 14
model, 148
Extrapolation, 111, 134
False-position method, 20, 25
Finite difference
mesh, 2
method, 1, 270, 299
model, 54, 277
Finite element
mesh, 2
method, 1
Finite volume method, 1
Floating points, 12
Flow rate, 140, 168
Fortran language, 321
Forward
differencing, 178
divided differences, 112
elimination, 58
Fourier’s law, 273, 283
Function
cosine, 10, 329
error, 16
exponential, 14, 166,
hyperbolic cosine, 19
sine, 10, 277
Gauss elimination method
improved, 63
naive, 61
problems, 62
Gauss integration, 197
n-point, 200
two-point, 200
locations, 204
weights, 204, 208
Gauss-Jordan method, 55, 68
Gauss-Legendre formulas, 201
Gauss-Seidel iteration method, 55, 85, 277
Graphical method, 20
Gravitational acceleration constant, 20, 227
Grid point, 55, 274, 280
Heat conduction
bar, 283
plate, 305, 311
Heat flux, 273
Heun’s method, 234
non-self-starting, 253
High-speed wind tunnel, 160, 171
Identity matrix, 63, 320
Ill-conditioned system, 63
Implicit method, 289
Increment function, 240
Independent variable, 228, 270
Infinite series, 10, 228Index 373
Initial condition, 273, 300
Insulated edge, 281
Integrand, 174
Integration
area under curve, 174
double, 205
error, 178 , 182, 196
error function, 174, 217
logarithmic function, 174
polynomial, 188, 217
summation, 174
Interpolation, 112
fourth-order, 124
linear, 119,112
Interval, 279
Jacobi iteration method, 82
Jacobian matrix, 44
Lagrange interpolating
polynomials, 118
functions, 119
Laplace’s equation, 270, 274
Least-squares regression, 141
Linear regression, linear data, 142, 146
Logarithm, 149, 162
natural, 149
LU decomposition method, 55, 72, 277
Mach number, 305
Mass density, 283
Mathematica, 9
MATLAB, 9
built-in functions, 38, 332, 336
contour plotting, 339
decision commands, 349
environment, 325
function file, 344
graph plotting, 336
icon symbol, 331
input/output commands, 344
iteration commands, 352
long commands, 334
long format, 331
mathematica operations, 335
m-file, 251, 327
programming commands, 349
programming, 347
scalar variables, 328
script file, 328
short format, 333
vector variables, 331
MATLAB command
cumtrapz, 209
ode113, 250
ode23, 250
ode45, 250
solver, 250
sysode, 252
trapz, 208
(back slash), 72
MATLAB function
chol, 81
fzero, 39
inline, 39
interpl, 133
lu, 83
polyfit, 154
polyval, 155
roots, 40
spline, 132
MATLAB window
command, 324
edit/debug, 326
figure, 330
help, 330
history, 329
workspace, 330
Matrix, 319
calculus, 323
coefficients, 320
column, 321
differentiation, 324
identity, 320
integration, 324
inverse, 322
inversion method, 71
multiplication, 326
non-positive definite, 100
partitioning, 323
positive definite, 91
row, 320
square, 320374 Numerical Methods in Science and Engineering
subtraction, 321
symmetric, 320
transpose, 322
vector, 331
Minimization, 143, 151
Modified Euler’s method, 237
Multiple integration, 205
area, 205
volume, 205
Multiple regression
linear data, 156
nonlinear data, 162
Multistep methods, 252
Newton’s divided differences, 112
Newton’s second law, 4, 227, 298
Newton-Cotes formulas, 188, 192, 198
Newton-Raphson method, 32, 43
Numerical integration, 173
Numerical methods
basic concept, 3
Oblique shock, 47
One-point iteration method, 20, 28
One-step method, 252
Open method, 28
Ordinary differential equation, 227
coefficients, 228
coupled, 246
first-order, 227, 239
higher-order, 228
linear, 4, 228, 260
nonlinear, 228, 260
second-order, 228, 264, 367
system, 252
Orthogonal property, 96
Partial differential equation, 227, 269
coefficients, 272
elliptic, 270, 271
hyperbolic, 271, 297
linear, 270
nonlinear, 270
parabolic, 271, 283
second-order, 270. 285
Pascal language, 9
Pi, 12
Pivoting, 65
Polynomial
first-order, 112
nth-order, 115, 135
second-order, 19, 120
Positive definite, 88
Power equation, 146
Predictor, 235, 253
Pressure, 156, 168
Regression
linear, 142
multiple, 142, 156
polynomial, 146, 150
Remainder, 178
Residual vector, 44, 92
Reynolds number, 161, 166
Romberg integration, 174, 178, 192
Root of equation, 19
Roots of nonlinear equations, 41, 51
Runge’s function, 123, 132
Runge-Kutta formula
derivation, 355
Runge-Kutta method, 255
first-order, 240
fourth-order, 243, 247, 355
second-order, 240
third-order, 242
Saturation-growth-rate equation, 149
Scaling, 63
Search direction vector, 92
Secant method, 20, 37
Second-order derivative, 213, 227
Separation of variables, 5, 231
Shock wave
bow, 111
propagation, 297
Significant figures, 12
Simpson’s rule, 176, 184
approximate error, 180, 182
composite, 188
error, 183, 218
segment, 174
Simpson’s three-eighth rule, 189, 191
error, 189
Simultaneous equations, 53, 151, 276
Slope, 244, 248, 252
average, 234, 237Index 375
Solution
accuracy, 239, 240
approximate, 174, 296
diverged, 289, 294
exact, 228, 231
nonlinear, 250
Space shuttle, 4
tiles, 161
Spacing, 289
Specific heat, 47, 152, 155, 285
Specified
heating, 270
tolerance, 83, 279
Spline interpolation, 112, 124, 131
cubic, 128
linear, 125
Stencil form, 275, 281
Step size, 216, 230, 333
Stopping tolerance, 23
Stress-strain data, 167, 169
Successive over-relaxation method, 55, 87
Supercomputer, 9
Supersonic flow, 125
Surface heating, 53
Swinging pendulum, 174, 227, 246, 249
System of equations, 44, 55
Taylor series, 32, 43, 240, 255, 355
two variables, 231, 341
Temperature, 269
transient, 16
Thermal conductivity, 54, 273, 283
coefficient, 273, 283
Time, 4
Time step, 7, 243, 260, 285, 293
critical, 286, 291
Total error
minimization, 142, 151, 163
Transcendental equation, 20,
Transient heat conduction, 271
Trapezoidal rule, 174, 175
approximate error, 180, 182
composite, 180
error, 176, 177, 180
segment, 180
Tridiagonal system, 67, 290
True error, 186, 188
Velocity, 4
Vibration of string, 288, 313
Weighting factor, 87
Weighting functions, 33

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