Numerical Methods in Science and Engineering

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
   3.16 Conjugate Gradient Method 88
   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
   4.2.2 Quadratic interpolation 113
   4.2.3 nth-order Polynomial interpolation 115
   4.3 Lagrange Interpolating Polynomials 118
   4.3.1 Linear interpolation 118
   4.3.2 Quadratic interpolation 120
   4.3.3 Polynomial interpolation 122
   4.4 Spline Interpolations 124
   4.4.1 Linear spline 125
   4.4.2 Quadratic spline 126
   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
   7.8.2 Adams-Bashforth method 255
   7.8.3 Adams-Moulton method 258
   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
   Adams-Bashforth method, 251
   formulas, 256
   fourth-order, 256
   Adams-Moulton method, 258
   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
   Conjugate gradient method, 55, 88
   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
   Gauss quadrature, 174, 197
   locations, 204
   weights, 204, 208
   Gauss-Jordan method, 55, 68
   Gauss-Legendre formulas, 201
   Gauss-Seidel iteration method, 55, 85, 277
   Gradient, 216
   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
   quadratic, 113, 118
   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
   read/write data files, 344
   scalar variables, 328
   script file, 328
   short format, 333
   vector variables, 331
   MATLAB command
   cumtrapz, 209
   dblquad, 209
   gradient, 216
   ode113, 250
   ode23, 250
   ode45, 250
   solver, 250
   sysode, 252
   trapz, 208
   triplequad, 209
   (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
   addition, 321
   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
   advantages, 4
   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 head, 168
   Pressure, 156, 168
   Quadratic function, 89
   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
   quadratic, 126
   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

كلمة سر فك الضغط : books-world.net
The Unzip Password : books-world.net