Numerical Methods in Science and Engineering

Numerical Methods in Science and Engineering
اسم المؤلف
P. Dechaumpwwhai, N. Wansophark
التاريخ
1 أبريل 2022
المشاهدات
111
التقييم
(لا توجد تقييمات)
Loading...

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

تحميل

يجب عليك التسجيل في الموقع لكي تتمكن من التحميل
تسجيل | تسجيل الدخول

التعليقات

اترك تعليقاً