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| موضوع: كتاب Numerical Methods in Science and Engineering الجمعة 01 أبريل 2022, 6:34 pm | |
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أخواني في الله أحضرت لكم كتاب Numerical Methods in Science and Engineering Theories with MATLAB, Mathematica, Fortran, C and Python Programs P. Dechaumpwwhai N. Wansophark
و المحتوى كما يلي :
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 #ماتلاب,#متلاب,#Matlab,
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