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Advanced Engineering Mathematics with MATLAB
Advances in Applied Mathematics
FIFTH EDITION
Dean G. Duffy
و المحتوى كما يلي :
Contents
Dedication v
Contents vii
Acknowledgments xiii
Author xv
Introduction xvii
List of Definitions xix
Chapter 1:
First-Order Ordinary
Differential Equations 1
1.1 Classification of Differential Equations 1
1.2 Separation of Variables 4
1.3 Homogeneous Equations 16
1.4 Exact Equations 17
1.5 Linear Equations 20
viiviii Advanced Engineering Mathematics with MATLAB
1.6 Graphical Solutions 31
1.7 Numerical Methods 34
Chapter 2:
Higher-Order Ordinary
Differential Equations 47
2.1 Homogeneous Linear Equations with Constant Coefficients 51
2.2 Simple Harmonic Motion 59
2.3 Damped Harmonic Motion 63
2.4 Method of Undetermined Coefficients 68
2.5 Forced Harmonic Motion 73
2.6 Variation of Parameters 80
2.7 Euler-Cauchy Equation 85
2.8 Phase Diagrams 88
2.9 Numerical Methods 93
am1 am2 · · · amn
Chapter 3:
Linear Algebra
101
3.1 Fundamentals 101
3.2 Determinants 109
3.3 Cramer’s Rule 113
3.4 Row Echelon Form and Gaussian Elimination 115
3.5 Eigenvalues and Eigenvectors 129
3.6 Systems of Linear Differential Equations 136
3.7 Matrix Exponential 141Table of Contents ix
z
(0,0,1)
n C
C 2
3
(0,1,0)
y
(1,0,0) C1
x
amplitude spectrum (ft) times 10000
10000.0
1000.0
100.0
10.0
1.0
0.1
Bay bridge and tunnel
1 10 100 1000 10000
k |G(ω)|
11.0
10.0
9.0
c /km = 0.01 2
8.0
7.0
6.0
5.0
4.0
3.0 c /km = 0.1 2
2.0
c /km = 1 2
1.0
0.0
0.0
0.5 1.0 1.5 2.0
ω/ω
0
Chapter 4:
Vector Calculus 147
4.1 Review 147
4.2 Divergence and Curl 154
4.3 Line Integrals 158
4.4 The Potential Function 163
4.5 Surface Integrals 164
4.6 Green’s Lemma 171
4.7 Stokes’ Theorem 174
4.8 Divergence Theorem 181
Chapter 5:
Fourier Series 189
5.1 Fourier Series 190
5.2 Properties of Fourier Series 202
5.3 Half-Range Expansions 211
5.4 Fourier Series with Phase Angles 216
5.5 Complex Fourier Series 220
5.6 The Use of Fourier Series in the Solution of Ordinary Differential Equations 225
5.7 Finite Fourier Series 232
Chapter 6:
The Fourier Transform 249
6.1 Fourier Transforms 249
6.2 Fourier Transforms Containing the Delta Function 262x Advanced Engineering Mathematics with MATLAB
time (seconds)
TIME 3.5
4 1
6.3 Properties of Fourier Transforms 264
6.4 Inversion of Fourier Transforms 275
6.5 Convolution 279
6.6 The Solution of Ordinary Differential Equations by Fourier Transforms 283
6.7 The Solution of Laplace’s Equation on the Upper Half-Plane 285
6.8 The Solution of the Heat Equation 287
Chapter 7:
The Laplace Transform 295
7.1 Definition and Elementary Properties 295
7.2 The Heaviside Step and Dirac Delta Functions 299
7.3 Some Useful Theorems 307
7.4 The Laplace Transform of a Periodic Function 315
7.5 Inversion by Partial Fractions: Heaviside’s Expansion Theorem 317
7.6 Convolution 324
7.7 Solution of Linear Differential Equations with Constant Coefficients 329
Chapter 8:
The Wave Equation 347
8.1 The Vibrating String 348
8.2 Initial Conditions: Cauchy Problem 351
8.3 Separation of Variables 351
8.4 D’Alembert’s Formula 365
8.5 Numerical Solution of the Wave Equation 372Table of Contents xi
DISTANCE
2
TIME
0 0
Chapter 9:
The Heat Equation 387
9.1 Derivation of the Heat Equation 387
9.2 Initial and Boundary Conditions 389
9.3 Separation of Variables 390
9.4 The Superposition Integral 405
9.5 Numerical Solution of the Heat Equation 409
u(R,θ )
Chapter 10:
Laplace’s Equation 419
10.1 Derivation of Laplace’s Equation 419
10.2 Boundary Conditions 421
10.3 Separation of Variables 422
10.4 Poisson’s Equation on a Rectangle 429
10.5 Numerical Solution of Laplace’s Equation 433
Chapter 11:
The Sturm-Liouville
Problem 443
11.1 Eigenvalues and Eigenfunctions 444
11.2 Orthogonality of Eigenfunctions 457
11.3 Expansion in Series of Eigenfunctions 461
11.4 Finite Element Method 485xii Advanced Engineering Mathematics with MATLAB
Chapter 12:
Special Functions 493
12.1 Legendre Polynomials 495
12.2 Bessel Functions 519
12.A Appendix A: Derivation of the Laplacian in Polar Coordinates 567
12.B Appendix B: Derivation of the Laplacian in Spherical Polar Coordinates 568
Answers to the Odd-Numbered Problems 571
Index 58
Index
abscissa of convergence, 296
Adams-Bashforth method, 40
addition
of matrices, 102
of vectors, 147
age of the earth, 289–290
aliasing, 239–241
amplitude spectrum, 251
Archimedes’ principle, 186–187
autonomous ordinary differential
equation, 4, 50
auxiliary equation, 52
back substitution, 107, 118
bandlimited Fourier transform, 255
Bernoulli equation, 28–29
Bessel
equation of order n, 519–524
function of the first kind, 521
expansion in, 529–534
function of the second kind, 521
function, modified, 524
recurrence formulas, 524–525
Bessel, Friedrich Wilhelm, 520
Biot number, 468
boundary condition
Cauchy, 351
Dirichlet, 389
Neumann, 389
Robin, 390
boundary-value problems, 48
carrier frequency, 268
Cauchy
boundary condition, 351
data, 351
problem, 351
centered finite differences, 373
characteristic
polynomial, 129
equation, 52
value, 129, 444
vector, 129
characteristic function, 444
characteristics, 365
chemical reaction, 12–13
circular frequency, 60
circulation, 161
closed
contour integral, 159
surface integral, 165
coefficient matrix, 117
cofactor, 109
column of a matrix, 102
column vector, 105
complementary error function, 301
complementary solution of an
ordinary differential equation, 68
589590 Advanced Engineering Mathematics with MATLAB
complex matrix, 102
components of a vector, 147
compound interest, 9
conformable
for addition of matrices, 102
for multiplication of matrices, 103
conservative field, 161
consistency in finite differencing
for the heat equation, 410
for the wave equation, 375
consistent system of linear eqns, 116
convergence
of a Fourier integral, 251
of finite difference solution
for heat equation, 412
for wave equation, 378
of Fourier series, 191
convolution theorem
for Fourier transforms, 279–282
for Laplace transforms, 324–327
Coriolis force, 149
Cramer’s rule, 113
Crank-Nicholson method, 414
critical points, 33, 90
stable, 33, 90
stable node, 92
unstable, 33, 91
cross product, 148
curl, 156
curve, space, 148
d’Alembert’s formula, 367
d’Alembert’s solution, 365–370
d’Alembert, Jean Le Rond, 365
damped harmonic motion, 63
damping constant, 63
degenerate eigenvalue problem, 453
del operator, 150
delay differential equation, 340–341
delta function, 252–255, 304–306
design of film projectors, 321–324
design of wind vane, 66–67
determinant, 109–112
diagonal, principal, 102
differential equations
nth order, 47–97
linear first-order, 1–46
nonlinear, 1
order, 1
ordinary, 1–100
partial, 1, 407
type, 1
differentiation of a Fourier series, 202
diffusivity, 388
dimension of a vector space, 130
direction fields, 31
Dirichlet conditions, 192
Dirichlet problem, 389
Dirichlet, Peter Gustav Lejeune, 193
dispersion, 358
divergence
of a vector, 155
theorem, 181–187
dot product, 148
double Fourier series, 431
Duhamel’s theorem
for the heat eqn, 405–408, 471–477
eigenfunctions, 444–478
expansion in, 461
orthogonality of, 458
eigenvalue(s)
of a matrix, 129
of a Sturm-Liouville problem, 444–453
eigenvalue problem, 129–132, 376–377
for ordinary differential eqns, 444–453
singular, 444
eigenvectors, 129–132, 376–377
orthogonality of, 457
electrical circuits, 24, 78, 335–340
electrostatic potential, 547
element of a matrix, 102
elementary row operations, 116
elliptic partial differential equation, 419
equilibrium points, 33, 90
equilibrium systems of linear eqns, 116
error function, 301
Euler’s method, 34–37
Euler-Cauchy equation, 85–88
exact ordinary differential equation, 17
existence of ordinary differential eqns
nth-order, 48
first-order, 8
explicit numerical methods
for the heat equation, 410
for the wave equation, 372–373
exponential order, 296Index 591
fast Fourier transform (FFT), 239
filter, 242
final-value theorem
for Laplace transforms, 311
finite difference approximation
to derivatives, 372–373
finite Fourier series, 232–242
first-order ordinary differential eqns, 1–46
linear, 20–31
flux lines, 152
folding frequency, 241
forced harmonic motion, 73–77
Fourier
coefficients, 190
cosine series, 196
cosine transform, 291
Joseph, 192
number, 395
series for a multivariable function, 224
series for an even function, 196
series for an odd function, 197
series in amplitude/phase form, 216–219
series on [−L, L], 190–201
sine series, 197
sine transform, 291
Fourier coefficients, 462
Fourier cosine series, 196
Fourier transform, 249–285
basic properties of, 264–274
convolution, 279–282
inverse of, 250, 275–276
method of solving heat eqn, 287–292
of a Bessel function, 254
of a constant, 262
of a derivative, 267
of a multivariable function, 255
of a sign function, 263
of a step function, 263
Fourier-Bessel
coefficients, 530
expansions, 529
Fourier-Legendre
coefficients, 502
expansion, 502
Fredholm integral eqn, 126
free underdamped motion, 60
frequency convolution, 282
frequency modulation, 270
frequency spectrum, 252
function
even extension of, 211
generalized, 306
odd extension of, 211
vector-valued, 150
fundamental of a Fourier series, 190
Gauss’s divergence theorem, 181–187
Gauss, Carl Friedrich, 182
Gauss-Jordan elimination, 119
Gauss-Seidel method, 434
general solution to an
ordinary differential equation, 4
generalized Fourier series, 462
generalized functions, 306
generating function
for Legendre polynomials, 498
Gibbs phenomenon, 206–208, 505
gradient, 150
graphical stability analysis, 33
Green’s lemma, 171–174
grid point, 372
groundwater flow, 422–426
half-range expansions, 211–214
Hankel transform, 554
harmonic functions, 420
harmonics of a Fourier series, 190
heat conduction
in a rotating satellite, 228–231
within a metallic sphere, 510–516
heat equation, 227–292, 387–416,
465–484, 538–562
for a semi-infinite bar, 287–289
for an infinite cylinder, 401, 538–541
nonhomogeneous, 389
one-dimensional, 390–393, 465
within a solid sphere, 399–401, 538
Heaviside
expansion theorem, 317–324
step function, 299–302
Heaviside, Oliver, 300
homogeneous
ordinary differential eqns, 16–17, 47
solution to ordinary differential eqn, 68
system of linear eqns, 106
hydraulic potential, 422
hydrostatic equation, 8
hyperbolic partial differential equation, 348592 Advanced Engineering Mathematics with MATLAB
impulse function
see (Dirac) delta function
inconsistent system of linear eqns, 116
indicial admittance for heat equation, 472
inertia supercharging, 213
initial
-value problem, 47, 329–342
conditions, 351
initial-boundary-value problem, 389
initial-value theorem
for Laplace transforms, 310
inner product, 103
integral curves, 90
integrals, line, 158–162
integrating factor, 19
integration of a Fourier series, 203–205
interest rate, 9
inverse
discrete Fourier transform, 233–235
Fourier transform, 250, 275–276
Laplace transform, 317–324
inverse formula for Fourier transform, 250
inversion of Fourier transform
by direct integration, 275–276
by partial fraction, 276
inversion of Laplace transform
by convolution, 324
by partial fractions, 317–319
in amplitude/phase form, 320–324
irrotational, 156
isoclines, 31
iterative methods
Gauss-Seidel, 434
successive over-relaxation, 436
Kirchhoff’s law, 24
Klein-Gordon equation, 358
Kutta, Martin Wilhelm, 39
Laplace integral, 295
Laplace transform, 295–342
basic properties of, 307–313
convolution for, 324–327
definition of, 295
derivative of, 310
in solving delay differential
equation, 340–341
integration of, 310
inverse of, 317–324
of derivatives, 298
of periodic functions, 315–317
of the delta function, 304–306
of the step function, 299–302
solving of ordinary
differential eqns, 329–342
Laplace’s eqn, 286–287, 419–438, 484,
508–519, 547–567
in cylindrical coordinates, 420
in spherical coordinates, 421
numerical solution of, 433–438
solution by separation
of variables, 422–427, 508–516,
547–554
solution on a half-plane, 285–287
Laplace’s expansion in cofactors, 109
Laplace, Pierre-Simon, 421
Laplacian, 155
Lax-Wendroff scheme, 381
Legendre polynomial, 497
expansion in, 502
generating function for, 498
orthogonality of, 501
recurrence formulas, 499
Legendre’s differential equation, 495
Legendre, Adrien-Marie, 495
length of a vector, 147
line integral, 158–162
line spectrum, 218–222
linear dependence
of eigenvectors, 129
of functions, 55
linear Fredholm integral equation, 505
linear transformation, 107
linearity
of Fourier transform, 264
of Laplace transform, 297
lines of force, 152
Liouville, Joseph, 446
logistic equation, 12
LU decomposition, 127
magnitude of a vector, 147
matrices
addition of, 102
equal, 102
multiplication, 103Index 593
matrix, 101
algebra, 101
amplification, 376–377
augmented, 117
banded, 106
coefficient, 117
complex, 102
diagonalization, 134
exponential, 141
identity, 102
inverse, 104
invertible, 104
method of stability
of a numerical scheme, 376
nonsingular, 104
null, 102
null space, 124
orthogonal, 128
real, 102
rectangular, 102
square, 102
symmetric, 102
tridiagonal, 106
unit, 102
upper triangular, 106
vector space, 105
zero, 102
maximum principle, 420
Maxwell’s field eqns, 158
mechanical filter, 324
method of partial fractions
for Fourier transform, 276
for Laplace transform, 317–324
method of undetermined coefficients, 69–72
minor, 110
mixed-boundary-value problems, 440–442
modified Bessel function,
first kind, 524
second kind, 524
modified Euler method, 34–37
modulation, 268–271
multiplication of matrices, 103
nabla operator, 150
natural vibrations, 357
Neumann problem, 389
Neumann’s Bessel function of order n, 522
Newton’s law of cooling, 467
non-local boundary conditions, 417
nondivergent, 155
nonhomogeneous
heat equation, 389
ordinary differential equation, 47
system of linear eqns, 106
norm of a vector, 105, 147
normal differential equation, 47
normal mode, 357
normal to a surface, 150
null space, 124
numerical solution
of heat equation, 409–416
of Laplace’s equation, 433–438
of the wave equation, 372–382
Nyquist frequency, 241
Nyquist sampling criteria, 239
one-sided finite difference, 373
order of a matrix, 102
orthogonal matrix, 128
orthogonality, 458
of eigenfunctions, 457–460
of eigenvectors, 457
orthonormal eigenfunction, 460
overdamped ordinary differential eqn, 64
overdetermined system of linear eqns, 121
parabolic partial differential eqn, 388
Parseval’s equality, 205–206
Parseval’s identity
for Fourier series, 205
for Fourier transform, 271–272
partial fraction expansion
for Fourier transform, 276
for Laplace transform, 317–324
particular solution to ordinary
differential equation, 3–4, 68
path in line integrals, 159
path independence in line integrals, 161
phase
angle in Fourier series, 216–219
diagram, 89
line, 33
path, 90
spectrum, 251
pivot, 117
pivotal row, 117594 Advanced Engineering Mathematics with MATLAB
Poisson’s
equation, 429–431
integral formula
for a circular disk, 426–427
for a upper half-plane, 287
summation formula, 272–275
Poisson, Sim´eon-Denis, 430
population growth and decay, 11
position vector, 147
potential flow theory, 157
potential function, 163–164
power content, 205
power spectrum, 272
principal diagonal, 102
principle of linear superposition, 52, 354
QR decomposition, 128
quieting snow tires, 197–201
radiation condition, 351, 467
rank of a matrix, 119
real matrix, 102
rectangular matrix, 102
recurrence relation
for Bessel functions, 524–525
for Legendre polynomial, 499–502
in finite differencing, 94
reduced row echelon, 119
reduction in order, 49
regular Sturm-Liouville problem, 444
relaxation methods, 434–438
resonance, 77, 226, 332
rest points, 33
Robin problem, 390
Rodrigues’s formula, 498
row echelon form, 118
row vector, 105
rows of a matrix, 102
Runge, Carl, 38
Runge-Kutta method, 37–40, 95–100
Saulyev’s method, 415–416
scalar, 147
Schwarz’s integral formula, 287
second shifting theorem, 301
secular term, 226
separation of variables
for heat equation, 390–401, 465–471,
538–547
for Laplace’s equation, 422–427,
508–516, 547–554
for ordinary differential eqns, 4–14
for Poisson’s equation, 429–431
for wave equation, 351–362, 536–538
shifting
in the ω variable, 268
in the s variable, 307
in the t variable, 264, 308
sifting property, 253
simple eigenvalue, 447
simple harmonic motion, 60, 331
simple harmonic oscillator, 59–63
sinc function, 251
singular
solutions to ordinary differential eqns, 6
Sturm-Liouville problem, 444
singular Sturm-Liouville problem, 444
singular value decomposition, 135
slope field, 31
solenoidal, 155
solution curve, 31
solution of ordinary differential eqns
by Fourier series, 225–231
by Fourier transform, 283–285
space curve, 148
spectral radius, 129
spectrum of a matrix, 129
square matrix, 102
stability of numerical methods
by Fourier method for heat eqn, 411
by Fourier method for wave eqn, 376
by matrix method for wave eqn, 376
steady-state heat equation, 10, 394
steady-state output, 33
steady-state solution to ordinary
differential eqns, 75
step function, 299–302
Stokes’ theorem, 174–180
Stokes, Sir George Gabriel, 175
streamlines, 152
Sturm, Charles-Fran¸cois, 445
Sturm-Liouville
equation, 444
problem, 444–453
subtraction
of matrices, 102
of vectors, 147
successive over-relaxation, 434Index 595
superposition integral
of heat equation, 405–408, 471–477
superposition principle, 354
surface conductance, 467
surface integral, 164–170
system of linear
differential eqns, 136–140
homogeneous eqns, 106
nonhomogeneous eqns, 106
tangent vector, 148
telegraph equation, 360
terminal velocity, 9, 27
thermal conductivity, 388
threadline equation, 350–351
time shifting, 264–265, 307
trace, 102
trajectories, 90
transform
Fourier, 249–285
Laplace, 295–342
transient solution to ordinary
differential equations, 75
transpose of a matrix, 104
tridiagonal matrix, solution of, 106–107
underdamped, 64
underdetermined system of linear eqns, 118
uniformitarianism, 290
uniqueness of ordinary differential eqns
nth-order, 48
first-order, 3
unit
normal, 151
step function, 299–302
vector, 147
Vandermonde’s determinant, 113
variation of parameters, 80–85
vector, 105, 147
vector element of area, 167
vector space, 105, 130
vibrating string, 348–350
vibrating threadline, 350–351
vibration of floating body, 62
volume integral, 181–187
wave equation, 347–382, 479, 535–538
damped, 359–362
for a circular membrane, 535–538
for an infinite domain, 365–371
one-dimensional, 350
weight function, 458
Wronskian, 56
zero vector, 147
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