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| Авторизация |
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| Поиск по указателям |
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| Braun M. — Differential Equations and Their Applications: An Introduction to Applied Mathematics |
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| Предметный указатель |
Adjoint matrix 313
Analytic functions 187
APL programming 516
APL programming, addition 516
APL programming, division 518
APL programming, exponential 518
APL programming, functions 520
APL programming, functions, absolute value 520
APL programming, functions, arc cosine 520
APL programming, functions, arc sine 520
APL programming, functions, arc tangent 520
APL programming, functions, cosine 520
APL programming, functions, exponential 520
APL programming, functions, factorial 520
APL programming, functions, natural logarithm 520
APL programming, functions, signum 520
APL programming, functions, sine 520
APL programming, functions, tangent 520
APL programming, identifiers 519
APL programming, logical expressions 522
APL programming, multiplication 518
APL programming, programs 522
APL programming, subtraction 516
Asymptotic stability 379
Atomic waste problem 46
Autonomous systems 374
Bernoulli equation 67
Bessel equation 135
Bessel equation order  216
Bessel equation order zero 218
Bessel function,  218
Bessel function,  216
Boundary conditions for elastic string 481
Boundary conditions for heat equation 480
Boundary conditions for Laplace equation 481
Boundary conditions for second order o.d.e. 474
Boundary-value problem 474 481
Boundary-value problem, heat equation 481
Boundary-value problem, Laplace equation 506
Boundary-value problem, wave equation 501
Carbon dating 19
Cauchy ratio test 187
Cayley — Hamilton theorem 349
Characteristic equation 136
Characteristic equation, complex roots 139
Characteristic equation, real and equal roots 143
Characteristic equation, real and unequal roots 136
Characteristic polynomial 332
Characteristic polynomial, complex roots 339
Characteristic polynomial, real distinct roots 332
Characteristic polynomial, repeated roots 343
Combat models 403
Complex exponentials 139
Compound interest 57
Convolution integral 250
Damping force 164
Detecting art forgeries 11
Detection of diabetes 176
Difference equations 91
Differential operators 127
Diffusion equation 479
Dirac delta function 242
Dirac delta function, Laplace transform of 244
Dirac, P.A.M. 242
Direction field 395
Dirichlet problem 481
Discontinuous forcing function 236
Eigenfunctions 476
Eigenvalues of boundary-value problem 476
Eigenvalues of matrix 331
Eigenvalues of matrix, multiplicity of 345
Eigenvectors of a matrix 331
Eigenvectors of a matrix, linear independence of 333
Elastic membrane, vibration of 480
Elastic string 480 501
Elastic string, boundary conditions for 481
Elastic string, fundamental frequencies of 503
Elastic string, initial conditions for 481
Elastic string, nonzero initial displacement 503
Elastic string, propagation of initial conditions 504
electrical circuits 173
Epidemic models 38
Epidemic models, gonorrhea 489
Epidemic models, plagues 39
Epidemic models, threshold theorem 458
Equilibrium points 371
Error effect of step size 102
Error for Euler’s method 100
Error for improved Euler’s method 109
Error for Runge — Kutta method 112
Error for three-term Taylor series 106
Error formula 101
Error, round-off 104
Escape velocity 52
Euler equation 143 147 196
Euler method 97
Euler method, formula error 101
Even functions 491
exact equations 58 60
Existence and uniqueness theorems for first-order equations 67
Existence and uniqueness theorems for orbits 312
Existence and uniqueness theorems for second-order equations 127
Existence and uniqueness theorems for systems of first-order equations 289 412
Extension of a function, even 493
Extension of a function, odd 494
Finding roots by iteration 81
First-order differential equations 1 2
First-order differential equations, exact 58 60
First-order differential equations, existence and uniqueness theorem for 67
First-order differential equations, general solutions of 4 21
First-order differential equations, homogeneous 3
First-order differential equations, initial-value problem 5 7 21
First-order differential equations, integrating factor for 8 64
First-order differential equations, linear 3
First-order differential equations, nonlinear 3
First-order differential equations, numerical solutions of see "Numerical methods"
First-order differential equations, separable 21
First-order differential equations, several variables 262
First-order differential equations, systems of 263
Fourier series 485 486
Fourier series, convergence of 485 486
Fourier series, cosine series 492
Fourier series, Fourier coefficients 487
Fourier series, Parseval identity 490
Fourier series, sine series 492
Frobenius method 201
Fundamental matrix 353
Fundamental set of solutions 131
General solutions, first-order linear equations 4 21
General solutions, higher-order equations 257
General solutions, second-order equations 130 140 146
General solutions, systems of first-order equations 332
Generalized functions 247
Gompertzian relation 53
Gonorrhea model 489
Green’s Theorem 435
Heat equation 479 497
Heat equation, bar with insulated ends 498
Heat equation, boundary conditions 481
Heat equation, initial conditions 481
Heat equation, nonhomogeneous boundary conditions 501
Heat equation, smoothing of discontinuities in initial conditions 497
Hermite equation 195
Hermite polynomial 195
Higher-order equations 257
Higher-order equations, general solution 257
Higher-order equations, variation of parameters 260
Homogeneous equations 25
Hypergeometric equations 216
Identity matrix 309
| Impedance 175
Improved Euler method 109
Indicial equation 207
Initial conditions 480
Initial conditions for elastic string 481
Initial conditions for heat equation 481
Initial conditions, propagation of initial conditions for wave equation 504
Initial conditions, smoothing of discontinuities for heat equation 497
Initial-value problem 5 7 21 126 257
Integral equation 70
Integrating factor 8 64
Inverse matrix 314
Judicious guessing of particular solutions 155
Jump discontinuities 225 236
Kirchoff's second law 174
Laguerre equation 216
Laplace equation 480 506
Laplace equation, boundary conditions 481
Laplace equation, Dirichlet problem 481
Laplace equation, Neumann problem 481
Laplace transform 223
Laplace transform of derivatives 227
Laplace transform of systems 366
Laplace transform of the convolution 250
Laplace transform of the Dirac delta function 244
Laplace transform, definition 224
Laplace transform, existence of 225
Laplace transform, inverse of 228
Legendre equation 195
Legendre polynomials 195
Limit cycles 434
Linear algebra 269 294
Linear differential equations see "First-order differential equations" "Second-order "Systems
Linear independence 134 279
Linear independence of eigenvectors 333
Linear independence of vector functions 291
Linear independence of vectors 279
Linear operators 129
Linear transformations 322
Logistic curves 30 43
Logistic law of population growth 28
Malthusian law of population growth 27
Matrices 265
Matrices, addition of 274
Matrices, adjoint 313
Matrices, characteristic polynomial 332
Matrices, determinant of 297
Matrices, diagonal 299
Matrices, diagonal, lower 299
Matrices, diagonal, upper 299
Matrices, eigenvalues of 331
Matrices, eigenvectors of 331
Matrices, fundamental 353
Matrices, identity 309
Matrices, inverse 314
Matrices, product of 308
Mechanical vibrations 163
Mechanical vibrations, damped forced vibrations 167
Mechanical vibrations, damped free vibrations 165
Mechanical vibrations, forced free vibrations 168
Mechanical vibrations, free vibrations 164
Mechanical vibrations, resonant frequency 171
Method of elimination 255
Mixing problems 53
Natural frequency 169
Neumann problem 481
Newtonian mechanics 46 125 164
Newton’s method 88
Nonlinear differential equations autonomous system 374
Numerical methods 95
Numerical methods, effect of step size 102
Numerical methods, error 100 102 106 109 112
Numerical methods, Euler 97
Numerical methods, Euler, formula error 101
Numerical methods, improved Euler 109
Numerical methods, Runge — Kutta 112
Numerical methods, three-term Taylor series 106
Odd functions 491
Ohm’s Law 174
Orbits 392 395
Order of a differential equation 1
Ordinary differential equations definition 1
Parseval’s identity 490
Partial differential equations definition 479
Particular solution 149 362
Periodic solutions 414
Phase plane 392
Phase portrait 416 417
Picard iterates 70
Piecewise continuous functions 225
Poincare — Bendixson theorem 431
Population models 27
Population models, competitive exclusion 449
Population models, Logistic law of population growth 28
Population models, Malthusian law of population growth 27
Population models, sharks 441 442
Potential equation 480
Power series 186
Qualitative theory 370
Radioactive dating 12
Reduction of order 145
Reduction to systems of equations 263
Regular singular points 202
Resonance 168
resonant frequency 171
Reynolds number 50
Runge — Kutta method 112
Schwartz, Laurent 243 246
Second-order differential equations 125 136
Second-order differential equations, characteristic equation 136
Second-order differential equations, characteristic equation, complex roots 139
Second-order differential equations, characteristic equation, real and equal roots 143
Second-order differential equations, characteristic equation, real and unequal roots 136
Second-order differential equations, Euler equation 143 148 196
Second-order differential equations, existence and uniqueness theorem 127
Second-order differential equations, fundamental set of solutions 131
Second-order differential equations, general solutions 130 140 146
Second-order differential equations, homogeneous equation 126
Second-order differential equations, judicious guessing of particular solution 155
Second-order differential equations, nonhomogeneous equation 149
Second-order differential equations, nonhomogeneous equation, discontinuous function 236
Second-order differential equations, particular solution 149
Second-order differential equations, reduction of order 145
Second-order differential equations, series solution 183 186
Second-order differential equations, singular points 196 202
Second-order differential equations, variation of parameters 153
separable equations 20
Separation of variables 481 482
Separation of variables for heat equation 482 498
Separation of variables for Laplace equation 506 508
Separation of variables for wave equation 501
Sequence of eigenvalues 476
Series solution 183 186
Series solution, recurrence formula 185
Series solution, when roots of the indicial equation are equal 210 214
Series solution, when roots of the indicial equation differ by an integer 210 214
Singular points 196 202
Solutions, definition 1
Solutions, definition, first-order equations 4 21 60
Solutions, definition, higher-order equations 257
Solutions, definition, second-order equations 130 140 146
Solutions, definition, systems of first-order equations 332
Spread of technological innovations 39
Spring-mass dashpot system 163
Spring-mass dashpot system, damped vibrations 165 167
Spring-mass dashpot system, forced vibrations 168
Spring-mass dashpot system, free vibrations 164
Stability of a solution 371 376
Stability of equilibrium points 383 384
Stability, asymptotic 379
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