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Differential Equations with Mathematica 5th edition [Pehme köide]

(Associate Professor Emeritus), (Professor Emerita)
  • Formaat: Paperback / softback, 608 pages, kõrgus x laius: 276x216 mm, kaal: 1590 g
  • Ilmumisaeg: 13-Jul-2022
  • Kirjastus: Academic Press Inc
  • ISBN-10: 0128241608
  • ISBN-13: 9780128241608
Teised raamatud teemal:
Differential Equations with Mathematica 5th editionSuurem pilt
  • Formaat: Paperback / softback, 608 pages, kõrgus x laius: 276x216 mm, kaal: 1590 g
  • Ilmumisaeg: 13-Jul-2022
  • Kirjastus: Academic Press Inc
  • ISBN-10: 0128241608
  • ISBN-13: 9780128241608
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780128241608.html
Märksõnad:

Differential Equations with Mathematica, Fifth Edition uses the fundamental concepts of the popular platform to solve (analytically, numerically, and/or graphically) differential equations of interest to students, instructors, and scientists.

Mathematica’s diversity makes it particularly well suited to performing calculations encountered when solving many ordinary and partial differential equations. In some cases, Mathematica’s built-in functions can immediately solve a differential equation by providing an explicit, implicit, or numerical solution. In other cases, Mathematica can be used to perform the calculations encountered when solving a differential equation.

Because one goal of elementary differential equations courses is to introduce students to basic methods and algorithms so that they gain proficiency in them, nearly every topic covered this book introduces basic commands, also including typical examples of their application. A study of differential equations relies on concepts from calculus and linear algebra, so this text also includes discussions of relevant commands useful in those areas. In many cases, seeing a solution graphically is most meaningful, so the book relies heavily on Mathematica’s outstanding graphics capabilities.

  • Demonstrates how to take advantage of the advanced features of Mathematica
  • Introduces the fundamental theory of ordinary and partial differential equations using Mathematica to solve typical problems of interest to students, instructors, scientists, and practitioners in many fields
  • Showcases practical applications and case studies drawn from biology, physics, and engineering
Preface ix
1 Introduction to differential equations
1.1 Definitions and concepts
1(3)
1.2 Solutions of differential equations
4(8)
1.3 Initial-and boundary-value problems
12(6)
1.4 Direction fields
18(11)
1.4.1 Creating interactive applications
27(2)
2 First-order ordinary differential equations
2.1 Theory of first-order equations: a brief discussion
29(4)
2.2 Separation of variables
33(9)
2.3 Homogeneous equations
42(4)
2.4 Exact equations
46(4)
2.5 Linear equations
50(10)
2.5.1 Integrating factor approach
51(5)
2.5.2 Variation of parameters and the method of undetermined coefficients
56(4)
2.6 Numerical approximations of solutions to first-order equations
60(23)
2.6.1 Built-in methods
60(3)
2.6.2 Other numerical methods
63(14)
Application: Modeling the spread of a disease
77(6)
3 Applications of first-order equations
3.1 Orthogonal trajectories
83(7)
3.2 Population growth and decay
90(14)
3.2.1 The Malthus model
90(5)
3.2.2 The logistic equation
95(9)
3.3 Newton's law of cooling
104(4)
3.4 Free-falling bodies
108(7)
4 Higher-order linear differential equations
4.1 Preliminary definitions and notation
115(16)
4.1.1 Introduction
115(4)
4.1.2 The nth-order ordinary linear differential equation
119(5)
4.1.3 Fundamental set of solutions
124(3)
4.1.4 Existence of a fundamental set of solutions
127(1)
4.1.5 Reduction of order
128(3)
4.2 Solving homogeneous equations with constant coefficients
131(10)
4.2.1 Second-order equations
131(4)
4.2.2 Higher-order equations
135(6)
4.3 Introduction to solving nonhomogeneous equations
141(4)
4.4 Nonhomogeneous equations with constant coefficients: the method of undetermined coefficients
145(19)
4.4.1 Second-order equations
147(11)
4.4.2 Higher-order equations
158(6)
4.5 Nonhomogeneous equations with constant coefficients: variation of parameters
164(6)
4.5.1 Second-order equations
164(3)
4.5.2 Higher-order nonhomogeneous equations
167(3)
4.6 Cauchy-Euler equations
170(9)
4.6.1 Second-order Cauchy-Euler equations
170(3)
4.6.2 Higher-order Cauchy-Euler equations
173(4)
4.6.3 Variation of parameters
177(2)
4.7 Series solutions
179(26)
4.7.1 Power series solutions about ordinary points
179(10)
4.7.2 Series solutions about regular singular points
189(1)
4.7.3 Method of Frobenius
190(10)
Application: Zeros of the Bessel functions of the first kind
200(5)
4.8 Nonlinear equations
205(16)
5 Applications of higher-order differential equations
5.1 Harmonic motion
221(35)
5.1.1 Simple harmonic motion
221(7)
5.1.2 Damped motion
228(10)
5.1.3 Forced motion
238(12)
5.1.4 Soft springs
250(3)
5.1.5 Hard springs
253(1)
5.1.6 Aging springs
254(1)
Application: Hearing beats and resonance
255(1)
5.2 The pendulum problem
256(9)
5.3 Other applications
265(18)
5.3.1 L-R-C circuits
265(3)
5.3.2 Deflection of a beam
268(2)
5.3.3 Bode plots
270(3)
5.3.4 The catenary
273(10)
6 Systems of ordinary differential equations
6.1 Review of matrix algebra and calculus
283(14)
6.1.1 Defining nested lists, matrices, and vectors
283(4)
6.1.2 Extracting elements of matrices
287(1)
6.1.3 Basic computations with matrices
288(2)
6.1.4 Systems of linear equations
290(2)
6.1.5 Eigenvalues and eigenvectors
292(4)
6.1.6 Matrix calculus
296(1)
6.2 Systems of equations: preliminary definitions and theory
297(18)
6.2.1 Preliminary theory
301(7)
6.2.2 Linear systems
308(7)
6.3 Homogeneous linear systems with constant coefficients
315(18)
6.3.1 Distinct real eigenvalues
315(5)
6.3.2 Complex conjugate eigenvalues
320(5)
6.3.3 Solving initial-value problems
325(2)
6.3.4 Repeated eigenvalues
327(6)
6.4 Nonhomogeneous first-order systems: undetermined coefficients, variation of parameters, and the matrix exponential
333(15)
6.4.1 Undetermined coefficients
334(3)
6.4.2 Variation of parameters
337(5)
6.4.3 The matrix exponential
342(6)
6.5 Numerical methods
348(14)
6.5.1 Built-in methods
349(7)
6.5.2 Euler's method
356(4)
6.5.3 Runge-Kutta method
360(2)
6.6 Nonlinear systems, linearization, and classification of equilibrium points
362(23)
6.6.1 Real distinct eigenvalues
363(5)
6.6.2 Repeated eigenvalues
368(3)
6.6.3 Complex conjugate eigenvalues
371(3)
6.6.4 Nonlinear systems
374(11)
7 Applications of systems of ordinary differential equations
7.1 Mechanical and electrical problems with first-order linear systems
385(6)
7.1.1 L-R-C circuits with loops
385(1)
7.1.2 L-R-C circuit with one loop
385(2)
7.1.3 L-R-C circuit with two loops
387(3)
7.1.4 Spring-mass systems
390(1)
7.2 Diffusion and population problems with first-order linear systems
391(8)
7.2.1 Diffusion through a membrane
391(2)
7.2.2 Diffusion through a double-walled membrane
393(3)
7.2.3 Population problems
396(3)
7.3 Applications that lead to nonlinear systems
399(24)
7.3.1 Biological systems: predator-prey interactions, the Lotka-Volterra system, and food chains in the chemostat
400(14)
7.3.2 Physical systems: variable damping
414(4)
7.3.3 Differential geometry: curvature
418(5)
8 Laplace transform methods
8.1 The Laplace transform
423(9)
8.1.1 Definition of the Laplace transform
423(3)
8.1.2 Exponential order, jump discontinuities, and piecewise continuous functions
426(2)
8.1.3 Properties of the Laplace transform
428(4)
8.2 The inverse Laplace transform
432(7)
8.2.1 Definition of the inverse Laplace transform
432(6)
8.2.2 Laplace transform of an integral
438(1)
8.3 Solving initial-value problems with the Laplace transform
439(5)
8.4 Laplace transforms of step and periodic functions
444(18)
8.4.1 Piecewise defined functions: the unit step function
444(3)
8.4.2 Solving initial-value problems with piecewise continuous forcing functions
447(2)
8.4.3 Periodic functions
449(8)
8.4.4 Impulse functions: the delta function
457(5)
8.5 The convolution theorem
462(4)
8.5.1 The convolution theorem
462(2)
8.5.2 Integral and integrodifferential equations
464(2)
8.6 Applications of Laplace transforms, Part I
466(12)
8.6.1 Spring-mass systems revisited
466(3)
8.6.2 L-R-C circuits revisited
469(5)
8.6.3 Population problems revisited
474(2)
Application: The tautochrone
476(2)
8.7 Laplace transform methods for systems
478(10)
8.8 Applications of Laplace transforms, Part II
488(15)
8.8.1 Coupled spring-mass systems
488(5)
8.8.2 The double pendulum
493(3)
Application: Free vibration of a three-story building
496(7)
9 Eigenvalue problems and Fourier series
9.1 Boundary-value problems, eigenvalue problems, and Sturm-Liouville problems
503(7)
9.1.1 Boundary-value problems
503(2)
9.1.2 Eigenvalue problems
505(3)
9.1.3 Sturm-Liouville problems
508(2)
9.2 Fourier sine series and cosine series
510(8)
9.2.1 Fourier sine series
510(5)
9.2.2 Fourier cosine series
515(3)
9.3 Fourier series
518(17)
9.3.1 Fourier series
518(7)
9.3.2 Even, odd, and periodic extensions
525(5)
9.3.3 Differentiation and integration of Fourier series
530(4)
9.3.4 Parseval's equality
534(1)
9.4 Generalized Fourier series
535(10)
10 Partial differential equations
10.1 Introduction to partial differential equations and separation of variables
545(3)
10.1.1 Introduction
545(1)
10.1.2 Separation of variables
546(2)
10.2 The one-dimensional heat equation
548(9)
10.2.1 The heat equation with homogeneous boundary conditions
548(3)
10.2.2 Nonhomogeneous boundary conditions
551(3)
10.2.3 Insulated boundary
554(3)
10.3 The one-dimensional wave equation
557(8)
10.3.1 The wave equation
557(5)
10.3.2 D'Alembert's solution
562(3)
10.4 Problems in two dimensions: Laplace's equation
565(5)
10.4.1 Laplace's equation
565(5)
10.5 Two-dimensional problems in a circular region
570(15)
10.5.1 Laplace's equation in a circular region
570(3)
10.5.2 The wave equation in a circular region
573(12)
Bibliography 585(2)
Index 587
Martha L. Abell and James P. Braselton are graduates of the Georgia Institute of Technology and the Ohio State University, respectively, and teach at Georgia Southern University, Statesboro where they have extensive experience instructing students at both the undergraduate and graduate levels. Other books by the authors include Differential Equations with Mathematica and Mathematica by Example. Martha L. Abell and James P. Braselton are graduates of the Georgia Institute of Technology and the Ohio State University, respectively, and teach at Georgia Southern University, Statesboro where they have extensive experience instructing students at both the undergraduate and graduate levels. Other books by the authors include Differential Equations with Mathematica and Mathematica by Example.