Articoli correlati a Differential Equations and Their Applications: An Introducti...
Sinossi
1 First-order differential equations.- 1.1 Introduction.- 1.2 First-order linear differential equations.- 1.3 The Van Meegeren art forgeries.- 1.4 Separable equations.- 1.5 Population models.- 1.6 The spread of technological innovations.- 1.7 An atomic waste disposal problem.- 1.8 The dynamics of tumor growth, mixing problems, and orthogonal trajectories.- 1.9 Exact equations, and why we cannot solve very many differential equations.- 1.10 The existence-uniqueness theorem; Picard iteration.- 1.11 Finding roots of equations by iteration.- 1.11.1 Newton's method.- 1.12 Difference equations, and how to compute the interest due on your student loans.- 1.13 Numerical approximations; Euler's method.- 1.13.1 Error analysis for Euler's method.- 1.14 The three term Taylor series method.- 1.15 An improved Euler method.- 1.16 The Runge-Kutta method.- 1.17 What to do in practice.- 2 Second-order linear differential equations.- 2.1 Algebraic properties of solutions.- 2.2 Linear equations with constant coefficients.- 2.2.1 Complex roots.- 2.2.2 Equal roots; reduction of order.- 2.3 The nonhomogeneous equation.- 2.4 The method of variation of parameters.- 2.5 The method of judicious guessing.- 2.6 Mechanical vibrations.- 2.6.1 The Tacoma Bridge disaster.- 2.6.2 Electrical networks.- 2.7 A model for the detection of diabetes.- 2.8 Series solutions.- 2.8.1 Singular points; Euler equations.- 2.8.2 Regular singular points; the method of Frobenius.- 2.8.3 Equal roots, and roots differing by an integer.- 2.9 The method of Laplace transforms.- 2.10 Some useful properties of Laplace transforms.- 2.11 Differential equations with discontinuous right-hand sides.- 2.12 The Dirac delta function.- 2.13 The convolution integral.- 2.14 The method of elimination for systems.- 2.15 Higher-order equations.- 3 Systems of differential equations.- 3.1 Algebraic properties of solutions of linear systems.- 3.2 Vector spaces.- 3.3 Dimension of a vector space.- 3.4 Applications of linear algebra to differential equations.- 3.5 The theory of determinants.- 3.6 Solutions of simultaneous linear equations.- 3.7 Linear transformations.- 3.8 The eigenvalue-eigenvector method of finding solutions.- 3.9 Complex roots.- 3.10 Equal roots.- 3.11 Fundamental matrix solutions; eAt.- 3.12 The nonhomogeneous equation; variation of parameters.- 3.13 Solving systems by Laplace transforms.- 4 Qualitative theory of differential equations.- 4.1 Introduction.- 4.2 Stability of linear systems.- 4.3 Stability of equilibrium solutions.- 4.4 The phase-plane.- 4.5 Mathematical theories of war.- 4.5.1 L. F. Richardson's theory of conflict.- 4.5.2 Lanchester's combat models and the battle of Iwo Jima.- 4.6 Qualitative properties of orbits.- 4.7 Phase portraits of linear systems.- 4.8 Long time behavior of solutions; the Poincaré-Bendixson Theorem.- 4.9 Introduction to bifurcation theory.- 4.10 Predator-prey problems; or why the percentage of sharks caught in the Mediterranean Sea rose dramatically during World War I.- 4.11 The principle of competitive exclusion in population biology.- 4.12 The Threshold Theorem of epidemiology.- 4.13 A model for the spread of gonorrhea.- 5 Separation of variables and Fourier series.- 5.1 Two point boundary-value problems.- 5.2 Introduction to partial differential equations.- 5.3 The heat equation; separation of variables.- 5.4 Fourier series.- 5.5 Even and odd functions.- 5.6 Return to the heat equation.- 5.7 The wave equation.- 5.8 Laplace's equation.- Appendix A.- Appendix B.- Appendix C.- Answers to odd-numbered exercises.
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