Trefethen L.N., Birkisson A., Driscoll T.A. Exploring ODEs

Philadelphia: Society for Industrial and Applied Mathematics, 2018. — 343 p. — ISBN: 1611975158.What if all you had to do to solve an ODE were just to write it down ? That is the line we will follow in this book. Our emphasis is not just on the mathematics of ODEs, but on how the solutions behave. Do they blow up, decay, oscillate ? Are there rapid transitions where they flip from one state to another ? Does the behavior change if a coefficient is perturbed or a new term is added ? And how can such variety be deployed to explain the world around us ? We shall not just talk about these matters but explore them in action. ODEs are among the core topics of mathematics, with applications so ubiquitous that listing examples almost seems inappropriate. (Heat conduction, chemical reactions, chaos, population dynamics, deformations of a beam, radioactivity, bifurcation theory, stability theory, differential geometry, quantum mechanics, economics, finance, infectious diseases, nerve signals, vibrations, optics, waves, dynamics of networks, special functions, ballistics, planetary dynamics...). ODEs are everywhere ! To solve ODEs by writing them down, we will use Chebfun, an open-source MatLAB package that is freely available at www.chebfun.org. Many textbooks on ODEs concentrate on linear problems, because nonlinear ones are rarely analytically solvable. Here, with analytical solutions playing a lesser role, we will be able to give a more balanced treatment and fully appreciate the remarkable effects that come with nonlinearity.First-order scalar linear ODEs.
First-order scalar nonlinear ODEs.
Second-order ODEs and damped oscillations.
Boundary-value problems.
Eigenvalues of linear BVPs.
Variable coefficients and adjoints.
Resonance.
Second-order equations in the phase plane.
Systems of equations.
The fundamental existence theorem.
Random functions and random ODEs.
Chaos.
Linear systems and linearization.
Stable and unstable fixed points.
Multiple solutions of nonlinear BVPs.
Bifurcation.
Continuation and path-following.
Periodic ODEs.
Boundary and interior layers.
Into the complex plane.
Time-dependent PDEs.
Appendix A: Chebfun and its ODE algorithms.
Appendix B: 100 more examples.True PDF