I will offer a new two-semester course on the numerical integration of time-dependent differential equations (DEs) with a special emphasis on applications. There is enough material for a two-semester sequence of courses. The second semester will be at a higher level than the first one, and it will complement it well. The second semester can be considered as being a preparatory step for doing research work on the numerical solution of DEs whereas the first semester will be at a lower level. In these courses I will emphasize on time-dependent DEs arising in applications, especially in engineering. I would like each student to solve an application of his/her own interest. The project will be computer-oriented and will make use of the current available software, such as Fortran or Matlab codes. Each project will be the subject of a presentation in class to develop oral skills. It will also be the subject of a short report to be written. Both the report and the oral presentation will count for the final grade. The numerical solution of time-dependent DEs is a sufficiently broad topic to attract many engineering students. This course will cover the development, the mathematical analysis, and the use of practical algorithms for the numerical solution of time-dependent DEs. While initially offered as M371-M372, the longer run plan is to have a sequence numbered like M177-M277, as suggested by Ken Atkinson.

Here is a tentative list of some possible applications to be discussed during the courses depending on the interests of the different departments and also of the students:

- mechanical systems and multibody dynamics
- electrical circuits
- molecular dynamics
- atmospheric chemistry
- chemical reaction systems
- convection-diffusion-reaction problems
- optimal control problems
- prescribed trajectory path control problems
- structural dynamics

- ordinary differential equations (ODEs)
- differential-algebraic equations (DAEs)

We will consider other classes of differential equations among the following ones depending on the interests raised and on the applications to be treated. We will give an emphasis on the classes of DEs with most interests in terms of applications and just a short introduction or no treatment at all on the ones with least interest. A detailed treatment will be given only in the more advanced course:

- time-dependent partial differential equations (PDEs)
- stochastic differential equations (SDEs)
- delay-differential equations (DDEs)
- differential inequalities (DIs)
- discontinuous differential equations
- differential inclusions

Some issues that will be discussed and which require different techniques for the diverse kinds of equations:

- initial/boundary value problems (IVP/BVP)
- autonomous/nonautonomous problems
- definition of a solution
- existence/uniqueness of a solution
- linearity/nonlinearity

Some more advanced topics which could be treated only in the more advanced course:

- stability of linear/nonlinear systems (Routh-Hurwitz criterion, Lyapunov functions)
- differentiability/sensitivity of solutions with respect to parameters/initial values (Wronskian matrices, variational equations, the resolvent, the Groebner-Alekseev formula)
- Sturm-Liouville eigenvalue problems
- optimal control problems (the Pontryagin maximum principle)
- parameter estimation

Certain of the following topics in variational calculus could be treated in relation with applications in mechanics:

- first/second variations
- geodesics
- Euler-Lagrange equations
- Lagrangian mechanics
- Hamiltonian mechanics (symplectic applications)

- Taylor series
- explicit/implicit Runge-Kutta (RK) methods

- explicit/implicit multistep methods (Adams/BDF methods)
- collocation methods
- Rosenbrock methods
- extrapolation methods

- the Euler method and its convergence
- construction of higher order methods
- existence/uniqueness of the numerical solution
- local error analysis
- order conditions
- stability (see more details below)
- global error analysis: consistency + stability => convergence
- convergence order
- nonstiff/stiff equations

- Butcher-tree analysis for the local error of Runge-Kutta methods
- the symplifying assumptions of Runge-Kutta coefficients
- use of the structure of the DEs to construct tailored methods
- error constants
- asymptotic expansion of global error
- adjoint, symmetric methods
- symplectic methods for Hamiltonian systems
- volume-preserving flows and numerical methods
- method of lines for time-dependent PDEs (leading to large-scale ODEs)
- the Courant-Friedrich-Levy (CFL) conditions
- single/multiple shooting for BVPs

- critical points
- periodic solutions
- perturbation calculus
- bifurcations point (e.g., Hopf bifurcations)
- strange attractors
- Poincare sections
- stable/instable/central manifolds
- homoclinic/heteroclinic orbits

- stability regions
- A-stability

- Pade approximations to the exponential function
- order stars theory and the Ehle conjecture
- logaritmic norms of matrices
- B-stability
- Dahlquist's barriers for multistep methods
- preservation of invariants
- backward error analysis
- shadowing of numerical solutions

- solution of nonlinear systems for implicit methods (Newton-type methods, iterative solvers for linear systems)
- error estimation (Richardson extrapolation, embedded methods)
- stepsize control for variable stepsizes
- order selection for multistep methods
- dense output
- event location
- codes performance

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