Ordinary differential equations I 1000-114aRRZI
Differential equation and its solution, first order and higher order equations, systems of differential equations, direction field, solution methods for simple types equations.
Simple numerical one- and multistep schemes. Runge-Kutta methods. Explicit and implicit schemes. Ways to derive numerical methods for ODEs.
Local existence and uniqueness theorems. Prolongation of the solution. Dependence on a parameter or on the initial condition; differentiability with respect to the parameter.
Systems of linear ODEs, the basis of the solutions. The fundamental matrix. Wronskian, Liouville's theorem. Systems with constant coefficients. Exponential of a matrix, nonhomogeneous systems. Higher order linear ODEs with constant coefficients.
Difference equations and their properties. Convergence theory for one-step methods. Consistency and stability. Stability and strong stability of multistep methods.
Nonlinear ODEs and stability. Lyapunov function. Phase plane and taxonomy of phase curves of autonomous systems. Singular points on a plane.
Absolute stability and the region of absolute stability. Stiffness and how to cope with it.
Computer lab experiments: numerical and symbolic ODE packages.
Type of course
Bibliography
- V.I.Arnold, R.Crooke "Ordinary differential equations", Springer
- Boyce, DiPrima, "Elementary differential equations", Wiley
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