The method of differential inequalities

Develop qualitative research methods of nonlinear differential equations of elliptic and parabolic type. Based on the method of monotone iterations prove theorems of comparison, allowing us to investigate the existence of solutions of nonlinear equations. As examples, the various applications described by reaction-diffusion equations.

It is read in the 9-th semester.
2 hours of lectures per week

• Nefedov Nikolai Nikolaevich

1. Initial value problems for ODEs. Chaplygin's theorem for scalar equations and systems. Examples and applications.
2. Boundary Value Problems for ODEs. Nagumo theorem. Applying Theorem Nagumo for a singularly perturbed Neumann problem - the notion of the asymptotic method of differential inequalities.
3. Partial differential equations of elliptic type. Statement of the problem. The upper and lower solutions. The theorem on differential inequalities. The method of monotone iterations for the elliptic problem. Existence theorem solutions. Uniqueness theorem for solutions. Examples and applications.
4. Partial differential equations of parabolic type. Statement of the problem. The upper and lower solutions. The theorem on differential inequalities. The method of monotone iterations for parabolic problems. Existence theorem solutions. Uniqueness theorem for solutions. The examples are not the only solution. Examples and applications.
5. Stability of stationary solutions. Lyapunov stability. The theorem on asymptotic stability and instability of the stationary solutions. Examples and applications.