# Theory of differential equations. PDEs

Closed trajectory and limit cycle. Linearization, Hartman-Grobman theorem. Stability theory. Invariant manifolds. Bendixson theorem, Duffing equation.

## Partial Differential Equations and Group Theory

Poincare index theory Dissipative and Conservative systems, energy method. Liouville theorem. Lyapunov stability and instability theorems.

Poincare Bendixson theorem. Exam Blowing-up method. Robustness of the system.

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Normal form. Hopf-Takens bifurcation. Examples of higher-dimensional systems.

### Outline of Course

Types of partial differential equations. Classification of partial differential equations of the second order. Nonstacionary heat conduction. Fouriers method. Method of decomposition by eigenfunctions. Application of Laplace transform to the equation of parabolic type. Equation of oscillation of a string. Application of Fouriers method of separation of variables to the wave equation.

Laplacian in Cartesian, cylindrical and spherical coordinates. Boundary-value problems for Laplace equation.

## Partial Differential Equations (PDEs) | Department of Mathematics

The operation of finding a partial derivative can be applied to a function that is itself a partial derivative of another function to get what is called a second-order partial derivative. The order and degree of partial differential equations are defined the same as for ordinary differential equations.

Many physically important partial differential equations are second-order and linear. For example:.

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8. Thus, the Laplace equation is elliptic, the heat equation is parabolic, and the wave equation is hyperbolic. Partial differential equation.

Using Green's Functions to Solve Nonhomogeneous ODEs

Info Print Cite. Submit Feedback. Thank you for your feedback. Quotes [ edit ] One morning early in my Hersh's years as a thesis student of Peter Lax , I entered my mentor's office to find him glowing in smiles. Louis, I wondered?

He had been on leave in England; now he was back home! At the time.

I didn't get it. Then they both stayed on to become famous faculty members there—Louis, a world master at elliptic partial differential equations, and Peter, a world master of hyperbolic PDEs. They hardly ever collaborated or produced joint publications. But their conversations and their intellectual and emotional interactions were a vital part of their creativity and success. Princeton University Press.

ISBN Keep in mind that there is in truth no central core theory of nonlinear partial differential equations, nor can there be. The sources of partial differential equations are so many - physical, probalistic, geometric etc.