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PARTIAL DIFFERENTIAL EQUATIONS

Sergei Borisenok


1. First order PDEs

The first order PDE from two independent variables. Lagrange’s method. Integral surfaces. Classification of solutions. Lagrange — Sharpit’s method. Jacobi’s method. Cauchy methods of characteristics.

2. Second order PDEs

Linear PDE with constant coefficients. Methods of solutions for linear equations.
Classification of second order PDEs. Canonical forms. Conjugated operators.
Riemann’s method. Monge’s method.

2a. Hyperbolic type equations

Wave equation. Canonical form of one-dimensional wave equation and its solution. d’Alembert solution for one-dimensional wave equation. Separation of variables. Method of eigenfunctions. Uniqueness of solution. Duhamel’s principle for wave equation solution. Two-dimensional wave equation.

2b. Parabolic type equations

Diffusion equation. Boundary conditions. Separation of variables. Diffusion equation in cylindrical coordinates. Diffusion equation in spherical coordinates. Extremum principle. Theorem of uniqueness.

2c. Elliptic type equations

Laplace equation. Poisson equation. General properties of harmonic functions. Boundary conditions. Separation of variables. Laplace equation in cylindrical coordinates. Laplace equation in spherical coordinates. Internal Dirichlet problem for circle. External Dirichiet problem for circle. Internal Neumann problem for circle. Internal Dirichiet problem for sphere. Periodic solutions for wave equation with symmetry.

3. Integral transforms and the Green function methods

Laplace transform and its application to PDEs. Fourier transform and its application to PDEs.
Green function method and its application to PDEs.

4. Non-linear dynamical equations of high orders.
Dynamical PDEs. Self-similar solutions. KdF equation. Burgers’ equation. Solitons.

5.* PDEs and the systems of computer algebra

Literature:
1. P. R. Garabedian. Partial Differential Equations. New York: John Wiley & Sons, 1964.
2. D. G. Duffy. Solutions of Partial Differential Equations. Delhi: CBS, 1988.
3. G. L. Lamb. Introductory Applications of Partial Differential Equations. New York: John Wiley & Sons, 1995.
4. P. K. Kythe, P. Pun, M. R. Schafcrkotter. Partial Differential Equations and Mathernatica. New York: CRC Press, 1997.
5. G. B. Folland. Introduction to Partial Differential Equations. New Delhi: Prentice Hall of India, 2001.
6. J. Jost. Partial Differential Equations. New York: Springer-Verlag, 2002.
7. M. A. Pinsky. Partial Differential Equations and Boundary-Value Problems with Applications. Singapore: McGrow-Hill, 1998.