Differential Equations and Boundary-Value Problems

Code: MA751 | L-T-P-C: 4-0-0-8

MA751 Differential Equations and Boundary-Value Problems [4-0-0-8]

Existence and uniqueness of solutions of ODEs, power series solution, singular points, some special functions. Nonlinear system of ODE : Preliminary concepts and definitions, the fundamental existence-uniqueness results, dependence on initial conditions and parameters, the maximum interval of existence, linearlization, stability and Liapunov functions, saddle, nodes, foci and centers, normal form theory and Hamiltonian systems. Boundary value problems : Green's function method, Sturm-Liouville problem.

First-order PDEs, Cauchy problem, method of characteristics, Second-order PDEs, classification, characteristics and canonical forms. Elliptic boundary value problems : Maximum principle, Green's function, Sobolev spaces, variational formulations, weak solutions, Lax-Milgram theorem, trace theorem, Poincarénequality, energy estimates, Fredholm alternative, regularity estimates, system of conservation laws, entropy criteria.

Texts/References:

1. L. Perko, Differential Equations and Dynamical Systems, Springer, 2001.
2. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
3. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990
4. Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, 1998
5. Robert C. McOwen, Partial Differential Equations - Methods and Applications, Pearson Education Inc., Indian Reprint 2004.
6. S.J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Publications,New York, 1982.