Differential Equations

Code: MA662 | L-T-P-C: 3-1-0-8

MA662 Differential Equations [3-1-0-8] Prerequisite: Nil

Existence and Uniqueness of Initial Value Problems: Picard's and Peano's Theorems, Gronwall's inequality, continuous dependence, maximal interval of existence. Second and Higher Order Linear Equations: Fundamental solutions, Wronskian, variation of constants, behaviour of solutions. Power series method with properties of Legendre polynomials and Bessel functions. Linear Systems: Autonomous Systems and Phase Space Analysis, matrix exponential solution, critical points, proper and improper nodes, spiral points and saddle points.

First Order Partial Differential Equations: Classification, Method of characteristics for quasi-linear and nonlinear equations, Cauchy's problem, Cauchy-Kowalewski's Theorem. Second-Order Partial Differential Equations: Classification, normal forms and characteristics, Well-posed problem, Stability theory, energy conservation, and dispersion, Adjoint differential operators. Laplace Equation: Maximum and Minimum principle, Green's identity and uniqueness by energy methods, Fundamental solution, Poisson's integral formula, Mean value property, Green's function. Heat Equation: Maximum and Minimum Principle, Duhamel's principle. Wave equation: D'Alembert solution, method of spherical means and Duhamel's principle. The Method of separation of variables for heat, Laplace and wave equations.

Texts/References:

1. G. F. Simmons and S. G. Krantz, Differential Equations: Theory, Technique, and Practice, McGraw Hill, 2006.
2. E. A. Coddington, An Introduction to Ordinary Differential Equations, Prentice Hall India, 1995.
3. E. L. Ince, Ordinary Differential Equations, Dover Publications, 1958.
4. S. L. Ross, Differential Equations, 3rd edition, Wiley India, 1984.
5. I. N. Sneddon, Elements of Partial Differential Equations, McGraw Hill, 1957.
6. S. J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Publications, 1993.
7. R. Haberman, Elementary Applied Partial Differential equations with Fourier Series and Boundary Value Problem, 4th Edition, Prentice Hall, 1998.
8. Fritz John, Partial Differential Equations, Springer-Verlag, Berlin, 1982.