Module I (10 hours)
Existence and Uniqueness of Initial Value Problems: Picard's and Peano's Theorems, Gronwall's inequality, continuous dependence, maximal interval of existence. Linear Systems: Autonomous Systems and Phase Space Analysis, matrix exponential solution, critical points, proper and improper nodes, spiral points and saddle points.
Module II (8 hours)
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.
Module III (12 hours)
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 parabolic, hyperbolic and elliptic equations.

The objective of this course is to present the main results in the context of differential equations that allow learning about these topics.

Differential equations allow deterministic mathematical formulations of phenomena in physics and engineering as well as biological processes among many other scenarios

To equip students with the concepts of ODEs and PDEs and how to solve them with different analytical methods. Students also will be introduced to some physical problems in Engineering models that result in differential equations.

CO 1. Students will learn the basic theory behind ordinary and partial differential equations, the fundamental principles and methods for analyzing various differential equations, identify real-world phenomena as models for differential equations.
CO2. Solve various types of ODEs and PDEs using standard techniques.
CO3. Study qualitative analysis for ODEs, such as stability analysis, phase plane analysis, and classification of equilibrium points.
CO4. They will understand and apply different techniques for solving specific types of PDEs (e.g., Laplace’s equation, wave equation, heat equation).

S. L. Ross,, Differential Equations, 3rd edition,, Wiley India.

I. N. Sneddon,, Elements of Partial Differential Equations,, Dover Publications, 2006.

G. F. Simmons and S. G. Krantz,, Differential Equations: Theory, Technique, and Practice,, McGraw Hill, 2006.

Fritz John,, Partial Differential Equations,, Springer-Verlag, Berlin.