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.

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 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 PDEs and how to solve PDEs with different analytical methods. Students also will be introduced to some physical problems in Engineering models that result in PDEs.

CO1 Students will learn the basic principles and methods for the analysis of various partial differential equations. Able to solve the most common PDEs, recurrent in engineering using standard techniques.
CO2 Apply some techniques/ methods to predict the behavior of certain phenomena. Identify real phenomena as models of PDEs.
CO3 Apply logical/mathematical thinking: the analytic process.
CO4 Understand the Fourier series for periodic functions and determine the Fourier coefficients and able to solve various PDEs using Fourier series.
CO5 Students will learn the Integral transform method for parabolic, hyperbolic and elliptic equations.

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

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

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

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