Kinematics of Fluid flow, Laws of fluid motion, Inviscid incompressible flows, two and three-dimensional motions, inviscid compressible flows; Viscous incompressible flows, Navier-Stokes equations of motion and some exact solutions; Flows at small Reynolds numbers; Boundary layer theory.
Ordinary Differential Equations: Initial value problems- basic theory and application of multistep methods (explicit and implicit), stability analysis- zero stability, absolute stability, relative stability and intervals of stability, eigenvalue problems, predictor- corrector methods, Runge-Kutta methods, boundary value problems-shooting methods.
Partial Differential Equations:
(a) Parabolic Equations:Explicit and implicit finite difference approximations to one-dimensional heat equation, Alternating Direction Implicit (ADI) methods.
(b) Hyperbolic equations and Characteristics: Numerical integration along a characteristic, equations, numerical solution by the method of characteristics, finite diference solution of second order wave equation.
(c) Elliptic equations: finite difference methods in polar coordinates, techniques near curved boundaries, improvement of accuracy- direct and iterative schemes to solve systems, methods to accelerate the convergence.
(d) Convergence, consistency and stability analysis.
Existence-Uniqueness for systems: Picard’s theorem, Non-local existence theorem. (6 lectures)
Second Order Equations: General solution of homogeneous equations, Non-homogeneous equations, Wronskian, Method of variation of parameters, Sturm comparison theorem, Sturm separation theorem, Boundary value problems, Green's functions, Sturm-Liouville problems. (15 lectures)
Series Solution of Second Order Linear Equations: ordinary points, regular singular points, Legendre polynomials and properties, Bessel functions and properties. (15 lectures)
Systems of Differential Equations: Algebraic properties of solutions of linear systems, The eigenvalueeigenvector method of finding solutions, Complex eigenvalues, Equal eigenvalues, Fundamental matrix solutions, Matrix exponential. (4 lectures)