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)

Floating point arithmetic (1 lecture), stability of algorithms (2 lectures), conditioning of a problem (2 lectures), perturbation analysis (2 lectures), algorithmic complexity (1 lecture), Matrix decomposition including LU, Cholesky, QR, SVD, etc. (12 lectures), Iterative techniques mainly focussing on Krylov subspace methods including Lanczos, Arnoldi, Conjugate Gradient, GMRES, etc. (12 lectures), Preconditioning (2 lectures), structured matrix computations (4 lectures), designing matrix algorithms on modern computer architectures (3 lectures).

Part 0 Introduction, Root finding: Fixed point iteration (Newton method, Secant mathod, etc.) – Part 1: Floating point arithmetic – Part 2: Orthogonal polynomials, Polynomial Interpolation and Approximation ∗ Weierstrass approximation theorem ∗ Minimax approximation ∗ Computing the best approximation ∗ Lebesgue constants ∗ Error analysis – Part 3: Numerical Differentiation ∗ Construction of finite difference schemes ∗ Pade Approximants ∗ Error analysis ∗ Non-uniform grids – Part 4: Numerical Integration ∗ Rectangular, Trapezoidal and Simpsons rule ∗ Romberg integration and Richardson extrapolation ∗ Gaussian quadrature ∗ Adaptive quadrature ∗ Error analysis – Part 5: Transform techniques ∗ Fourier, Laplace and Chebyshev transforms ∗ Fast algorithms for above