Course description

Direct Solution of Linear systems : Gauss elimination, Gauss Jordan elimination,Pivoting, inaccuracies due to pivoting,Factorization, Cholesky decomposition,Diagonal dominance, condition number, ill conditioned matrices, singularity and singular value decomposition,Banded matrices, storage schemes for banded matrices, skyline solver - Iterative solution of Linear systems : Jacobi iteration,Gauss Seidel iteration,Convergence criteria - Direct Solution of Non Linear systems : Newton Raphson iterations to find roots of a 1D nonlinear equation,Generalization to multiple dimensions,Newton Iterations, Quasi Newton iterations,Local and global minimum, rates of convergence, convergence criteria - Iterative Solution of Non Linear systems : Conjugate gradient,Preconditioning - Partial Differential Equations : Introduction to partial differential equations,Definitions & classifications of first and second order equations,Examples of analytical solutions,Method of characteristics.

Numerical Differentiation : Difference operators (forward, backward and central difference),Stability and accuracy of solutions,Application of finite difference operators to solve initial and boundary value problems - Introduction to the Finite Element Method as a method to solve partial differential equations : Strong form of the differential equation,Weak form,Galerkin method: the finite element approximation,Interpolation functions: smoothness, continuity, completeness, Lagrange polynomials,Numerical quadrature: Trapezoidal rule, simpsons rule,Gauss quadrature - Numerical integration of time dependent partial differential equations:Parabolic equations : algorithms - stability, consistency and convergence, Lax equivalence theorem - Hyperbolic equations : algorithms - Newmark's method,stability and accuracy, convergence, multi-step methods - Numerical solutions of integral equations : Types of integral equations,Fredholm integral equations of the first and second kind,Fredholm's Alternative theorem,Collocation and Galerkin methods for solving integral equations.

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