Introduction to Numerical Methods : Why study numerical methods,Sources of error in numerical solutions: truncation error, round off error.,Order of accuracy – Taylor series expansion – 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.

### Course Curriculum

 Introduction to Numerical Methods Details 46:6 Error Analysis Details 55:15 Introduction to Linear Systems Details 54:2 Linear Systems – II Details 52:47 Linear Systems – III Details 55:9 Linear Systems – Error Bounds Details 54:54 Error Bounds and Iterative Methods for Solving Linear Systems Details 56:43 Iterative Methods for Solving Linear Systems Details 54:23 Iterative Methods – II Details 53:42 Iterative Methods – III Details 56:42 Iterative Methods for Eigen Value Extraction Details 53:49 Solving Nonlinear Equations Details 56:25 Solving Nonlinear Equations – II Details 55:9 Solving Multi Dimensional Nonlinear Equations Details 55:3 Solving Multi Dimensional Nonlinear Equations – II Details 54:58 ARC Length and Gradient Based Methods Details 55:8 Gradient Based Methods Details 53:36 Conjugate Gradient Method Details 55:24 Conjugate Gradient Method – II Details 55:49 Nonlinear Conjugate Gradient and Introduction to PDEs Details 54:7 Eigenfunction Solutions for the Wave Equation Details 53:49 Analytical Methods for Solving the Wave Equation Details 54:55 Analytical Methods for Hyoerbolic and Parabolic PDEs Details 54:43 Analytical Methods for Parabolic and Elliptic PDEs Details 53:58 Analytical Methods for Elliptic PDEs Details 57:13 Series Solutions for Elliptic PDE’s and Introduction to Differential Operators Details 54:46 Differential Operators Details 53:16 Differential Operators – II Details 54:47 Differential Operators – III Details 54:34 Interpolation Details 56:46 Polynomial Fitting Details 56:48 Orthogonal Polynomials Details 54:50 Orthogonal Polynomials – II Details 54:10 Orthogonal Polynomials – III Details 53:57 Spline Functions Details 55:53 Orthogonal Basis Functions for Solving PDE’s Details 55:21 Orthogonal Basis Functions for Solving PDE\’s – II Details 52:28 Integral Equations Details 55:57 Integral Equations – II Details 52:36 Integral Equations – III Details 45:29

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