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.

Other Resources

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

These video tutorials are delivered by IIT Kharagpur as part of NPTEL online courses program.

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