Syllabus

Introduction to Numerical Methods(2 hours)
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 (4 hours)
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 (3 hours)
Jacobi iteration Gauss Seidel iteration Convergence criteria

Direct Solution of Non Linear systems (4 hours)
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 (3 hours)
Conjugate gradient Preconditioning.

Partial Differential Equations (4 hours)
Introduction to partial differential equations Definitions & classifications of first and second order equations Examples of analytical solutions Method of characteristics.

Numerical Differentiation (4 hours)
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 (6 hours)
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 (4 hours)
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 (6 hours)
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

Understanding core concepts of error estimate and accuracy of numerical solutions. It then introduces the student to methods of solution of linear and non-linear equations.Both direct and iterative solution methods are discussed.

Introduction to the numerical solution of partial differential equations, after a brief review of canonical partial differential equations and well known analytical techniques for their solution, stressing when and why numerical solutions are necessary Introduction to Finite difference operators to solve typical initial and boundary value problems.

Introduction to the finite element method as a generic method for the numerical solution of partial differential equations. The concepts of weak form, finite element discretization, polynomial interpolation using Lagrange polynomials and numerical quadrature are introduced.

Numerical integration in the time domain emphasizing the key requirements of stability and accuracy of time integration algorithms. Introduction to integral equations and numerical techniques for their solution.

Understand and apply various numerical methods for solving mathematical problems.

Analyze the accuracy, efficiency, and stability of numerical algorithms.

Implement numerical methods using programming languages and software tools.

Students will learn how to apply, analyze, and implement various iterative methods.

Students will be equipped to handle a wide variety of real-world computational problems involving linear systems of equations

Timothy Sauer, Numerical Analysis, Pearson Education Inc, Boston, MA

D. Dahlquist, and A. Bork, Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ

Jorge Nocedal, and Stephen J. Wright, Numerical Optimization, Spring-Verlag New York, Inc.

I. Stakgold, Green's functions and Boundary Value Problems, Wiley

Not Applicable