**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.

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