Numerical Analysis introduces numerical techniques for obtaining approximate solutions to mathematical problems that are difficult or impossible to solve analytically. The course covers direct and iterative methods for solving linear systems, numerical algorithms for ordinary differential equations (ODEs), and finite difference schemes for partial differential equations (PDEs). Emphasis is placed on convergence, stability, error analysis, and the practical implementation of numerical methods.
Course Content
Solving Linear systems: Matrix notation, Direct methods, Pivotal and scaling techniques, Gauss and Jordan eliminations, LU decomposition techniques.
Iterative Methods - Theorems related to convergence of iterative sequences and convergence criteria, Jacobi, Gauss Seidel, SOR Methods.
Numerical solutions of ordinary differential equations (ODE): Lipschitz conditions and constants, Picard Iteration technique with applications. Euler (explicit and implicit) and Modified Euler methods, Runge Kutta method. Errors and error propagation. Higher order Taylor expansion for solving ordinary differential equations and Higher order Differential equations.
Numerical solutions of partial differential equations : Parabolic type, Elliptic type, Hyperbolic type using finite difference methods.