Numerical Methods - Engineering Mathematics

This course is designed to equip students with the fundamental concepts and techniques of numerical methods, which are essential for solving engineering and scientific problems. The course focuses on both theoretical understanding and practical implementation of numerical algorithms. It is tailored to meet the requirements of GATE examinations in various engineering disciplines.

1. **Introduction to Numerical Methods**:

  - Role of numerical methods in engineering and science.

  - Sources of error in numerical computations.

2. **Root Finding Methods**:

  - Bisection method.

  - Newton-Raphson method.

  - Secant method.

3. **Linear Systems of Equations**:

  - Gaussian elimination.

  - LU decomposition.

  - Iterative methods (Jacobi, Gauss-Seidel).

4. **Interpolation and Approximation**:

  - Lagrange interpolation.

  - Newton's divided difference interpolation.

  - Curve fitting using least squares.

5. **Numerical Differentiation and Integration**:

  - Numerical differentiation methods.

  - Numerical integration methods (Trapezoidal rule, Simpson's rule).

6. **Numerical Solutions of Ordinary Differential Equations (ODEs)**:

  - Euler's method.

  - Runge-Kutta methods.

  - Multistep methods (e.g., Adams-Bashforth).

7. **Partial Differential Equations (PDEs)**:

  - Finite difference methods for PDEs.

  - Boundary value problems (e.g., Laplace and Poisson equations).

8. **Numerical Techniques for Optimization**:

  - Gradient descent method.

  - Conjugate gradient method.

9. **Error Analysis and Stability**:

  - Analysis of errors in numerical methods.

  - Stability and convergence of numerical algorithms.

10. **Applications**:

  - Practical engineering and scientific problems solved using numerical methods.

  - Case studies from various engineering disciplines.