NATIONAL UNIVERSITY OF SCIENCES & TECHNOLOGY

MTH 851 Numerical Analysis (M Sc)

Title: Numerical Analysis Catalogue No: MTH 851Credit Hours: 03 Contact Hours: (3,0)

Text Books: a) R.L.Burden and J.D. Faires: Numerical Analysis; Prindle, Weber & Smith b) E.Kreyszing: Advanced Engineering Mathematics (8th ed)

References: a) Curtis F.Gerald Patrick O. Wheatley: Applied Numerical Analysis, Addison- Wesley

b) Donald Greenspan & Vincenzo Casulli: Numerical Analysis For Applied Mathematics, Science, and Engineering, Addison-Wesley

c) David Kahaner: Numerical Methods and Software, Prentice Hall.

Prepared by: Brig Dr. Muhammad Rafique

Goals: To teach numerical techniques for solving algebraic equations, ODEs and PDEs. Pre-requisites by Topic: Linear Algebra, ODEs and PDEs.

Topics:

1. Review of Calculus 2

hrs

2. Algorithms and Convergence 1 hr

3. Selections of equations in one variable by Bisection Method, Fixed point iteration, Newton- 3

hrs

Raphson Method, Secant Method, Method of False Position

4. Error Analysis for interactive methods 1 hr

5. Interpolation and polynomial approximation: Weierstrass Approximation Theorem, Lagrange 3 hrs

Interpolating Polynomial, Newton’s Interpolating Divided Difference Formulae

6. Cubic Spline Interpolation 2

hrs

7. Numerical Differentiations 1

hr

8. Numerical Integration: Simpson’s’ Rule, Trapezoidal Rule 1

hr

9. Composite Trapezoidal Rule, Composite Simpson’s’ Rule, Adaptive Quadrature Method. 2

hrs

10. Multiple Integrals 1

hr

11. Improper Integrals

12. Iterative Techniques in Matrix Algebra:

Norms of Vectors and Matrices, Schwarz Inequality 1 hr

Eigenvalues and Eigenvectors 1 hr

Iterative Technique for Solving Linear Systems: Jacobi Iterative Method, Guess Seidel Iterative 2 hrs

Error Estimates and Iterative Refinement 1 hr/tt/file_convert/55cf9c08550346d033a8516a/document.doc

Least Square Approximation, Orthogonal Polynomials and Least Square Approximations 3 hrs

13. Elementary Theory of Initial Value Problems for Ordinary Differential Equations: Euler’s 2

hrs

Method, Higher Order Taylor’s Method

14. Runga–Kutta Method 1 hr

15. Error Control and the Runga-Kutta-Felberg Method 1

hr

16. Runga Kutta Method for a System of Differential Equations and Error Analysis 2

hrs

17. Linear Shooting Method for Boundary Value Problems for ODEs. 1 hr

18. Shooting Method for Non-Linear Problems 1

hr

19. Finite Difference Methods for Linear and Nonlinear Problems 3

hrs

20. Numerical Solutions to Partial Differential Equations 4

hrs

Elliptic Partial Differential Equations (Poisson Equation)

Parabolic Partial Differential Equations (Heat Equation)

Hyperbolic Partial Differential equations (Wave Equation)

21. Finite Element Method 2

hrs

Computer Usage: Practice of different numerical techniques on computer.

Class Test:

Class Test # 1 6th weekClass Test # 2 12th weekFinal Test 16st week

Grading Criteria:

One hours tests 35%Quizzes/ Assignments 15%Final test 50%

/tt/file_convert/55cf9c08550346d033a8516a/document.doc