THIS IS THE FIRST COURSE IN NUMERICAL ANALYSIS FOR GRADUATE STUDENTS. THE MAIN OBJECTIVES OF THIS COURSE INCLUDE:

(1) DEVELOPMENT AND APPLICATIONS OF NUMERICAL METHODS WHEN ANALYTICAL TECHNIQUES ARE NOT AVAILABLE;

(2) DEVELOPMENT OF A CONCEPTUAL FRAMEWORK FOR ANALYSIS OF METHODS TO FIX THE PROBLEM;

(3) DISCRETE CALCULUS AND APPROXIMATIONS;

(4) TRADEOFFS BETWEEN ACCURACY AND COMPUTATIONAL COST;

1. INTERPOLATION (3 HRS)

(1) LAGRANGE POLYNOMIALS
(2) POLYNOMIAL INTERPOLATIONS; SPLINES

2. NUMERICAL DIFFERENTIATION (4 HRS)

(1) CONSTRUCTION OF FINITE DIFFERENCE SCHEME, ORDER OF ACCURACY
(2) MODIFIED WAVENUMBER AS A MEASURE OF ACCURACY
(3) PADE APPROXIMATION
(4) MATRIX REPRESENTATION OF FINITE DIFFERENCE SCHEMES

3. NUMERICAL INTEGRATION (8 HRS)

(1) TRAPEZOIDAL RULE; SIMPSON’S RULE; ERROR ANALYSIS AND MID-POINT RULE
(2) ROMBERG INTEGRATION AND RICHARDSON’S EXTRAPOLATION
(3) ADAPTIVE QUADRATURE; GAUSS QUADRATURE

4. NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS (10 HRS)

(1) INITIAL VALUE PROBLEMS; NUMERICAL STABILITY ANALYSIS, MODEL EQUATION
(2) ACCURACY; PHASE AND AMPLITUDE ERRORS
(3) RUNGE-KUTTA TYPE FORMULAS, MULTI-STEP METHODS; IMPLICIT METHODS
(4) SYSTEM OF DIFFERENTIAL EQUATIONS; STIFFNESS
(5) LINEARIZATION FOR IMPLICIT SOLUTION OF NON-LINEAR DIFFERENTIAL EQUATIONS
(6) BOUNDARY VALUE PROBLEMS, SHOOTING, DIRECT METHODS, NON-UNIFORM GRIDS, EIGENVALUE PROBLEMS

5. PARTIAL DIFFERENTIAL EQUATIONS (10 HRS)

(1) FINITE-DIFFERENCE SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
(2) MODIFIED WAVENUMBER AND VON NEUMANN STABILITY ANALYSIS, MODIFIED EQUATIONS ANALYSIS
(3) ALTERNATING DIRECTION IMPLICIT METHODS; NON-LINEAR EQUATIONS; ITERATIVE METHODS FOR ELLIPTIC PDE’S