Lecture 
Contents 

(1 hr) 
(Sections from a module, which are relevant for each lecture, are indicated in the last column) 


Module 1: Equation Forms in Process Modeling 
Sections 
1 
Introduction and Motivation, Linear and Nonlinear Algebraic Equation 
1,2.12.2 
2 
Optimization based Formulations, ODEIVPs and Differential Algebraic Equations 
2.3,2.4, 3 
3 
ODEBVPs and PDEs 
4 
4 
ODEBVPs and PDEs, Abstract model forms 
4,5 




Module 2: Fundamentals of Vector Spaces 
Sections 
5 
Generalized concepts of vector space, subspace, linear dependence 
1,2 
6 
Concept of basis, dimension, norms defined on general vector spaces 
2 
7 
Examples of norms defined on different vector spaces, Cauchy sequence and convergence, introduction to concept of completeness and Banach spaces 
3 
8 
Inner product in a general vector space, Innerproduct spaces and their examples, 
4 
9 
CauchySchwartz inequality and orthogonal sets 
4 
10 
GramSchmidt process and generation of orthogonal basis, well known orthogonal basis 
5 
11 
Matrix norms 
6 




Module 3: Problem Discretization Using Approximation Theory 
Sections 
12 
Transformations and unified view of problems through the concept of transformations, classification of problems in numerical analysis, Problem discretization using approximation theory 
1,2 
13 
Weierstrass theorem and polynomial approximations, Taylor series approximation 
2, 3.1 
14 
Finite difference method for solving ODEBVPs with examples 
3.2 
15 
Finite difference method for solving PDEs with examples 
3.3 
16 
Newton’s Method for solving nonlinear algebraic equation as an application of multivariable Taylor series, Introduction to polynomial interpolation 
3.4 
17 
Polynomial and function interpolations, Orthogonal Collocations method for solving ODEBVPs 
4.1,4.2,4.3 
18 
Orthogonal Collocations method for solving ODEBVPs with examples 
4.4 
19 
Orthogonal Collocations method for solving PDEs with examples 
4.5 
20 
Necessary and sufficient conditions for unconstrained multivariate optimization, Least square approximations 
8 
21 
Formulation and derivation of weighted linear least square estimation, Geormtraic interpretation of least squares 
5.1,5.2 
22 
Projections and least square solution, Function approximations and normal equation in any inner product space 
5.3 
23 
Model Parameter Estimation using linear least squares method, Gauss Newton Method 
5.4 
24 
Method of least squares for solving ODEBVP 
5.5 
25 
Gelarkin’s method and generic equation forms arising in problem discretization 
5.5 
26 
Errors in Discretization, Generaic equation forms in transformed problems 
6,7 




Module 4: Solving Linear Algebraic Equations 
Sections 
27 
System of linear algebraic equations, conditions for existence of solution  geometric interpretations (row picture and column picture), review of concepts of rank and fundamental theorem of linear algebra 
1,2 
28 
Classification of solution approaches as direct and iterative, review of Gaussian elimination 
3 
29 
Introduction to methods for solving sparse linear systems: Thomas algorithm for tridiagonal and block tridiagonal matrices 
4 
30 
Blockdiagonal, triangular and blocktriangular systems, solution by matrix decomposition 
4 
31 
Iterative methods: Derivation of Jacobi, GaussSiedel and successive overrelaxation methods 
5.1 
32 
Convergence of iterative solution schemes: analysis of asymptotic behavior of linear difference equations using eigen values 
9 
33 
Convergence of iterative solution schemes with examples 
5.2 
34 
Convergence of iterative solution schemes, Optimization based solution of linear algebraic equations 

35 
Matrix conditioning, examples of well conditioned and illconditioned linear systems 
7 




Module 5: Solving Nonlinear Algebraic Equations 
Sections 
36 
Method of successive substitutions derivative free iterative solution approaches 
1,2 
37 
Secant method, regula falsi method and Wegsteine iterations 
3.1,3.2 
38 
Modified Newton’s method and qausiNewton method with Broyden’s update 
3.3, 3.4, 3.5 
39 
Optimization based formulations and LeverbergMarquardt method 
4 
40 
Contraction mapping principle and introduction to convergence analysis (Optional lecture) 
6 




Module 6: Solving Ordinary Differential Equations – Initial Value Problems (ODEIVPs) 
Sections 
39 
Introduction, Existence of Solutions (optional topic), 

40 
Analytical Solutions of Linear ODEIVPs 
3 
41 
Analytical Solutions of Linear ODEIVPs (contd.), Basic concepts in numerical solutions of ODEIVP: step size and marching, concept of implicit and explicit methods 
4 
42 
Taylor series based and RungeKutta methods: derivation and examples 
5 
43 
RungeKutta methods 
5 
44 
Multistep (predictorcorrector) approaches: derivations and examples 
6.1 
45 
Multistep (predictorcorrector) approaches: derivations and examples 
6.1 
46 
Stability of ODEIVP solvers, choice of step size and stability envelopes 
7.1,7.2 
47 
Stability of ODEIVP solvers (contd.), stiffness and variable step size implementation 
7.3,7.4 
48 
Introduction to solution methods for differential algebraic equations (DAEs) 
8 
49 
Single shooting method for solving ODEBVPs 
9 
50 
Review 
