The exhaustive list of topics in Computational Techniques in which we provide Help with Homework Assignment and Help with Project is as follows:

- Mathematical Models in Chemical Engineering
- Examples of linear and nonlinear algebraic equations
- Examples of ODE-IVP and ODE-BVP.
- PDEs: examples, classification.
- Model parameter estimation problem
- Review of abstract equation forms
- Concept of iterative solution approach.
- Fundamentals of Analysis
- Generalized concepts of vector space, sub-space, linear dependence.
- Concept of basis, dimension, norm defined on a general vector spaces.
- Examples of norms defined on different vector spaces, matrix norms.
- Inner product in a general vector space and orthogonal sets.
- Gram-Schmidt process and generation of orthogonal basis
- Well known orthogonal basis (Legandre polynomials, Laguerre polynomials etc.).
- Taylor series and polynomial approximations and their applications in numerical analysis.
- Problems classification, transformation and basic tools of numerical analysis.
- Linear Algebraic Equations and Related Numerical Schemes
- System of linear algebraic equations
- Conditions for existence of solution - geometric interpretations
- Classification of solution approaches.
- Direct methods: Review of Gaussian elimination
- L-U decomposition and Gauss-Jordan method.
- Motivation for sparse linear systems: solution of linear ODE-BVP / PDE using finite difference method.
- Motivation for sparse linear systems: Interpolation
- Cubic spline interpolation.
- Methods for sparse linear systems: Thomas algorithm
- Triangular systems.
- Iterative methods: Jacobi, Gauss-Siedel and successive over-relaxation methods.
- Convergence of iterative solution scheme for linear algebraic equations.
- Matrix conditioning, well conditioned and ill-conditioned linear systems.
- Nonlinear algebraic equations- Motivation: basics of orthogonal collocation.
- Nonlinear algebraic equations- Motivation: Solution of nonlinear ODE-BVP / PDE using orthogonal collocation.
- Nonlinear algebraic equations: derivative free iterative solution approaches (successive substitutions, Wegsteine iterations etc.).
- Newton Raphson method and its variations.
- ODE-IVP and Related Numerical Schemes
- Motivation: dynamic modeling and simulation of lumped parameter systems
- Motivation: Solving ODE-BVP using shooting method, solving PDE by converting to ODE-IVP using finite difference / orthogonal collocations
- Basic concepts in numerical solutions of ODE-IVP: step size, variable step size with accuracy monitoring, stiffness, concept of implicit and explicit methods
- Taylor series approximation and Runge-Kutta methods: derivation and examples
- Multi-step (predictor-corrector) approaches: derivations and examples
- Stability of ODE-IVP solvers, choice of step size and stability envelopes
- Introduction to Differential Algebraic system of equations
- Unconstrained Optimization and Related Numerical Schemes
- Parameter estimation problems in chemical engineering and their classification, approximations and interpolation, formulation of least square parameter estimation problem
- Necessary and sufficient conditions for unconstrained multivariate optimization
- Derivation of linear least square method (multivariate regression) through algebraic and geometric viewpoint (through projections), weighted least square
- Statistical interpretations of linear least square solution
- Nonlinear in parameter models: Gauss - Newton method
- Nonlinear in parameter models: Gradient and conjugate gradient method
- Solving linear and nonlinear algebraic equations using optimization (Conjugate gradient method for linear equation solving, Leverberg-Marquardt method for nonlinear equation solving)
- Finite element method for solving ODE-BVP / PDEs: Raleigh Ritz method, Gelarkin's method