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