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 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