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ES4C3 - Mathematical & Computational Modelling

(15 Credits)

Aims

To provide the theoretical background to advanced modelling techniques as used in industry and to provide the necessary software application skills.

Learning Outcomes

By the end of the module the student should be able to:

  • Reproduce theoretical derivations used in the computer solution of ordinary differential equations and partial differential equations.
  • Be able to use modern computer software such as Matlab and Maple to solve engineering problems that involve a variety of ordinary differential equations and partial differential equations.

Syllabus

Introduction to symbolic computation.

  • Introduction to computer algebra systems (e.g. in Matlab and Maple).
  • Ordinary Linear Differential Equations.
  • Finite difference methods
  • Runge-Kutta methods
  • Stiff ordinary differential equations
  • Finite difference methods for parabolic, hyperbolic and elliptic partial differential equations.
  • Basic derivatives
  • Explicit methods for parabolic equations
  • Implicit methods for parabolic equations
  • Stability and convergence
  • Characteristic methods for first-order hyperbolic equations
  • Lax-Wendroff explicit method for first-order hyperbolic equations
  • Second-order hyperbolic equations
  • Methods for elliptic equations
  • Finite element methods for generalised partial differential equations.
  • Variational methods
  • Forming element equations
  • Obtaining solutions
  • Boundary element methods
  • Non-Homogeneous Partial Differential Equations.
  • Laplace Transform Methods in Partial Differential Equations.