Differential Equations - 2020 entry

This module introduces various types of ordinary and partial differential equations and a number of analytical and numerical techniques used to solve them. Differential equations are at the heart of countless modern applications of mathematics to natural phenomena and man-made technology. Computational implementation plays a vital role in many areas of engineering, science, finance, health care, etc.

Differential equations develop ideas from Dynamics further, considering rates of change of a model’s variables (in one or multiple dimensions) in systems of equations, which relate these rates of change to expressions (functions) of the model’s variables. For example, in mechanical systems the rate of change of position, that is velocity, and the rate of change of velocity, that is acceleration, may be set in relation through physical laws. Building on your knowledge of dynamics, calculus and advanced calculus, and using algebraic methods, you will model systems of differential equations, develop an understanding on how to find solutions applying analytical or numerical methods.

The development of an understanding of the theoretical foundation will be accompanied by applications including the growth of plants and organisms, the spread of diseases, physical forces acting on an object or models describing the fluctuations of financial markets. In this, the module will enable you to demonstrate an understanding of, and the competence in, a range of analytical tools for posing and solving differential equations.

Pre-requisite modules: “Calculus and Geometry” (ECM1901) and “Advanced Calculus” (ECM1905), or equivalent

The aim of this module is to introduce you to some representative types of ordinary and partial differential equations, how these are relevant in many fields of applied sciences and engineering, and will introduce you to a number of techniques used to solve differential equations exactly (analytical methods) or approximately (numerical algorithms). You will also develop the computational skills to implement numerical algorithms in MATLAB/Python and use these to solve applied problems.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 Demonstrate an understanding of analytic and numerical techniques for solving basic forms of ordinary differential equations;

2 Demonstrate a basic understanding of analytic and numerical techniques for solving low order partial differential equations;

3 Demonstrate competency in modelling basic applied problems with differential equations;

4 Demonstrate competency in developing and applying quantitative and computational techniques for differential equations;

Discipline Specific Skills and Knowledge:

5 Demonstrate a clear understanding of fundamental mathematical concepts and analytical techniques for ordinary and partial differential equations;

6 Demonstrate competency in the development of numerical techniques for differential equations;

7 Demonstrate a basic understanding of the relevance of differential equations within the mathematical sciences, and skills to use differential equations for modelling and solving applied problems from engineering and science;

Personal and Key Transferable/ Employment Skills and Knowledge:

8 Reason using abstract ideas;

9 Formulate and solve problems and communicate reasoning and solutions effectively in writing and presentation;

10 Demonstrate appropriate use of learning resources;

11 Demonstrate self-management and time management skills.

Section A: Analytical Solutions:

- Review of integration methods for separable ODEs, analytical and numerical methods for solving first and second order ordinary differential equations (ODEs), integrating factors, homogeneous and non-homogeneous ODEs, general solutions, particular solutions, reduction of order, variation of parameters method, Existence and uniqueness of ODEs [6 hours];

- Higher order linear ODEs, systems of ODEs, stability and qualitative methods for ODEs, dynamical systems analysis of ODEs [6 hours];

- Special functions and their use in series solution of ODEs, Orthogonal functions including Legendre, Bessel and trigonometric functions, Sturm-Liouville Boundary value problems [6 hours];

- Examples of partial differential equations (PDEs) and their solutions; examples: Laplace's equation, heat conduction equation and the wave equation, separation of variables, Cartesian, spherical and cylindrical coordinate systems in 1D, 2D; Solution of PDEs using series and orthogonal functions; applications to boundary value problems [3 hours];

Section B: Numerical Solutions:

- Numerical methods for ODEs: Euler methods, Runge-Kutta methods, Adams-Bashforth methods, Implementation in MATLAB/Python, initial value problems (IVPs), boundary value problems (BVPs) [3 hours];

- Numerical solution of special ODEs in MATLAB/Python e.g. stiff and non-stiff ODEs, delay differential equations (DDEs), implicit differential equations, switching differential equations and stochastic differential equations (SDEs) [3 hours];

- Numerical methods for PDEs: elliptic, parabolic and hyperbolic PDEs, finite difference method, Dirichlet, Neumann and mixed boundary conditions, Implementation in MATLAB /Python [3 hours].

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

Basic Reading:

ELE -  http://vle.exeter.ac.uk

Reading list for this module: