20 credits at level HE6
This module will develop the solution methods for differential equations covered in Further Mathematical Methods. Numerical techniques for the solution of both ordinary and partial differential equations are developed in this module, along with analytical methods for partial differential equations. There is an emphasis on applications throughout the module.
Numerical Solution of First-Order Ordinary Differential Equations (ODEs):
Eulerís and Eulerís Improved Method.
Predictor-Corrector form: Runge-Kutta methods.
Higher order ordinary differential equations.
Applications of ODEs.
Fourier series: full and half range.
Linear Partial Differential Equations (PDEs) : Separation of Variables.
Solution using Laplace transforms.
Applications to wave propagation and heat conduction problems.
Potential theory and Laplaceís equation.
Applications to fluid dynamics and water waves.
Finite Differences and Application to PDEs.
Linear Wave Equation : Time marching.
Conservation form: the Lax-Wendroff method.
Approximately two-thirds of the available time will be devoted to lectures based on printed notes. Class discussion and participation will be encouraged. The remainder of the time will be devoted to supervised tutorial work and practical computer laboratory sessions.
Two pieces of coursework will be set, each to be completed by a prescribed date outside class contact time. There will be a formal closed-book examination of 2ľ hours duration at the end of the module. The weighting of the two components of assessment is as follows:
Coursework: 30% Examination: 70%
when you have successfully completed this module you will:
to demonstrate that you have achieved the learning outcome you will:
|1.||Have an understanding of a variety of numerical methods for the solution of ordinary differential equations.||Derive a variety of numerical methods for the solution of ordinary differential equations; apply the methods to obtain the numerical solution of various ordinary differential equations.|
|2.||Have an understanding of a variety of analytical and numerical methods for the solution of partial differential equations.||
Apply a variety of methods to obtain closed-form analytical solution of partial differential equations;
derive and analyse various numerical methods for the solution of partial differential equations.
|3.||Have an understanding of the applications of differential equations.||Use differential equations to develop a variety of mathematical models.|
Your achievement of the learning outcomes for this module will be tested as follows:
|Description||Assignment on numerical methods and applications for the solution of ordinary differental equations and analytical methods in partial differential equations.||Assignment on analytical methods and applications in partial differential equations.||2.25 hour examination covering syllabus above.|
Before taking this module you must have successfully completed the following:
No restrictions apply.
Haberman R. Applied Partial Differential Equations
Pearson Prentice Hall (2004).
Jeffrey A. Applied Partial Differential Equations: An Introduction
Elsevier Science (2003).
Morton K.W. and Mayers D.F. Numerical Solution of Partial Differential Equations : An Introduction, Cambridge (1994).
Pinsky M.A. Partial Differential Equations and Boundary-Value Problems with Applications. (1998) (out of print, but available in library)
Shampine F. Numerical Solution of Ordinary Differential Equations.(1994)
Haberman R. Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems. (2003)
Smith G.D. Numerical Solution of Partial Differential Equations : Finite Difference Methods., Oxford (1985).
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