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In this module we will study linear partial differential equations, we will explore their properties and discuss the physical interpretation of certain equations and their solutions. We will learn how to solve first order equations using the method of characteristics and second order equations using the method of separation of variables.
Introduction to linear PDEs: Review of partial differentiation; first-order linear PDEs, the heat equation, Laplace's equation and the wave equation, with simple models that lead to these equations; the superposition principle; initial and boundary conditions
Separation of variables and series solutions: The method of separation of variables; simple separable solutions of the heat equation and Laplace’s equation; Fourier series; orthogonality of the Fourier basis; examples and interpretation of solutions
Solution by characteristics: the method of characteristics for first-order linear PDEs; examples and interpretation of solutions; characteristics of the wave equation; d’Alembert’s solution, with examples; domains of influence and dependence; causality.
Total contact hours: 42
Private study hours: 108
Total study hours: 150
Assessment 1 Exercises, requiring on average between 10 and 15 hours to complete 20%
Assessment 2 Exercises, requiring on average between 10 and 15 hours to complete 20%
Examination 2 hours 60%
The coursework mark alone will not be sufficient to demonstrate the student's level of achievement on the module.
T. Myint-U, L. Debnath, Linear Partial Differential Equations for Scientists and Engineers, Birkhäuser 2007 (online)
L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, 3rd edition, Birkhäuser 2012 (online)
E. Kreyszig, Advanced Engineering Mathematics, Wiley 2011
See the library reading list for this module (Canterbury)
The intended subject specific learning outcomes.
On successfully completing the module students will be able to:
1 demonstrate knowledge and critical understanding of the well-established principles within linear partial differential equations (PDEs);
2 demonstrate the capability to use a range of established techniques and a reasonable level of skill in calculation and manipulation of the material to solve problems in the
following areas: separation of variables, Fourier series, the method of characteristics;
3 apply the concepts and principles in basic linear PDE methods in well-defined contexts beyond those in which they were first studied, showing the ability to evaluate
critically the appropriateness of different tools and techniques.
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