Course Materials
Main content blocks
- Lecturer: Profile: Melanie Rupflin
Much of the study of differential equations in the first year consisted of finding explicit solutions of particular ODEs or PDEs. However, for many differential equations which arise in practice one is unable to give explicit solutions and, for the most part, this course considers what information one can discover about solutions without actually finding the solution. Does a solution exist? Is it unique? Does it depend continuously on the initial data? How does it behave asymptotically? What are appropriate data? In this course some techniques will be developed for answering such questions.
The course will furnish undergraduates with the necessary skills to pursue any of the applied options in the third year and will also form the foundation for deeper and more rigorous courses in differential equations, the part B courses on Distribution Theory and on Fourier Analysis and the Part C courses Functional Analytic Methods for PDEs and Fixed Point Methods for Nonlinear PDEs.
Phase plane analysis: Phase planes, critical points, definition of stability, classification of critical points and linearisation, Bendixson-Dulac criterion. (4 lectures)
PDEs in two independent variables: First order semi-linear PDEs (using parameterisation). Classification of second order, semilinear PDEs; Normal form; Ideas of uniqueness and wellposedness. Illustration of suitable boundary conditions by example. Poisson's Equation and the Heat Equation: Maximum Principle leading to uniqueness and continuous dependence on the initial data.
(7 lectures)
Section outline
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The lecture notes consist of 4 main chapters on
1 ODEs and Picard’s Theorem
2 Plane autonomous systems of ODEs
3 First order semi-linear PDEs: the method of characteristics
4 Second order semi-linear PDEs
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This problem sheet is on ODEs and Picard's Theorem.
The relevant sections of the lecture notes are 1.1-1.5.
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The first half of the problem sheet is on Picard's theorem for systems and higher order ODEs covered in sections 1.6 and 1.7 of the lecture notes.
The second half of the problem sheet is on plane autonomous systems, in particular the classification of critical points, covered in sections 2.1-2.3 of the lecture note.
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Questions 1 and 2 are further examples of plane autonomous systems covered in chapter 2 of the lecture notes.
Questions 3-6 are on the method of characteristics for first order PDEs and correspond to chapter 3 from the lecture notes. -
Sheet 4 is on second order PDEs.
Questions 1-4 correspond to material covered in sections 4.1-4.3 while questions 5 and 6 are on the maximum principle which is covered in section 4.4 of the lecture notes.
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