B5.2 Applied Partial Differential Equations (2025-26)
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- Lecturer: Profile: Peter Howell
Course information
General prerequisites:
Differential Equations 1 and Differential Equations 2 from Part A are prerequisites, and the material in these courses will be assumed to be known. Calculus of Variations and Fluids and Waves from Part A are desirable but not essential. Integral Transforms from Part A is strongly desirable.
Course term: Michaelmas
Course lecture information: 16 lectures
Course weight: 1
Course level: H
Assessment type: Written Examination
Course overview:
This course continues the Part A Differential Equations courses. In particular, first-order conservation laws are solved and the idea of a shock is introduced; general nonlinear and quasi-linear first-order partial differential equations are solved, the classification of second-order partial differential equations is extended to systems, with hyperbolic systems being solved by characteristic variables. Then Riemann's function, Green's function and similarity variable methods are demonstrated.
Learning outcomes:
Students will know a range of techniques to characterise and solve PDEs including non-linear first-order systems, and second-order. They will be able to demonstrate various principles for solving PDEs including the method of characteristics, Green's functions, similarity solutions and Riemann functions.
Course synopsis:
First-order equations; applications. Characteristics, domain of definition. [2 lectures]
Weak solutions, conservation laws, shocks. [2 lectures]
Non-linear equations; Charpit's equations; eikonal equation. [3 lectures]
Systems of partial differential equations, characteristics. Shocks; weak solutions. [3 lectures]
2nd order semilinear equations. Hyperbolic equations, Riemann functions. [2 lectures]
Elliptic equations, parabolic equations. Well-posed problems, Green's function, similarity solutions. [4 lectures]
Weak solutions, conservation laws, shocks. [2 lectures]
Non-linear equations; Charpit's equations; eikonal equation. [3 lectures]
Systems of partial differential equations, characteristics. Shocks; weak solutions. [3 lectures]
2nd order semilinear equations. Hyperbolic equations, Riemann functions. [2 lectures]
Elliptic equations, parabolic equations. Well-posed problems, Green's function, similarity solutions. [4 lectures]
Section outline
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Opened: Sunday, 19 October 2025, 3:50 PMSimilarity solutions; first-order quasilinear PDEs
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Opened: Sunday, 19 October 2025, 3:50 PMQuasilinear systems; linear 2nd order PDEs
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Sheet 4 solutions Assignment
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Work submission deadline is 72 hours before each class.
- Class 1: Tutor Peter Howell; TA Connor McAllister
- 9-10:30am Wednesday weeks 4, 6, 8 in C1; submission deadline 9am Sunday before each class.
- 9-10:30am Tuesday in HT week 1 in C6; submission deadline 9am Saturday before the class.
- Class 2: Tutor Peter Howell; TA Connor McAllister
- 10:30am-12noon Wednesday weeks 4, 6, 8 in C1; submission deadline 10:30am Sunday before each class.
- 10:30am-12noon Tuesday in HT week 1 in C6; submission deadline 10:30am Saturday before the class.
- Class 3: Tutor Andreas Münch; TA Tomas Gillanders
- 3-4:30pm Thursday weeks 3, 5, 7 in C3; submission deadline 3pm Monday before each class.
- 2-3:30pm Thursday in HT week 1 in C3; submission deadline 2pm Monday before the class.
- Class 4: Tutor Andreas Münch; TA Tomas Gillanders
- 4:30-6:00pm Thursday weeks 3, 5, 7 in C3; submission deadline 4:30pm Monday before each class.
- 4-5:30pm Thursday in HT week 1 in C3; submission deadline 4pm Monday before the class.
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Registration start: Monday, 6 October 2025, 12:00 PMRegistration end: Friday, 7 November 2025, 12:00 PM
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Class Tutor's Comments Assignment
Class tutors will use this activity to provide overall feedback to students at the end of the course.
- Class 1: Tutor Peter Howell; TA Connor McAllister