F11MP - Partial Differential Equations

Course leader(s):

Daniel Coutand (Edinburgh, January)

Aims

The module aims to provide a critical understanding of the basic theory of PDE’s, the main properties of the classical equations in mathematical physics, the different concepts required to study nonlinear equations and the uses of PDE's in mathematical modelling.

Syllabus

1. Linear hyperbolic equations in the plane. (1.1 Chain rule and change of variables in the plane. , 1.2 , 1.3 Classification of linear second order PDE in the plane. , 1.4 , 1.5 Cauchy data along a curve. , 1.6 , 1.7 The general Cauchy problem. , 1.8 , 1.9 Method of characteristics in the hyperbolic case to find the general solution. , 1.10 , 1.11 Finding a particular solution by specifying the Cauchy data along a curve.)

2. The linear heat equation. (2.1 Reminders on integrals depending on a parameter. , 2.2 , 2.3 Heat kernel and solution to initial value problem of heat equation in the full space. , 2.4 , 2.5 Separation of variables for the heat equation on a segment. , 2.6 , 2.7 The parabolic boundary of a space-time cylinder. , 2.8 , 2.9 Maximum and minimum principles for the heat equation.)

3. Elliptic equations in dimension n. (3.1 Harmonic, subharmonic and superharmonic functions. , 3.2 , 3.3 Maximum and minimum principles for the Laplacian. , 3.4 , 3.5 Dirichlet and Neumann problems associated with the Laplacian. , 3.6 , 3.7 Reminder from vector calculus: Integration by parts in dimension n. , 3.8 , 3.9 Uniqueness to nonlinear elliptic problems. , 3.10 , 3.11 Dirichlet problem and minimisation of a functional.)

4. Nonlinear conservation laws. (4.1 Characteristic curves and solution to the problem. , 4.2 , 4.3 Finite time of existence when characteristic curves intersect at a positive time.)

5. Uniqueness and finite time singularity formation for nonlinear parabolic and hyperbolic PDE. (5.1 Standard inequalities for integrals, Gronwall’s inequality. , 5.2 , 5.3 Finite time blow-up for parabolic semilinear PDE. , 5.4 , 5.5 Uniqueness for parabolic semilinear PDE. , 5.6 , 5.7 Uniqueness for hyperbolic semilinear PDE.)

Learning outcomes

By the end of the course, students should be able to do the following:

Solve linear hyperbolic PDEs in the plane by the method of characteristics, define the notion of well-posedness for the wave equation.
Solve the linear heat equation in the full space by the method of the Gaussian kernel and draw conclusions based on the kernel formula.
Solve the linear heat equation on a segment by the method of separation of variables and draw conclusions about convergence to equilibrium.
Compare the solution of the heat equation to appropriate standard functions by the maximum principle in bounded spatial domains.
Compare the solution to Laplace’s equation in a bounded domain to appropriate standard functions, demonstrate uniqueness of the solution (including in semi linear situations).
For nonlinear conservation laws, define the characteristic lines and draw conclusions based on the formula for characteristics.
Derive appropriate energies and differential inequalities for the solutions to semi-linear heat and wave equations in a bounded spatial domain and draw conclusions from it.
Professional Practice: Demonstrate the ability to learn independently, manage time, work to deadlines and prioritise workloads.

Further details

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SCQF Level: 11

Credits: 15