The course aims to provide basic knowledge in the theory of partial differential equations, to study main properties of the classical equations of Mathematical Physics, to discuss the notion of weak solutions for simplest nonlinear PDEs and some applications to mathematical modelling.
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.)
By the end of the course, students should be able to do the following:
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SCQF Level: 10
Credits: 15