###### Learning Outcomes:

Upon completion of this module the student should be able to:

Explain the terms existence, uniqueness and stability for PDEs and how this relates to initial and boundary conditions.

Explain the difference between, and be able to give examples of, Dirichlet and Neumann boundary conditions.

Determine whether a PDE is linear, nonlinear, homogeneous, inhomogeneous for given examples.

Solve ﬁrst order PDEs using the method of characteristics.

Classify a second order partial differential equation by type (Hyperbolic, Elliptic, Parabolic) and classify it by order or linearity.

Complete (given a partial proof) proofs for uniqueness of solutions for the wave, diffusion and Laplace equations.

Demonstrate the use of energy methods and the maximum principle to determine uniqueness and stability for a given PDE.

Construct ﬁnite difference formulations for diffusion equation and Laplace equation and interpret the output of computational simulations in terms of stability and accuracy.

Construct Fourier series expansions for a given periodic function.

Construct half period expansions, both sine and cosine.

Apply the method of separation of variables to solve the diffusion equation, wave equation and Laplace equation with both Dirichlet and Neumann conditions.

Extend the method of separation of variables to inhomogeneous PDEs and apply in applications.

Complete in clearly argued mathematics (given a partial proof) the proof of convergence of Fourier series This includes the Riemann Lebesque Lemma.

Determine the Fourier transform of basic functions.

Determine the Fourier transform of a derivative and the convolution of two functions.

Apply the Fourier transform to the diffusion equation and interpret the solutions generated.

Model the pricing of a European option by providing a commentary on the Black-Scholes equation. Reduce the Black-Scholes equation to the diffusion equation and solve for a call and put option.

Apply the Fourier transform to Laplace and wave equation and for the latter derive the D’Alembert form of the solution to the wave equation.

Implement the Laplace transform to solve the diffusion equation and wave equation for a semi-inﬁnite domain with standard initial conditions.

Construct a proof of the Fourier Integral Theorem.

Express the solution of boundary value problems for the Laplacian in two and three dimensions using Green’s functions.

Prove that the Laplacian of the free-space Green’s functions in two and three dimensions equals the Dirac delta function, and apply the free-space Green’s functions and the method of images to solve boundary value problems.