Show/hide contentOpenClose All
Curricular information is subject to change
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 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 BlackScholes equation. Reduce the BlackScholes 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 semiinﬁnite 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 freespace Green’s functions in two and three dimensions equals the Dirac delta function, and apply the freespace Green’s functions and the method of images to solve boundary value problems.
Student Effort Type  Hours 

Lectures  30 
Specified Learning Activities  36 
Autonomous Student Learning  36 
Total  102 
It is recommended that students should be familiar with material in vector integral and differential calculus.
Description  Timing  Component Scale  % of Final Grade  

Assignment(Including Essay): Exercise sheet problems to be submitted weekly during tutorials  n/a  Standard conversion grade scale 40%  No  20 

Exam (Inperson): Two InClass Midterm Exams  n/a  Standard conversion grade scale 40%  No  40 

Exam (Inperson): 2 Hour Final Exam  n/a  Standard conversion grade scale 40%  No  40 
Resit In  Terminal Exam 

Spring  Yes  2 Hour 
• Group/class feedback, postassessment
Not yet recorded.
Name  Role 

Dr José Miguel Zapata García  Lecturer / CoLecturer 
Constantinos Menelaou  Tutor 