# Distance Learning - Ordinary and Partial Differential Equations

Models, methods and applications

## DESCRIPTION

Originator and trainer: Dr. Daniel J. Duffy.

The goal of this hands-on and self-contained course is to introduce ordinary and partial differential equations (ODEs and PDEs). These equations form the basis for modelling many kinds of phenomena in areas such as science, engineering, computational finance and more generally, mathematical physics. The approach is to examine representative model ODEs and PDEs and then to extend them to more complex and larger problems.

This course introduces the most important differential equations that are needed in applications as well as qualitative and modern quantitative methods to understand and find solutions to these equations.  The focus is on gaining insights into this area of mathematics and applying them to computational finance (for example, the Black Scholes PDEs in finance), science and engineering. We take a step-by-step approach by first focusing on understandable model problems and then moving to more challenging problems. In this way you develop skills in order to analyse a wide range of differential equations.

This course contains both standard undergraduate material as well as more advanced material (some of which has been developed by Daniel Duffy) that is needed and can be used in real life applications.

For MFE students, the essential modules are:

• Part A: Ordinary Differential Equations (ODEs).
• Part B: Two-Point Boundary Value Problems.
• Part F: Parabolic Time-dependent PDEs.
• Part G: PDEs in Computational Finance.

We estimate that these four modules can be completed (submitting answers to exercises) in 2 to 3 months. We also advise you to study the other modules as well because they contain relevant supporting material.

We introduce the topics in the course using a building-block approach: we first take a scoped version of a differential equation and we examine it from a number of viewpoints until we understand its properties and behaviour. Then we decide how to extend and generalise the differential equation to new situations including:

• From linear to nonlinear equations.
• From one dimension to higher dimensions.
• From diffusion equations to convection-diffusion-reaction equations.
• Applying your ODE/PDE skills to real-life applications.

The approach is similar to the following statement by Paul Halmos:

..the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.

Topics Covered

• ODE analytics,  numerics and applications.
• Boundary value problems.
• Elliptic, hyperbolic and parabolic PDE.
• Supporting techniques: Fourier series and transforms; special equations.
• From heat equation to convection-diffusion-reaction: attention points.
• Detailed analysis of PDEs in computational finance.

Benefits and Special Features of the Course

• Comprehensive, relevant  and accurate.
• Unique course in terms of content, applications and delivery.
• Monitor your progress by scaled examples, exercises and feedback from Daniel Duffy.
• Award of certificate for successful completion of the course.
• Seamless integration with PDEs in finance (Black Scholes and its variants).
• Optionally, the student can decide to do a small project based on the material at no extra cost.  It is a good learning experience and it might impress potential employers and universities.

For whom is the Course and what are the Prerequisites?

This course is suitable for professionals in industry and finance as well as for advanced undergraduate and MSc/MFE students. The prerequisite knowledge is calculus to the level  of differential calculus of one variable, for example. It is also an advantage if you are exposed to applications in which differential equations play a role or if you are writing a thesis in which they are needed.

The level of the course is similar to that in second and third year university courses in applied mathematics and engineering, for example. The main difference is that this focused course is dedicated to differential equations and their applications.

If you have any queries please do not hesitate to contact me, Daniel Duffy dduffy@datasim.nl to discuss the course.

Structure of Course and Student-Trainer Interaction

This course takes an incremental/inductive approach by first examining model problems in detail and then extending them to more advanced applications. Student progress is measured by working on graded exercises.  Regular communication takes places by e-mail and Skype.

• The e-book version of NUMERICAL METHODS IN COMPUTATIONAL FINANCE, A Partial Differential Equation (PDE/FDM) Approach (2021) by Daniel J. Duffy.
• Source code that implements PDE and ODE models (where applicable). The follow-on courses from Datasim are concerned with the numerical approximation of PDEs by the Finite Difference Method (FDM).
• Students send solutions to exercises using LATEX, ideally (LATEX is easy to learn).
• A signed certificate on successful completion of the course.
• An initial Skype meeting to set up the course plan and a final Skype meeting prior to certification.

Students receive a certificate on successful completion of the course.

The optimal way to learn in our opinion is by executing the following steps. This discussion pertains to studying and learning the contents of a single module:
1. Listen to the audio show and use the printed PowerPoint slides as backup.
2. Read the relevant material in the provided book(s).
3. Do the exercises; compile and run the programs (if applicable).
4. If you are having problems, go back to one of more of steps 1, 2, 3.
5. If step 4 has been unsuccessful then post your problem on the Datasim forum.
6. Go to next module.

Some Testimonials

“FDM is one of the most useful and advanced courses that I have ever taken.  The tools that I learnt were directly applied to value complex derivatives contracts, in an efficient manner using C++, such as convertible bonds and Bermudan callable bonds.  On the other hand, Daniel is an extraordinary tutor that helped me out in every step of the final applied project.”

- Jaime Maihuire Irigoyen - Sustainable Investing Research Associate, BlackRock, Inc.

"Professor Duffy’s course provides a comprehensive look at the Finite Difference Method in financial applications. He was readily accessible and provided a great balance of clear direction with room for independent research."

- Thomas M. - TDM Trading.

"Actually I've searched a few PDE online courses, most of them are not closely connected to computational finance, while the Datasim course is designed for it and very well structured."

- Prospective MFE student

The student price is Euro 1500  (no VAT if you are outside EU). Please contact dduffy@datasim.nl The courses remain accessible forever and you can plan your own schedule but in general it should be possible to complete a course (including exercises and mini-project) in 2-4 months.  See also our Frequently Asked Questions (FAQ) on the Datasim site.

The price for non-students is Euro 2395.00 excluding VAT.

If you are a college student please ignore the price below and contact dduffy@datasim.nl for your student price.

## Course price

Price (excl. VAT) € 2395.00 Start online course directly