# KC5008 - Ordinary & Partial Differential Equations

## What will I learn on this module?

The module is designed to introduce you to a first mathematical treatment of ordinary and partial differential equations. You will learn fundamental techniques for solving first- and second-order equations as well as approximation methods. These are used to explore the question of the existence of solutions and provide a qualitative behaviour of the solutions. Examples are drawn from applications to physics, engineering, biology, economics and finance and modelling of complex systems.
The module will cover topics such as:

Ordinary Differential Equations (ODEs)

1. First-order ODEs: Classification of ODEs, separable, Bernoulli, Riccati and exact equations as well as integrating factors. Picard iterations and existence of solutions.
2. Second-order ODEs: Solutions of linear equations, independence of solutions, linear stability, initial and boundary value problems, series solutions about ordinary and singular points, special functions

Partial Differential Equations (PDEs)

1. Introduction and classification of PDEs.
2. Method of characteristics for first order linear PDEs.
3. The method of separation of variables and Fourier series.
4. Solutions of Laplace, diffusion/heat and wave equations.
5. Applications

### How will I learn on this module?

You will learn through a combination of lectures and skills periods focussing on problem solving where you will be able to obtain help. Lectures allow students to experience and understand the formalism of the required mathematical techniques as well as include relevant examples and guided in-class exercise-solving sessions between more theoretical expositions. Students have an opportunity to enhance their understanding of the subject through seminars which promote both independent learning and problem solving within peer groups. The seminars will also be an opportunity to present you with open research problems, and will strengthen your transferable skills and employability. Northumbria’s computer labs and facilities are fully equipped with the latest industry-standard software such as Matlab and the computer algebra system Mathematica which will be used to support independent study and learning. Further technology-enhanced resources such as e-lecture notes, seminar sheets with answers and solution and past-paper questions will be provided via the e-learning portal.

### How will I be supported academically on this module?

In addition to academic contact with the module team during lectures and seminars, students are encouraged to develop their curiosity by making direct contact with the module team either via email or the open door policy operated throughout the programme. Students will also be regularly referred to supporting resources including relevant texts and relevant multimedia materials. References to these resources will be made available through the e-learning portal and in lectures and seminars.
You will be assessed by a coursework and a formal examination (30% and 70%, respectively). The examinations will cover all topics from the module. Formative feedback will be provided on seminar work which will include problems designed to aid your understanding.

### What will I be expected to read on this module?

All modules at Northumbria include a range of reading materials that students are expected to engage with. The reading list for this module can be found at: http://readinglists.northumbria.ac.uk
(Reading List service online guide for academic staff this containing contact details for the Reading List team – http://library.northumbria.ac.uk/readinglists)

### What will I be expected to achieve?

Knowledge & Understanding:
1. Apply fundamental techniques for solving ODEs about ordinary and singular points.
2. Apply the method of separation of variables and Fourier series to the Laplace, heat/diffusion and wave equations.

Intellectual / Professional skills & abilities:
3. Apply ODEs, PDEs and computer algebra to model and provide a comprehensive solution to applied problems.

Personal Values Attributes (Global / Cultural awareness, Ethics, Curiosity) (PVA):
4. Demonstrate critical enquiry and the ability to learn independently (see PVA1 from Programme Learning Outcomes).
5. Manifest the ability to contribute to discussions and to communicate new knowledge and research findings in mathematics and statistics (see PVA3 from Programme Learning Outcomes).

### How will I be assessed?

SUMMATIVE
1. Coursework (30%) – 1, 3, 4, 5
2. Examination (70%) – 1, 2, 3, 4, 5

FORMATIVE
Seminars – 1, 2, 3, 4, 5

Students will be assessed by a coursework and a formal examination (30% and 70%, respectively).

Formative feedback will be provided on seminar work which will include problems designed to aid student understanding.

None

None

### Module abstract

The study of ordinary and partial differential equations is a central theme in mathematics since it represents a cornerstone which takes advantage of your abilities from more pure topics to provide you with indispensable tools to tackle equations arising from research problems in engineering, physics, life sciences and many branches of mathematics. The topic requires a dual approach, both during the lectures and the tutorials, which combines theoretical and computational aspects and prepares you for more concrete subjects as well as more advanced ones. The module dynamically employs state-of-the-art computer algebra software to complement your analytical skills as well as obtain a visual glimpse of the solutions and their geometry. You will be assessed by a coursework and a formal examination (30% and 70%, respectively). The examinations will cover all topics from the module. Formative feedback will be provided on seminar work which will include problems designed to aid your understanding.

### Course info

UCAS Code G100

Credits 20

Level of Study Undergraduate

Mode of Study 3 years full-time or 4 years with a placement (sandwich)/study abroad

Department Mathematics, Physics and Electrical Engineering

Location City Campus, Northumbria University

City Newcastle

Start September 2024 or September 2025

## Mathematics BSc (Hons)

All information is accurate at the time of sharing.

Full time Courses starting in 2023 are primarily delivered via on-campus face to face learning but may include elements of online learning. We continue to monitor government and local authority guidance in relation to Covid-19 and we are ready and able to flex accordingly to ensure the health and safety of our students and staff.

Contact time is subject to increase or decrease in line with additional restrictions imposed by the government or the University in the interest of maintaining the health and safety and wellbeing of students, staff, and visitors, potentially to a full online offer, should further restrictions be deemed necessary in future. Our online activity will be delivered through Blackboard Ultra, enabling collaboration, connection and engagement with materials and people.

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