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# MATH2038 Partial Differential Equations

## Module Overview

Differential equations occupy a central role in mathematics because they allow us to describe a wide variety of real-world systems. The module will aim to stress the importance of both theory and applications of differential equations. The module begins by revisiting some of the material from the first year module on differential equations focussing attention on boundary value problems and also on equations with a source term. We then look at how one can express a general periodic function in terms of Fourier series of sine and cosine functions. The second section of the module introduces some of the basic concepts of partial differential equations (PDEs). It is shown how PDEs may be used to model situations in a wide variety of situations including biology, finance and applied mathematics. The three important classes of second order PDE appropriate for modelling different sorts of phenomena are introduced and the appropriate boundary conditions for each of these are considered. The technique of separation of variables will be used to reduce the problem to that of solving the sort of ordinary differential equations seen at the start of the module and writing the general solution using Fourier series. Throughout the module there will be a strong emphasis on problem solving and examples.

### Aims and Objectives

#### Module Aims

To stress the importance of both theory and applications of differential equations.

#### Learning Outcomes

##### Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• Solve simple second order differential equations
• Be able to calculate Fourier series
• Understand the wide applications of differential equation
##### Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• Prove the Orthogonality of eigenfunctions of boundary value problems
• Be able to classify second order partial differential equations and choose the appropriate boundary conditions
##### Subject Specific Practical Skills

Having successfully completed this module you will be able to:

• Apply the method of separation of variables to standard PDEs
• Use Laplace transforms to solve simple linear differential equations

### Syllabus

2ndOrder ODEâ€™s Revision of material from variation of parameters, Greenâ€™s Functions, boundary value problems, Eigenvalues and Eienfunctions, Orthogonality of eigenfunctions Fourier Series Periodic functions Odd and even functions, Half-range series, Convergence Introduction to PDEâ€™s 2nd order PDEâ€™s Type of PDE: Hyperbolic, Parabolic and Elliptic and corresponding boundary conditions Examples: Wave equations, Diffusion in physical and biological systems, Black-Scholes equation, applications in finance, Potential theory applications in biology and physics. Hyperbolic PDEâ€™s Wave equation Cauchy problem Dâ€™Alembertâ€™s solution Characteristics Separation of variables Parabolic PDEâ€™s Heat equation Separation of variables Applications to diffusion problems in Finance and Biology Elliptic PDEâ€™s Laplaceâ€™s equation Dirichlet problem 2-dim: Harmonic functions 3-dim: Fundamental solution Laplace transforms Definition and basic theory Examples of transforms Shift theorem Transforms of derivatives Applications to ODEâ€™s and systems Applications to PDEâ€™s

### Learning and Teaching

#### Teaching and learning methods

Lectures, surgeries, private study

TypeHours
Teaching54
Independent Study96
Total study time150

#### Resources & Reading list

BOYCE, W.E. & DI PRIMA, R.C.. Elementary Differential Equations and Boundary-Value Problems.

### Assessment

#### Summative

MethodPercentage contribution
Coursework and class tests 20%
Written exam 80%

#### Referral

MethodPercentage contribution
Exam %

#### Repeat Information

Repeat type: Internal & External

### Linked modules

Prerequisites (MATH1056 or MATH1059 and MATH1060) and MATH2039 or (MATH1006 and MATH1007)

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.

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