## MATH4A1 Techniques in Applied Mathematics 1

First Semester |

This paper is an introduction to techniques for solving problems in applied mathematics. The focus of MATH4A1 will be on solving scalar equations. The topics will include techniques for ordinary differential equations, partial differential equations, difference equations, integral equations. We will discuss problems posed on continuum and discrete domains. We will also describe how mathematical models for real-world phenomena are derived.

### Prerequisites

Successful completion of at least MATH 202, MATH 203, COMO 204, or permission of the lecturer. There will be a variety of topics and we will not strongly rely on any one earlier paper.

### Topics

I. Modelling using differential equations; discrete versus continuum domains; ordinary versus partial differential equations; how to construct a mathematical model; solution of differential equations arising from various applications

II. Exact solution methods for ordinary differential equations; linear and nonlinear separable equations; first order linear equations, second order constant coefficient equations; initial value problems

III. Second order variable coefficient ordinary differential equations; Cauchy-Euler equations; fundamental solutions and linear independence; the Wronskian; inhomogeneous second order equations; reduction of order; variation of parameters

IV. Approximate and iterative solution methods for ordinary differential equations; Taylor series solutions; Taylor polynomial approximations; methods for solving discrete / difference equations; small parameter perturbation theory

V. Boundary value problems in one variable; Sturm–Liouville theory; eigenvalues and eigenfunctions; construction of an orthonormal basis; singular Sturm-Liouville problems

VI. Partial differential equations on unbounded domains; method of characteristics, d'Alembert's formula for the wave equation on the real line; the Fourier transform; heat kernels and the diffusion equation on the real line; hyperbolic versus parabolic versus elliptic PDEs

VII. Boundary value problems in multiple variables; general separation of space and time variables on bounded space domains; spectral methods and their convergence, Laplace's equation and Poisson's equation on bounded space domains; the diffusion equation on bounded space domains; the wave equation on bounded space domains; the Schrödinger equation on bounded space domains

VIII. Diffusion processes on semi-infinite domains; self-similar solutions on the half-line; self-similar solutions for piecewise initial conditions on the real line

IX. Integral equations; Fredholm integral equations; Volterra integral equations; Neumann series; iterated kernels and the Resolvent method.

### Lecturer

Dr R. A. Van Gorder

You will be responsible for all material covered in lecture, but no other references are needed. I may provide additional optional references for specific topics, as appropriate.

### Assessment

Assessment will be through assignments only. These will be take-home problem sets.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = A**

where:

- A is the Assignments mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic integrity** means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

**Academic misconduct** is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

**Plagiarism**

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

- Although not intended,
*unintentional plagiarism*is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided. *Intentional plagiarism*is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

**Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

**Impersonation**

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

**Falsiﬁcation**

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

**Use of Unauthorised Materials**

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

**Assisting Others to Commit Academic Misconduct**

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.