## MAS253 Mathematics for Engineering Modelling

Note: Information for future academic years is provisional. Timetable information and teaching staff are especially likely to change, but other details may also be altered, some courses may not run at all, and other courses may be added.

 Semester 1, 2022/23 10 Credits Lecturer: Prof Robert von Fay-Siebenburgen Timetable Reading List Aims Outcomes Assessment Full Syllabus

Prerequisites: MAS152 (Essential Mathematical Skills and Techniques) 1st Year Mathematics course – suitable qualifications for overseas students.

The following modules have this module as a prerequisite:

 MAS254 Computational and Numerical Methods

## Outline syllabus

Taylor and Fourier series; Laplace transforms; partial differential equations; Laplace, wave and heat equations; multiple integrals.

## Aims

To further extend the student’s understanding, developed in Level 1, of a variety of mathematical techniques and the application of these techniques in modelling engineering problems.

## Learning outcomes

By the end of the semester the student should:
1. Be aware of the properties and uses of Taylor and Fourier series.
2. Be familiar with the technique of Laplace transforms and their use to solve ordinary differential equations, including impulsive inputs.
3. Understand the forms of solution of some partial differential equations.
4. Be able to perform multiple integration appropriate to centroids, moments of inertia, etc.
24 lectures, 12 tutorials

## Assessment

One two-hour written examination.

## Full syllabus

.

Week 1: SERIES, TAYLOR AND MACLAURIN SERIES: Finite and infinite series; nth partial sum; notion of convergence and divergence, power series; mention of radius of convergence; Taylor's theorem; Taylor and Maclaurin series; L'Hopital’s rule (deduced from Taylor's theorem).
Weeks 2-3: FOURIER SERIES: Definitions and properties of even and odd functions; Definitions and properties of even and odd functions; orthogonality relations for sine and cosine functions; Fourier expansions over the interval −l < x < l and 0 < x < l; derivation of formulae for coefficients (informal treatment assuming summation and integration interchangeable); special case when l=π. Dirichlet’s conditions and statement of Fourier’s theorem; Fourier sine and cosine series.
Weeks 4-5: LAPLACE TRANSFORMS. Introduction to Laplace transforms; linear systems of ordinary differential equations with application to normal modes; Heaviside step function; the t-shift theorem, second order linear ordinary differential equations with discontinuous forcing function. The Dirac delta function with applications to bending beam equation.
Week 6: PARTIAL DIFFERENTIATION: Small increments of functions of two variables; the chain rule; Taylor’s theorem for functions of two variables.
Weeks 7-9: PARTIAL DIFFERENTIAL EQUATIONS: their applications; initial and boundary value problems. Application of method of separation of variables to the problem of the finite string; normal modes. Use of the method of separation of variables to 1-D heat conduction problems and the 2-D Laplace equation. d'Alembert’s solution u=f(x−ct)+g(x+ct) of the 1-D wave equation; interpretation of c as a wave velocity, simple examples involving travelling waves.
Week 9-end: MULTIPLE INTEGRALS: Repeated integrals; double integrals, geometric interpretation; change of variables (element of area for polar coordinate systems shown geometrically); triple integrals (brief mention). Applications to mass, centroids, second moments of area (for beam theory), moments of inertia (tie in with strip method).