Mathematical Modelling - Block 4 (Mathematical Methods and Models: Block 4)

This block contains units 13 - 16 Unit 13 Modelling with Non-linear Differential Equations; In this unit we study the mathematical models associated with two physical systems: the growth of two interacting populations, one a predator and the other its prey and the motion of a rigid pendulum. This is demonstrated through the use of Lotka-Volterra equations, which apply to a pair of interacting populations. How these equations can be linearized near an equilibrium state and the graphical representation of the solutions are discussed. Unit 14 Modelling Motion in Two and Three Dimensions; This unit turns the attention to motion and forces in more than one dimension. It draws on ideas about vectors, forces and component forces and fundamental ideas about mechanics of particles, in particular Newton's second law. There is also mention of kinetic energy and potential energy. Unit 15 Modelling Heat Transfer; This unit makes use of ideas relating to energy and first-order differential equations. It begins by developing models that could be used to answer questions such as the following. How much does it cost to heat up a tank full of hot water? What thickness of insulation should be applied to a hot-water tank? What thickness of insulation should I use in my loft, and what savings would I make over a year? What should the gap be in double-glazing? Is it better to insulate the roof, insulate the walls or double-glaze the windows of my house? The common factor in answering all these questions is the need to consider the transfer of heat energy between different regions of space. This unit introduces the basic ideas of heat energy and temperature. It discusses conduction and convection as well as a third mode of heat energy transfer, radiation. Unit 16 Interpretation of Mathematical Models; This unit introduces the ideas of mathematical modelling. It discusses the five key stages of the mathematical modelling process in detail and looks at dimensions and units of physical quantities to see how they can be used to predict and check the outcomes of the modelling process.