Mathematical Modelling Techniques
Many real situations are very complex and thus modeled approximate on a computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws.
Building Models from Data and Scientific Principles
In engineering , physics models are often made by mathematical methods such as finite element analysis. Different mathematical models use different geometries that are not necessarily accurate descriptions of the geometry of the universe. Euclidean geometry is much used in classical physics, while special relativity and general relativity are examples of theories that use geometries which are not Euclidean. Since prehistorical times simple models such as maps and diagrams have been used.
Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers can build a descriptive model of the system as a hypothesis of how the system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try out different control approaches in simulations.
A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. Variables may be of many types; real or integer numbers, boolean values or strings , for example. The actual model is the set of functions that describe the relations between the different variables.
In business and engineering , mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs.
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The system relating inputs to outputs depends on other variables too: decision variables , state variables , exogenous variables, and random variables. Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system represented by the state variables. Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables.
MSc Mathematical Modelling Module Information | Mathematics - UCL - London's Global University
The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance , as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved computationally as the number increases. For example, economists often apply linear algebra when using input-output models. Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables.
Mathematical Modelling Conference in Engineering & Human Behaviour
Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information on the system is available. A black-box model is a system of which there is no a priori information available. A white-box model also called glass box or clear box is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take.
Usually it is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly.
Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying function. But we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model.
In black-box models one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks which usually do not make assumptions about incoming data.
Alternatively the NARMAX Nonlinear AutoRegressive Moving Average model with eXogenous inputs algorithms which were developed as part of nonlinear system identification  can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise. The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to the underlying process, whereas neural networks produce an approximation that is opaque.
Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intuition , experience , or expert opinion , or based on convenience of mathematical form. Bayesian statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify a prior probability distribution which can be subjective , and then update this distribution based on empirical data.
Synopsis MTH is designed to equip you with standard mathematical modelling techniques that can be applied to model the real world scenarios.
Newtonians mechanics in more that one-dimension, heat transfer, circular motion of particles as well as rotational of rigid bodies introduced. The course also covered the key stages in the mathematical modelling process. Topics Oscillations.
Explaining the physical behaviour of the world around us through mathematics
Oscillations and Energy. Newtonian mechanics in more than one-dimension. Resisted projectiles. Stages in mathematical modelling process. The fluid flows studied range from environmental buoyancy-driven flows to technologically important micro- and nanofluidic flows. The modelling of materials involves the use of mathematical and computational techniques to solve a wide and varied class of problems; this includes nanoscale devices where the fate of individual atoms is important.
It spans length scales and time scales that vary over many orders of magnitude and involves the solution of equations that range from continuum to quantum mechanical descriptions.
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