# 数学代写|偏微分方程代写partial difference equations代考|MATH4310

## 数学代写|偏微分方程代写partial difference equations代考|Mathematical Modelling

The human body is an important large physical system consisting of many different types of subsystems down to the cellular levels. Each subsystem involves complex natural processes governed by certain universal laws. ${ }^1$ It is a perfect example of an efficient system that has inbuilt optimisation schema for each of its component subsystems mostly controlled by the brain, which uses (chemical) controls all the time to perform a multilayer nonlinear analysis at a very high speed to ensure enhanced functioning of each vital organ for better survival chances against all odds. Some of the differential equations derived in this chapter also find applications in studies related to certain important problems in medical sciences. For example, according to recent studies, developmental biology uses differential equation models to study molecular mechanisms responsible for cell signalling and aggregation when nutrients are scarce. The motivated reader may refer the text [1] to know more.

In general, application of mathematical tools in analysing a practical situation requires a proper articulation of scientific facts so that the solution of the associated problem admits a convenient physical interpretations. Further, an approximation technique used in a particular situation depends on how appropriately one may choose the applicable postulates or laws such as Newton’s for classical mechanics, Schrödinger’s for quantum mechanics, Navier-Stokes’ for fluid flow, and Maxwell’s for electrodynamics.

In most cases, a mathematical description of processes involved with a physical system leads to problem involving an unknown function of certain number of independent variables, and also some known functions representing the system parameters. It is common to make some simplifying assumptions and reduce the number of parameters involved, subject to that the state of the system under study do not change drastically. As a rule of thumb, we first formulate an easy-to-deal-with model and upgrade to a more realistic model as we go along. The equation (or a system of equations) so obtained is used to analyse the associated practical problems. In most applications, the ultimate aim is to facilitate the analysis and design of controls for the related dynamical systems.

## 数学代写|偏微分方程代写partial difference equations代考|Model Formulation

At the first stage of model formulation of a phenomenon, we make some simplifying assumptions considering all related scientific facts about the involved natural processes. Each such assumption has to have some relevance to the purpose of the study, and their precision and correctness help in choosing an appropriate set of postulates and laws governing the involved processes. For example, mass is constant for models based on Newtonian physics, but it must be assumed a variable while using Einstein’s special theory of relativity. Next, we identify important system parameters concerning all relevant processes occurring in the system. Accordingly, dependent and independent variables are identified. For example, while modelling for weather forecast, the position and time are the independent variables, and dependent variables specify temperature and humidity. Notice that, in this case, earth’s gravity field and rotational speed act as nonadjustable parameters.

In actual practice, modelling of a dynamic system takes as input large volume of real data for the involved variables and parameters. Also, a good assessment of system environment and natural constraints is needed to undertake a meaningful study of the phenomena. So, at the final stage, we collect data for chosen variables from diverse sources, if possible, and subsequently use postulates and/or laws governing the involved processes to derive model equation(s). In particular, to formulate a statistical model of best fit, suitable interpolation techniques are applied to the source data collected earlier for variables and system parameters. In this latter situation, type of data used classifies a model as deterministic or probabilistic. If the independent variables are random, and probabilistic data is known, then the associated distributions related equation(s) gives a stochastic model.

All through the process, in general, a model needs to be calibrated several times using the collected real data. Such an exercise is important because it ensures that the difference between the model outputs and the real values largely remains below the acceptable error tolerance, and hence validates the model.

A fruitful way of formulating a model involving only dimensionless quantities is known as nondimensionalising a model, which provides an insight into how to scale relations of the system so that the total number of variables and/or parameters are minimal. Since variables and parameters in general have physical dimensions, the techniques of nondimensionalising and scaling in terms of their transformations are useful tools to simplify and analyse a mathematical model (mainly due to Buckingham $\pi$-theorem). As the present book is not about modelling techniques per se, we will not discuss dimension analysis and scaling aspects any further. The interested reader may like to read more from the book [5] or [6].

# 偏微分方程代考

## 数学代写|偏微分方程代写partial difference equations代考|Model Formulation

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