Mathematical Models and Methods for Real World Systems

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Intuitive introduction of advanced mathematical techniques in the context of model building, for an understanding of real-life application.

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An emphasis on an overall philosophy of building models, drawing repeatedly from the key elements of data, physical processes and mathematical methods to develop and refine models. Senior undergraduate and postgraduate students of the natural sciences taking courses in, for example: mathematical modelling, ecological modelling, mathematics for biologists, mathematics for environmental sciences, computational biology, modelling natural systems, mathematical methods.

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Applied Mathematical Modelling

Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Close Find out more. Modeling methods We will now discuss some of the commonly used techniques for developing the equations and data sets for a mathematical model. A spacecraft orbital model based on Newton's laws of motion is an example of a physics-based model. Many systems have behavior that is too complex to represent easily in terms of the laws of physics. The aerodynamics of a supersonic aircraft, for example, tend to be complex and nonlinear.

In this case, the only reasonable approach may be to measure the system's behavior with a sub-scale model in a wind tunnel and then create a set of lookup tables to serve as the model. Let's look at an example of a physics-based model. Figure 1 shows a pendulum suspended from a string of length l under the influence of gravitational acceleration g. The pendulum angular deflection with respect to the vertical is q, given in radians. The mass of the pendulum bob is defined as m.

The goal for this model will be to determine the period of oscillation of the pendulum as a function of the initial deflection angle q0 assuming that the initial velocity, q0, is zero.

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Given the goal of determining the oscillation period, the effects that must be modeled will now be considered. We will look at the relevant physical effects and determine which to include in the model and which to ignore:. We know that a real pendulum will eventually slow down and stop due to friction in the string and air resistance. This isn't the kind of behavior we are interested in, so we will modify our goal to be the determination of the oscillation period at the time its motion is started.

This assumes that the pendulum slows gradually and the oscillation period changes slowly. We have made several simplifying assumptions that will make the model development task easier. Next, we will apply the laws of physics to the system to develop the dynamic equations. The force of interest due to gravity will act on the pendulum bob in the direction perpendicular to the string at any moment in time. This force is defined as shown in Equation This leads to the final dynamic equation given in Equation Note that this equation does not depend on the mass of the bob m; however, it does depend on the assumptions we listed previously.

It is also a nonlinear differential equation because the term sinq appears in it. To completely determine a solution for this equation the initial conditions of the system must be specified as shown in Equation 8. The parameter q0 in Equation 8 is the angle from which the bob is released at time zero with an initial velocity of zero. To make Equation 7 suitable for simulation, it must be restated as a set of first-order differential equations and corresponding initial conditions as shown in Equation Using the techniques of numerical integration, these equations can be solved for various values of q0 and the oscillation period can be determined from examining the solutions.

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This example has demonstrated the basic approach for developing a physics-based mathematical model of a dynamic system. Similar techniques can be used in the development of models from other disciplines such as electronics and chemistry, as long as the dynamic equations describing the system are well defined and the data values appearing in the equations can be determined with sufficient precision.

In this example, the only data values required were the gravitational acceleration g, the string length l, and the initial displacement q0. Of course, other data would have to be examined to verify that the assumptions are reasonable, such as the assumption that the mass of the string is small relative to the mass of the bob. In situations where the dynamic equations or data values for a mathematical model are not known to the desired degree of precision, alternative methods for model development are necessary.

These techniques are discussed in the next section. Empirical modeling Empirical modeling techniques use measured data from various types of experiments to develop a mathematical model of a system. In reality, all mathematical models are empirical to some degree. For example, the pendulum model in the previous section includes some empirically determined constants.

However, our interest in this section is on the development of models for systems with substantial dynamic behavior that is not readily modeled by well-defined equations. Table interpolation is a static modeling technique used to evaluate functions of the form shown in Equation This is a static method in the sense that it does not permit the direct implementation of dynamic equations.

However, table interpolation functions can be used in the construction of dynamic equations. For example, coefficients appearing in the dynamic equations can be computed using table interpolation. This approach is normally used when the output of the function must be determined experimentally.

It is also applicable as a speed optimization technique if a lengthy computation perhaps an iterative procedure is required to evaluate the function.

In this case, a table interpolation to estimate the result of the computation may execute many times faster than a direct computation. The function inputs x1, x2, and so on can be any variable in the simulation, such as time, a state variable, or even a constant.

The number of function inputs is arbitrary, but in practical applications it is usually five or less.

Modern mathematical models, methods and algorithms for real world systems - DTU Orbit

As more inputs are added to the function, its memory requirements and execution time will tend to increase. The output y depends only on the values of these inputs when the function is evaluated. A one-dimensional example of the data set for an interpolation table with eight equally spaced breakpoints is shown in Figure 2. The span of the input variable x is [0, 0. If the input variable precisely matches the x location of one of the breakpoints it is a simple matter to return the corresponding y value as the result of the function evaluation.

If the input value falls between the breakpoints an interpolation must be performed. Many different techniques are available for interpolating functions between breakpoints. We will discuss the commonly used method of linear interpolation. One-dimensional linear interpolation is performed graphically by drawing straight lines between adjacent points as shown in Figure 3.

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The interpolated function changes continuously between breakpoints, but its derivative is discontinuous at the breakpoints. If this not acceptable, other interpolation techniques that produce smoother results such as cubic spline interpolation 1 can be used instead. Unequally spaced breakpoints If the x coordinates of the breakpoints are not equally spaced, it takes more work to determine which point interval contains the function input value.

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  6. A general approach for locating the correct interval is the technique of bisection. This is an algorithm for performing an efficient search of an ordered list. Linear Interpolation with Equally Spaced Breakpoints We will assume that the y coordinates of each point is stored in an array of length N with indices that begin at zero. The x coordinates of the points begin at x 0 with a separation of Dx between them. First, ensure that the input variable has a value greater than or equal to the first point in the table and less than the last point.

    A limit function can be used if that is appropriate. It is also possible to linearly extrapolate outside the table using the first or last two data points in the table to define a straight line. However, this approach may produce significant errors if the extrapolation does not accurately model the system's performance outside the table's range and will not be considered here. It may make more sense to issue an error message and abort the simulation run if the input variable is outside the valid input range of the table.

    Determine the array index of the closest point with an x coordinate that is less than or equal to the function input value, but not equal to the last point in the table. Eykhoff defined a mathematical model as 'a representation of the essential aspects of an existing system or a system to be constructed which presents knowledge of that system in usable form'.

    Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. There are six basic groups of variables: decision variables, input variables, state variables, exogenous variables, random variables, and output variables.

    Since there can be many variables of each type, the variables are generally represented by vectors. Mathematical modelling problems are often classified into black box or white box models, according to how much a priori information is available of the system.

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    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 only works as an intuitive guide for approach.