Mathematical modeling Newton equation

In this chapter we are focusing on a particular technique, the Gauss-Newton method, for the estimation of the unknown parameters that appear in a model described by a set of algebraic equations. Namely, it is assumed that both the structure of the mathematical model and the objective function to be minimized are known. In mathematical terms, we are given the model... 

Mathematics has been defined as the logic of drawing unambiguous conclusions from arbitrary assumptions. The assumptions do not come from the mathematics but arise from the engineering or scientific discipline. Mathematical models of real world phenomena have been remarkably useful. The simplest of these express various physical laws in mathematical form. For example, Ohms law V = IR), Newton s second law (f = ma) and the ideal gas law (pv = nRT) provide starting points for many theories. More complicated models employ difference, differential, or integral equations. Whatever the model, the conclusions drawn from it are unambiguous ... 

If the mathematical model involves two (or more) simultaneous nonlinear equations in two (or more) unknowns, the Newton-Raphson method can be extended to solve these equations simultaneously. In what follows, we will first develop the Newton-Raphson method for two equations and then expand the algorithm to a system of k equations. 

Hydraulic calculation is made using iterative method. The results of this calculation are provided for heat transfer analysis by the HEATHYD code. The physical and mathematical model of the heat transfer of HEATHYD includes the equation for thermal conduction and Newton s law of cooling. 

Hooke s law, the direct proportionality between stress and strain in tension or shear, is often assumed such that the constitutive equations for a purely elastic solid are o = fjs for unidirectional extension and x = qy in simple shear flow. The latter expression is recognized from Chapter 7 as the constitutive relationship for a Newtonian fluid and, in analogy to Hooke s law for elastic solids, is sometimes termed Newton s law of viscosity. For cross-linked, amorphous polymers above 7, a nonlinear relationship can be derived theoretically. For such materials v = 0.5. When v is not 0.5, it is an indication that voids are forming in the sample or that crystallization is taking place. In either case, neither the theoretical equation nor Hooke s law generally applies. Before turning to one of the simplest mathematical models of viscoelasticity, it is important to recall that the constitutive equations of a purely viscous fluid are a = fj for elongational flow and x = qy for shear flow. 

Many models in the physical sciences take the form of mathematical relationships, equations connecting some property with other parameters of the system. Some of these relationships are quite simple, e.g., Newton s second law of motion, which says that force = mass x acceleration F = ma. Newton s gravitational law for the attractive force F between two masses m and m2 also takes a rather simple form... 

The first difficulty derives from the fact that given any values of the macroscopic expected values (restricted only by broad moment inequality conditions), a probability density always exists (mathematically) giving rise to these expected values. This means that as far as the mathematical framework of dynamics and probability goes, the macroscopic variables could have values violating the laws of phenomenological physics (e.g., the equation of state, Newton s law of heat conduction, Stokes law of viscosity, etc.). In other words, there is a macroscopic dependence of macroscopic variables which reflects nothing in the microscopic model. Clearly, there must exist a principle whereby nature restricts the class of probability density functions, SF, so as to ensure the observed phenomenological dependences. 

At all but the simplest level, treatment of the results from a time-domain experiment involves some mathematical procedure such as non-linear least squares analysis. Least squares analysis is generally carried out by some modification of the Newton-Raphson method, that proposed by Marquardt currently being popular [21, 22]. There is a fundamental difficulty in that the normal equations that must be solved as part of the procedure are often ill-conditioned. This means that rather than having a single well-defined solution, there is a group of solutions all of which are equally valid. This is particularly troublesome where there are exponential components whose time constants differ by less than a factor of about three. It is easy to demonstrate that the behaviour is multi-exponential, but much more difficult to extract reliable parameters. The fitting procedure is also dependent on the model used and it is often quite difficult to determine the number of exponentials needed to adequately represent the data. Various procedures have been suggested to overcome these difficulties, but none has yet received wide acceptance in solid-state NMR [23-26]. 

The balance equations for column reactors that operate in a concurrent mode as well as for semibatch reactors are mathematically described by ordinary differential equations. Basically, it is an initial value problem, which can be solved by, for example, Runge-Kutta, Adams-Moulton, or BD methods (Appendix 2). Countercurrent column reactor models result in boundary value problems, and they can be solved, for example, by orthogonal collocation [3]. The backmixed model consists of an algebraic equation system that is solved by the Newton-Raphson method (Appendix 1). 

These last two equations are Hamilton s equations of motion. There are two equations for each of the three spatial dimensions. For one particle in three dimensions, equations 9.14 and 9.15 give six first-order differential equations that need to be solved in order to understand the behavior of the particle. Both Newton s equations and Lagrange s equations require the solution of three second-order differential equations for each particle, so that the amount of calculus required to understand the system is the same. The only difference lies in what information one knows to model the system or what information one wants to get about the system. This determines which set of equations to use. Otherwise, they are all mathematically equivalent. 

Before we leave this topic, it is important to recognize what these equations of motion provided. If one could indeed specify the forces acting on a particle, or a group of particles, one could predict how those particles would behave. Or if one knows the exact form of the potential energy of the particles in the system, or if one wants to know what the total energy of the system is, one could still model the system. Nineteenth-century scientists were complacent in their feeling that if the proper mathematical expressions for the potential energy or forces were known, then the complete mechanical behavior of the system could be predicted. Newton s,... 

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