Solving Scalar Equations

A scalar equation is an equation of the form f(x) = 0 for a function / that depends on one variable x. In mathematical notation, we express functions / that map one real variable x to the real value f(x) symbolically by writing / R — R, where R denotes the set of all real numbers. 

What mathematicians call functions is often referred to as state variables by engineers. These are parallel languages. If t is the independent time variable and T(t) represents the temperature at timet, then in engineering language , T(t) is a dependent state variable in the independent variable or parameter t, while mathematically speaking, T(t) is the temperature function dependent on time t. Use of the function terminology is more recent and allows for treating multi-variable, multi-output functions such as 

Polynomial equations such as x3 — 2.x2 +4 = 0, for example, have been studied for many centuries. Over the last hundred years, there have been over 4,000 research publications and many books written on how to solve the general polynomial equation 

Let us look at the process of finding polynomial roots as an input to output process  

INPUT polynomial p of degree n — root finder OUTPUT 

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Another method to solve scalar equations in one real variable x uses inclusion and bisection. Assume that for a given one variable continuous function / R —> R we know of two points X < xup G R with f xi) f(xup) < 0, i.e., / has opposite signs at X and xup. Then by the intermediate value theorem for continuous functions, there must be at least one value x included in the open interval (x ,xup) with f(x ) = 0. The art of inclusion/bisection root finders is to make judicious choices for the location of the root x G (x , xup) from the previously evaluated / values and thereby to bisect the interval of inclusion [x , xup] to find closer values v < u e [x , xup] with v — u < x — xup and f(v) f u) < 0, thereby closing in on the actual root. Inclusion and bisection methods are very efficient if there is a clear intersection of the graph of / with the x axis, but for slanted, near-multiple root situations, both Newton s method and the inclusion/bisection... 

If we could solve this equation to find w t and hence z t we could insert the result into the equation of motion and take the scalar product with a horizontal unit vector, i say, to obtain... 

Chapter 3 tries to give students the essential tools to solve lumped systems that are governed by scalar equations. It starts with the simplest continuous-start reactor, a CSTR in the adiabatic case. The first section should be studied carefully since it represents the basis of what follows. Our students should write their own codes by studying and eventually rewriting the codes that are given in the book. These personal codes should be run and tested before the codes on the CD are actually used to solve the unsolved problems in the book. Section 3.2 treats the nonadiabatic case. 

Nonlinear models described by nonlinear equations (these can be transcendental systems of scalar equations, ODEs, or PDEs). These models can generally only be solved numerically. 

For steady-state solutions, the set of the four differential equations (7.189) to (7.192) (or equivalently the DEs (7.190) to (7.193)) reduces to a set of four coupled rational equations in the unknown variables E (or e), Cx, Cs, and Cp. To solve the corresponding steady-state equations, we interpret the equations (7.189) to (7.192) as a system of four coupled scalar homogeneous equations for the right-hand sides of the DE system in the form F(E, Cx, Cs, Cp) =0. The resulting coupled system of four scalar equations is best solved via Newton s22 method after finding the Jacobian23 DF by partial differentiation of the right-hand-side functions / of the equations (7.189) to (7.192). I.e.,... 

Finally, let us note that the tensor part HLJ is itself gauge invariant. As we said, one has to choose a gauge (i.e., a coordinate system) in order to solve the equations. It is well-known that the vector modes are generally irrelevant for cosmology, so that most of the coordinate systems which are used only deal with scalar perturbations. Let us review some of the most used coordinate systems. 

Solve the differential equation (D.ll), with the aid of the normal ordering procedure according to which it is possible to pass from operators equations that are functions of the noncommutative Bosons to scalar equations. This is possible with the help of the N 1 operator, which us allow to write the following transformations [54] ... 

Equation (5.80), Equation (5.81), and Equation (5.82) may be solved by analytical methods similar to those described in the previous sections and by the Laplace transform method, which will be dealt with in Section 5.8. In this section, the eigenvalue method is discussed. Equation (5.84) is a first-order equation, the solution of which is similar to that of the corresponding scalar equation, Equation (5.12) ... 

Scalar equations (if any) are then solved using the corrected velocity field (for example, k and s equations when solving the k-s model of turbulence or the enthalpy equation when solving non-isothermal flows). 

The fractional step concept is frequently used to solve scalar transport equations on the generic form [10, 85, 137] ... 

It is perhaps surprising that it is possible to solve the SSOZ equation for a number of simple molecules. For diatomic symmetry molecules with hard sphere pair interactions, the SSOZ equation with PY closure has been solved analytically using a Weiner-Hopf technique introduced by Baxter. We consider a diatomic molecule consisting of two fused hard spheres of diameter a, with their centres a distance L apart. For a fluid composed of these molecules, each of the four correlation functions is the same by symmetry, and the SSOZ equation reduces to a scalar equation... 

Note the definition of the parameter p in Eq. 12.1786. Since our main concern is to solve elliptic equation, that is, the steady-state solution to Eq. 12.176, the solution can be obtained effectively by choosing a sequence of parameters p (Peaceman and Rachford 1955). The optimum sequence of this parameter is only found for simple problems. For other problems, it is suggested that this optimum sequence is obtained by numerical experiments. However, if the scalar parameter p is kept constant, the method has been proved to converge for all values of p. Interested readers should refer to Wachspress and Habetler (1960) for further exposition of this method. 

Finally, a system of N scalar equations has to be solved numerically, where N is the number of the unknown scalars T or P, ipg etc. In many cases,... 

Equations 6.26 provide a set of linear equations for each of the atoms, a = 1,..., N, which are to be solved to obtain the atomic charges and atomic dipoles. However, one has to define the parameters involved. Here, the quantity J.o( is the same for all the atoms and is equal to the chemical potential of the cluster which is unknown and hence is to be determined during the solution process. One has N atomic charges, 3N atomic dipole components and the quantity a as the unknowns and there are N scalar equations, N vector equations and one equation = 0, for the charge conservation for the neutral cluster. Among the other parameters, (ta is an atomic chemical potential parameter, and q(a, a) = r ° is the atomic self-hardness term. Among the other quantities, r (a,P) is the mutual atom-atom hardness which can be approximated... 

In one of the earliest attempts, Schuetz and Piesche [73] calculated the flow field in a stirred tank first to determine the local energy dissipation rate and then solved the PBEs using the finite volume method [74] to predict the local aggregate size distribution. Heath and Koh [75] have solved the population balances as scalar equations in the commercial CFD software CFX for simulating flocculation of suspensions by polymers. They employed 35 discrete sectional equations to represent the aggregate size distribution. 

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