Differential equations, partial derivation

If the dynamic process behavior has to be considered, Eqs. (3)—(5), (8)-(9) become partial differential equations including derivatives of the hold-up with respect to time (see more details in Section 9.5.2.6). 

This equation delivers the value of dy corresponding to arbitrary infinitesimal changes in x and z, so it is still correct if we choose values of dz and dx such that dy vanishes. We now divide nonrigorously by dx, and interpret the quotients of differentials as partial derivatives, remembering that y is held fixed by our choice that dy vanishes. 

In what follows, we always assume y>0. The solution of (7.2.19) turns out chaotic for sufficiently large system size, and will be analyzed in Sect. 7.4. This equation may be called the phase turbulence equation. Recently, the same partial differential equation was derived by Sivashinsky in connection with the dynamics of combustion, and was used in discussing the turbulization of flame fronts (Sivashinsky, 1977, 1979 Michelson and Sivashinsky, 1977). 

The identification of the models involves the repeated solving of large systems of nonlinear differential or partial derivatives equations, for the necessity of sensitivity analysis and optimization processes. These goals could be betto reached by using more efficient algorithms (progresses in this way are expected) or supercomputers (some works are in progress in this Une). 

Deterministic models. Analogous to fluid mixing, the deterministic mathematical models of solids mixing are derived from the continuity (mass conservation) of a key component being mixed in a certain volume element in the mixer. If this volume element is finite in size, a lumped model is obtained, and if it is infinitesimally small, a continuous model. Under steady-state conditions prevailing in a continuous mixer, the lumped model manifests itself as a set of difference equations, and the eontinuous model, a set of ordinary differential equations naturally, a combined or hybrid model is expressed in the form of a set of differential-difference equations. The corresponding mathematical expressions under unsteady-state conditions prevailing in a batch mixer are sets of ordinary differential equations, partial differential equations, and ordinary differential-partial differential equations. 

Then, a partial differential equation is derived to express the behavior of a current and a voltage in a single conductor by applying Kirchhoff s law based on a lumped-parameter equivalence of the distributed-parameter line. The current and voltage solutions of the differential equation are derived by assuming (1) sinusoidal excitation and (2) a lossless conductor. From the solutions, the behaviors of the current and the voltage are discussed. 

Equation (A2.1.26) is equivalent to equation (A2.1.25) and serves to identify T, p, and p. as appropriate partial derivatives of tire energy U, a result that also follows directly from equation (A2.1.23) and the fact that dt/ is an exact differential. 

All of these quantities are state fiinctions, i.e. the differentials are exact, so each of the coefficients is a partial derivative. For example, from equation (A2.1.35) p = —while from equation (A2.1.36)... 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

Partial Derivative The abbreviation z =f x, y) means that is a function of the two variables x and y. The derivative of z with respect to X, treating y as a constant, is called the partial derivative with respecd to x and is usually denoted as dz/dx or of x, y)/dx or simply/. Partial differentiation, hke full differentiation, is quite simple to apply. Conversely, the solution of partial differential equations is appreciably more difficult than that of differential equations. 

When the function involved in the equation depends upon only one variable, its derivatives are ordinary derivatives and the differential equation is called an ordinaiy differential equation. When the function depends upon several independent variables, then the equation is called a partial differential equation. The theories of ordinaiy and partial differential equations are quite different. In almost eveiy respect the latter is more difficult. 

Discretization of the governing equations. In this step, the exact partial differential equations to be solved are replaced by approximate algebraic equations written in terms of the nodal values of the dependent variables. Among the numerous discretization methods, finite difference, finite volume, and finite element methods are the most common. Tlxe finite difference method estimates spatial derivatives in terms of the nodal values and spacing between nodes. The governing equations are then written in terms of... 

This review of the foregoing simple derivation will help you to understand the following derivation of the plate equilibrium equations. The major difference between plate and beam problems is that beams are one-dimensional and plates are two-dimensional. Therefore, beams have ordinary differential equations as governing equations whereas plates have partial differential equations. Moreover, in the derivation of the governing differential equations, there will necessarily be more force equilibrium and moment equilibrium equations for plates than for beams. 

An ordinary differential equation contains a single independent variable and a single unknown function of that variable, with its derivatives. A partial differential... 

Setting the partial derivatives of E with respect to each of the coefficients of g(x) equal to zero, differentiating and summing over 1,. . . , n forms a set of m + 1 equations [9] so that... 

If equation (2.51) is the total differential for as a function of two variables, 1 and 2, we can expect that its partial derivatives (d E/d Zi) and (<9 /c> 2)5 can be expressed as functions of only those two variables. That is, — ( , 2). Thus, derivatives of (<9 /<9 ) and (d E/d Zi)- with respect to variables other than 1 and 2 should be zero. As we consider the implications of this statement, it is important to note that a change can be made independently in the r variable of one subsystem without affecting that of the other, but a change in 0 will affect both subsystems (since 0 is the same in both subsystems). Therefore, we must consider the implications for c and 0 separately in the analysis that follows. 

We can substitute partial derivatives for the ratio of differentials in equation (3.19). For example,... 

The quantities dX and d Y are called differentials, the coefficients in front of dX and dT are called partial derivatives,11 and dZ is referred to as a total differential because it gives the total change in Z arising from changes in both X and Y. If Z were to depend upon additional variables, additional terms would be included in equation (A 1.1) to represent the changes in Z arising from changes in those variables. For much of our discussion, two variables describe the processes of interest, and therefore, we will limit our discussion to two independent variables, with the exception of the description of Pfaffian differentials in... 

In our thermodynamic derivations, we will routinely make use of equations of the type represented by equation (A1.3) to replace AX7jdYz with (dX/dY)z. We may also represent this ratio of differentials as (dA/d Y)z in which the direct equality with the partial derivative (dX/0Y)Z is more immediately evident. That is... 

Derive the partial differential equation for uosieady-slate unidirectional diffusion accompanied by an /Uli-order chemical reaction irate constant k) ... 

Equation (8.12) is a partial differential equation that includes a first derivative in the axial direction and first and second derivatives in the radial direction. Three boundary conditions are needed one axial and two radial. The axial boundary condition is... 

This section describes a number of finite difference approximations useful for solving second-order partial differential equations that is, equations containing terms such as d f jd d. The basic idea is to approximate f 2 z. polynomial in x and then to differentiate the polynomial to obtain estimates for derivatives such as df jdx and d f jdx -. The polynomial approximation is a local one that applies to some region of space centered about point x. When the point changes, the polynomial approximation will change as well. We begin by fitting a quadratic to the three points shown below. 

Undoubtedly, the reader comes across difference operators Ba of the structure Ba = E — arAa, where Aq. approximates the differential operator La with partial derivatives of one argument x. For example, if La U = d u/dx, then A y = 2/ is a three-point operator, who.se use permits us to solve equation (3) by the elimination method. It is worth mentioning here that any difference scheme can be reduced to a sequence of simpler schemes in a number of different ways. This is certainly so with scheme (1), implying that... 

Finally, 3 " (j)[f (x )] is a short symbol expressing the m-th order partial derivative operators, acting first over the function f (x) and then, the resultant function, evaluated at the point x . The differential operators can be defined in the same manner as the terms present in equation (9), but using as second argument the nabla vector ... 

Estimation of parameters present in partial differential equations is a very complex issue. Quite often by proper discretization of the spatial derivatives we transform the governing PDEs into a large number of ODEs. Hence, the problem can be transformed into one described by ODEs and be tackled with similar techniques. However, the fact that in such cases we have a system of high dimensionality requires particular attention. Parameter estimation for systems described by PDEs is examined in Chapter 11. 

We will first consider the simple case of diffusion of a non-electrolyte. The course of the diffusion (i.e. the dependence of the concentration of the diffusing substance on time and spatial coordinates) cannot be derived directly from Eq. (2.3.18) or Eq. (2.3.19) it is necessary to obtain a differential equation where the dependent variable is the concentration c while the time and the spatial coordinates are independent variables. The derivation is thus based on Eq. (2.2.10) or Eq. (2.2.5), where we set xj> = c and substitute from Eq. (2.3.18) or Eq. (2.3.19) for the fluxes. This yields Fick s second law (in fact, this is only a consequence of Fick s first law respecting the material balance—Eq. 2.2.10), which has the form of a partial differential equation... 

Differential equations are usually classified as ordinary or partial . In the former case only one independent variable is involved and its differential is exact. Thus there is a relation between the dependent variable, say y(x), its various derivatives, as well as functions of the independent variable x. Partial differential equations contain several independent variables, and hence partial derivatives. 

---