First-Order Partial Differential Equations

Analysis of receptor/ligand interactions is made much more complicated by explicit consideration of these physical features. Without such considerations, we are often able to model receptor phenomena with first-order ordinary differential equations based on straightforward mass-action kinetic species balances. When diffusion and probabilistic effects are taken into account, the models can easily give rise to partial differential equations, second-order ordinary differential equations, extremely large sets of first-order ordinary differential equations, and/or probabilistic differential equations. 

The assumptions shown In Table 1 are from George s(2) work, and are fairly typical. The first two reduce the equations from second-order partial differential to first-order ordinary dlfferen-... 

On subsciCuLlng (12.49) into uhe dynamical equations we may expand each term in powers of the perturbations and retain only terms of the zeroth and first orders. The terms of order zero can then be eliminated by subtracting the steady state equations, and what remains is a set of linear partial differential equations in the perturbations. Thus equations (12.46) and (12.47) yield the following pair of linearized perturbation equations... 

These coupled second-order partial differential equations do not have a closed-form solution. Accordingly, the approximate numerical technique of finite differences is employed. First, however, the boundary conditions must be prescribed in order to complete the formulation of the problem. Symmetry of the laminate about several planes permits reduction of the region of consideration to a quarter of the laminate cross section in the y-z plane at any value of x as shown in Figure 4-52. There, along the stress-free upper surface. 

Unlike stirred tanks, piston flow reactors are distributed systems with one-dimensional gradients in composition and physical properties. Steady-state performance is governed by ordinary differential equations, and dynamic performance is governed by partial differential equations, albeit simple, first-order PDEs. Figure 14.6 illustrates a component balance for a differential volume element. 

A 5-point finite difference scheme along with method of lines was used to transform the partial differential Equations 4-6 into a system of first-order differential and algebraic equations. The final form of the governing equations is given below with the terms defined in the notation section. 

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

Aris, R. and Amundson, N. R. (1973) First Order Partial Differential Equations with Applications, Prentice-Hall. 

These equations are integrated from some initial conditions. For a specified value of s, the value of x and y shows the location where the solution is u. The equation is semilinear if a and b depend just on x and y (and not u), and the equation is linear if a, b, and/all depend on x and y, but not u. Such equations give rise to shock propagation, and conditions have been derived to deduce the presence of shocks. Courant and Hilbert (1953, 1962) Rhee, H. K., R. Aris, and N. R. Amundson, First-Order Partial Differential Equations, vol. I, Theory and Applications of Single Equations, Prentice-Hall, Englewood Cliffs, N.J. (1986) and LeVeque (1992), ibid. 

Several years ago Baer proposed the use of a matrix A, that transforms the adiabatic electronic set to a diabatic one [72], (For a special twofold set this was discussed in [286,287].) Computations performed with the diabatic set are much simpler than those with the adiabatic set. Subject to certain conditions, A is the solution of a set of first order partial differential equations. A is unitary and has the form of a path-ordered phase factor, in which the phase can be formally written as... 

Rhee, Aris, and Amundson, First-Order Partial Differential Equations Volume 1. Theory and Application of Single Equations Volume 2. Theory and Application of Hyperbolic Systems of Quasi-Linear Equations, Prentice Hall, Englewood Cliffs, New Jersey, 1986,1989. 

These are the Hamiltonian or canonical form of the equations of motion. The advantage of the Hamiltonian equations over the Lagrangian is that they contain 6n partial differential equations of the first order rather than 3n of the second order. 

Aris, R. Amundson, N. "First order partial differential equations with applications", Prentice Hall, Englewood Cliffs 1973... 

Analytical solution is possible only for first or zero order. Otherwise a numerical solution by finite differences, method of lines or finite elements is required. The analytical solution proceeds by the method of separation of variables which converts the PDE into one ODE with variables separable and the other a Bessel equation. The final solution is an infinite series whose development is quite elaborate and should be sought in books on Fourier series or partial differential equations. 

The partial differential equations representing material and energy balances of a reaction in a packed bed are rarely solvable by analytical means, except perhaps when the reaction is of zero or first order. Two examples of derivation of the equations and their analytical solutions are P8.0.1.01 and P8.01.02. In more complex cases analytical, approximations can be made (by "Collocation" or "Perturbation", for instance), but these usually are quite sophisticated to apply. Numerical solutions, on the other hand, are simple in concept and are readily implemented on a computer. Two such methods that are suited to nonlinear kinetics problems will be described. 

For the one-dimensional equation with x as the space variable, the diffusion equation is a partial differential equation of the first order in time and the second order in x. It therefore requires concentration to be known everywhere at a given time (in general =0) and, at any time t>0, concentration, flux, or a combination of both, to be known in two points (boundary conditions). In the most general case, the diffusion equation is a partial differential equation of the first order in time and the second order in the three space coordinates x, y, z. Concentration or flux conditions valid at any time >0 must then be given along the entire boundary. 

The problem in obtaining a state space model for the dynamics of the CSD from this physical model is that the population balance is a (nonlinear) first-order partial differential equation. Consequently, to obtain a state space model the population balance must be transformed into a set of ordinary differential equations. After this transformation, the state space model is easily obtained by substitution of the algebraic relations and linearization of the ordinary differential equations. 

The model for this crystallizer configuration has been shown to consist of the well known population balance (4), coupled with an ordinary differential equation, the concentration balance, and a set of algebraic equations for the vapour flow rate, the growth and nucleatlon kinetics (4). The population balance is a first-order hyperbolic partial differential equation ... 

Lax-Wendroff. This is a well known method to solve first-order hyperbolic partial differential equations in boundary value problems. The two step Richtmeyer implementation of the explicit Lax-Wendroff differential scheme is used (8). 

First-order error analysis is a method for propagating uncertainty in the random parameters of a model into the model predictions using a fixed-form equation. This method is not a simulation like Monte Carlo but uses statistical theory to develop an equation that can easily be solved on a calculator. The method works well for linear models, but the accuracy of the method decreases as the model becomes more nonlinear. As a general rule, linear models that can be written down on a piece of paper work well with Ist-order error analysis. Complicated models that consist of a large number of pieced equations (like large exposure models) cannot be evaluated using Ist-order analysis. To use the technique, each partial differential equation of each random parameter with respect to the model must be solvable. 

The behavior of a reactive wave depen ds on the flow of its reacting and product-gases. The conservation laws lead to systems of partial differential equations of the first order which are quasilinear, ie, equations in which partial derivatives appear linearly. In practical cases special symmetry of boundary and initial conditions is often invoked to reduce the number of independent variables. 

Provided we consider the Euclidean case, = js xp xp, (p, v, X, p = 1,2,3,4), where is the completely antisymmetric tensor, and equations (2) form the system of four real first-order partial differential equations. 

Summarizing we conclude that the problem of constructing conformally invariant ansatzes reduces to finding the fundamental solution of the system of linear partial differential equations (33) and particular solutions of first-order systems of nonlinear partial differential equations (39). 

Subalgebras listed in Assertion 1 give rise to P(l, 3) (Poincare)-invariant ansatzes. Analysis of the structure of these subalgebras shows that we can put 0 = 1,04 = 05 = 0 in formula (30) for the matrix H. Moreover, the form of the basis elements of these subalgebras imply that in formulas (32) and (38) /" f- 0, for all the values of a = 1,2,3. Therefore system (39) for the matrix H takes the form of 12 first-order partial differential equations for the functions 00, 01,02, 03... 

Summarizing, we conclude that the problem of construction of P(l,3)-invariant ansatzes reduces to finding solutions of linear systems of first-order partial differential equations that are integrated by rather standard methods of the general theory of partial differential equations. 

The first paper devoted to nonclassical symmetry of partial differential equations was published by Bluman and Cole [57]. However, the real importance of these symmetries was understood much later after the explanations given in several papers [31,32,58-61] where the method of conditional symmetries had been used in order to construct new exact solutions of a number of nonlinear partial differential equations. 

It is important to determine the partial-differential-equation order. One of the most important reasons to understand order relates to consistent boundary-condition assignment. All the equations are first order in time. The spatial behavior can be a bit trickier. The continuity equation is first order in the velocity and density. The momentum equations are second order on the velocity and first order in the pressure. The species continuity equations are essentially second order in the composition (mass fraction Yy), since (see Eq. 3.128)... 

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