Initial value differential equations

Extending time scales of Molecular Dynamics simulations is therefore one of the prime challenges of computational biophysics and attracted considerable attention [2-5]. Most efforts focus on improving algorithms for solving the initial value differential equations, which are in many cases, the Newton s equations of motion. 

A problem with the solution of initial-value differential equations is that they always have to be solved iteratively from the defined initial conditions. Each time a parameter value is changed, the solution has to be recalculated from scratch. When simulations involve uptake by root systems with different root orders and hence many different root radii, the calculations become prohibitive. An alternative approach is to try to solve the equations analytically, allowing the calculation of uptake at any time directly. This has proved difficult becau.se of the nonlinearity in the boundary condition, where the uptake depends on the solute concentration at the root-soil interface. Another approach is to seek relevant model simplifications that allow approximate analytical solutions to be obtained. 

CONP Kee, R. J., Rupley, F. and Miller, J. A. Sandia National Laboratories, Livermore, CA. A Fortran program (conp.f) that solves the time-dependent kinetics of a homogeneous, constant pressure, adiabatic system. The program runs in conjunction with CHEMKIN and a stiff ordinary differential equation solver such as LSODE (lsode.f, Hindmarsh, A. C. LSODE and LSODI, Two Initial Value Differential Equation Solvers, ACM SIGNUM Newsletter, 15, 4, (1980)). The simplicity of the code is particularly valuable for those not familiar with CHEMKIN. 

Solve the initial-value differential equation to obtain an analytical expression for [A] and [B] as a function of time. (For simplicity in the solution, you may assume that [A]o ... 

The Unsteady-State Behavior of the Digester and the Solution of the Initial Value Differential Equations... 

Alternately, the problem can be viewed as an initial value differential equation with the boundary condition X(0) = X, on the right-hand side of equation (6.130), once the boundary value X(Ht) = Xf is specified. 

Since the cooling jacket has cocurrent flow, the model consists of the set of four coupled initial value differential equations (7.5) to (7.8). Note that the first three DEs (7.5) to (7.7) contain the variable catalyst effectiveness factor rj. Thus there are other equations to be solved at each point along the length 0 < / < Lt of the reactor tube, namely the equations for the catalyst pellet s effectiveness factor rj. 

Figure 7.4 gives a flow diagram for the overall solution procedure. It starts with solving the reactor initial value differential equation (7.19) for ys, followed by substituting yB into the equations (7.16) and (7.18) to obtain xab and xbb, then solving equation (7.24) to obtain y and then xa and xb from (7.21) and (7.22), and finally obtaining y from equation (7.23). 

In this section we develop a dynamic model from the same basis and assumptions as the steady-state model developed earlier. The model will include the necessarily unsteady-state dynamic terms, giving a set of initial value differential equations that describe the dynamic behavior of the system. Both the heat and coke capacitances are taken into consideration, while the vapor phase capacitances in both the dense and bubble phase are assumed negligible and therefore the corresponding mass-balance equations are assumed to be at pseudosteady state. This last assumption will be relaxed in the next subsection where the chemisorption capacities of gas oil and gasoline on the surface of the catalyst will be accounted for, albeit in a simple manner. In addition, the heat and mass capacities of the bubble phases are assumed to be negligible and thus the bubble phases of both the reactor and regenerator are assumed to be in a pseudosteady state. Based on these assumptions, the dynamics of the system are controlled by the thermal and coke dynamics in the dense phases of the reactor and of the regenerator. 

The segregated-flow model described by (P2) forms a basis to generate an AR. We now develop conditions for the closure of this space with respect to the operations of mixing and reaction by means of a PFR, a CSTR, or a recycle PFR (RR). Consider the region depicted by the constraints of (P2). Our aim is to develop conditions that can be checked easily for the reaction system in question so that, if these conditions are satisfied, we need to solve only (P2) for the reactor targeting problem. We will analyze these conditions based on PFR trajectories projected into two dimensions. Here, a PFR, which is an n-dimen-sional trajectory in concentration space and parametric in time, is generated by the solution of the initial value differential equation system in (PI). Figure 3 illustrates a PFR trajectory and its projections in three-dimensional space, where the solid line represents the actual PFR trajectory and the dotted lines represent the projected trajectories. 

It is obvious that we obtain a stability condition that is not much different from the stability condition of the initial value equation. If At is larger than 2 y/MfK, (the cosine is smaller than —1), the solution grows exponentially and is numerically unstable. Hence, in the straightforward boundary value formulation of classical mechanics, we gain very little in terms of stability and step size compared to the solution of the initial value differential equation. The difficulty is not in the philosophical view (global or local) but in the estimate of the time derivative, which is approximated by a local finite difference expression. 

To conclude the discussion on the length-dependent action, we note that an initial value differential equation as a function of length also exists [6] and is... 

These are systems where the state variables are varying in one or more directions of the space coordinates. The simplest chemical reaction engineering example is the plug flow reactor. These systems are described at steady state either by an ordinary differential equation (where the variation of the state variables is only in one direction of the space coordinates, i.e. one dimensional models, and the independent variable is this space direction), or partial differential equations (when the variation of the state variables is in more than one direction of the space coordinates, i.e. two dimensional models, and the independent variables are these space directions). The ordinary differential equations of the steady state of the one-dimensional distributed model can be either initial value differential equations (e.g. plug flow models) or two-point boundary value differential equations (e.g. models with superimposed axial dispersion). The equations describing the unsteady state of distributed models are invariably partial difierential equations. 

Various simplified models can be used with varying degrees of accuracy for the simulation of the transient behaviour of non-porous catalyst pellets. The most suitable unsteady state model for this problem is that with infinite thermal conductivity. This simplified model is quite accurate for metal and metal oxide catalysts. In this model, equation (5.45) disappears and the model becomes strictly lumped parameter described only by ordinary initial value differential equations. 

The non-linear two-point boundary value differential equation of the catalyst pellet is best solved using the orthogonal collocation method as described by Elnashaie et al. (1988a). The bulk phase initial value differential equations should be solved using standard integration routines with automatic step size to ensure accuracy (e.g. DGEAR-IMSL library). 

The reactor simulation equations representing the heterogeneous one dimensional model are non-linear and therefore must be solved numerically. The model equations (6.228-6.234) are initial value differential equations. The Runge-Kutta method is the simplest algorithm and the most popular for solving such equations. 

In Chapter 3, we covered the mathematical modeling of lumped systems as well as some preliminary examples of distributed systems using a systematic, generalized approach. The examples for distributed systems were preliminary and they were not sufficiently generalized. In this chapter, we introduce sufficient generalization for distributed systems and give more fundamentally and practically important examples, such as the axial dispersion model resulting in two-point boundary-value differential equations. These types of model equations are much more difficult to solve than models described by initial-value differential equations, specially for nonlinear cases, which are solved numerically and iteratively. Also, examples of diffusion (with and without chemical reaction) in porous structures of different shapes will be presented, explained, and solved for both linear and nonlinear cases in Chapter 6. 

If we want to start the integration using the marching technique (Euler, Runge-Kutta, or using any subroutine from Polymath, Matlab, or IMSL libraries for the solution of initial-value differential equations, see Appendix B), then we need to assume a i(0) or X2(0). Suppose we choose to assume xi(0) = x"(0), where n refers to the n h iteration (for the initial guess, it will be = 1). Then, >C2(0) can be easily and directly computed from the relation... 

How can we compute yi (o>) so that we can get x + (0) This can be achieved by formulating adjoint equations for yi(co) that are solved simultaneously with the xj and xj differential equations. However, the adjoint equations must be initial-value differential equations. The method of doing this... 

A very efficient method is to precalculate steady-state solutions and save the results in look-up tables. Interpolation in multidimensional look-up tables is most often much faster than iterative solutions of non-linear equations. It is also possible to use look-up tables for initial-value differential equations using time as one variable in the look-up table. 

Let us consider the initial-value differential equation in the linear form ... 

In addition to the examples in the previous sections, several new examples of initial value differential equations will be presented in this section. The emphasis will be on nonlinear first and second order differential equations. 

Many coupled time dependent differential equations exhibit types of solutions known as chaotie behavior . In principle, the solution of an initial value differential equation problem is eompletely determined by the differential equation and the set of initial eonditions. However, for some types of coupled nonlinear differential equations an extremely small change in the initial conditions produces a very distinguishably different time behavior. Such systems are said to exhibit chaotic behavior. One sueh system of equations is the Lorentz equations defined by ... 

The fbevalQ function reverses this process by extracting out the sets flie initial values of the functions and derivatives on lines 15 and 16 from the single array of input values as well as saving the left boundary values and derivatives in the uL[] and upL[] arrays. The initial value differential equation solver odeivQ is then called with the proper argument arrays on line 17 to solve the differential equation as an initial value problem. The reader is referred to the previous chapter for a discussion of this function which normally returns a set of solution values. However, in this case, the only important values as far as the nsolvQ routine is concerned are the solution values and derivatives at the last spatial point which are returned by the odeivQ function in the u[] and up[] arrays. These are the right hand boundary values computed by the differential equation solver. The two sets of solution values are then passed to the user defined boundary value function on line 18 in order to evaluate the error values between the actual values and the specified values of the boundary conditions. These errors are returned from the user supplied fboundQ function in the beqs[] array which is in turn passed back to the nsolvQ function so the initial values can be appropriately updated. 

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