Differential equations, simultaneous

The steady-state problem yields a system of simultaneous linear algebraic equations that can be solved by Gaussian elimination and back substitution. I shall turn now to calculating the time evolution of this system, starting from a phosphate distribution that is not in steady state. In this calculation, assume that the phosphate concentration is initially the same in all reservoirs and equal to the value in river water, 10 I 3 mole P/m3. How do the concentrations evolve from this starting value to the steady-state values just calculated  

The phosphate content of each reservoir is the volume of that reservoir multiplied by the concentration of phosphate in that reservoir. Volumes are constant, so the rate of change of the content is the volume multiplied by the rate of change of the concentration. But for each reservoir, the rate of change of the content is the sum of all the fluxes of phosphorus in (the sources) minus the sum of all the fluxes of phosphorus out (the sinks). Thus, the content of the first reservoir, Atlantic surface, changes at the rate 

I divide both sides of the equation by the volume to obtain an equation 

The equations for the other reservoirs have exactly the same form  

The problem is to calculate new values of the variables, y, at a new time, x 4- delx, given the known values of y at time jc. In the reverse Euler method described in Section 2.4, 

It was mentioned at the beginning of Sec. 5.5 that the methods of solution of a single differential equation are readily adaptable for solving sets of simultaneous differential equations. To illustrate this, we use the set of n simultaneous ordinary differential equations  

This method is easily programmable using nested loops. In MATLAB, the values of k and y can be put in vectors and easily perform Eq. (5.99) in matrix form. 

Numerical Solution of Ordinary Differential Equations Chapter 5 

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The solution of the simultaneous differential equations implied by the mechanism can be expressed to give the time-varying concentrations of reactants, products, and intermediates in terms of increasing and decreasing exponential functions (8). Expressions for each component become comphcated very rapidly and thus approximations are built in at the level of the differential equations so that these may be treated at various limiting cases. In equations 2222 and 2323, the first reaction may reach equiUbrium for [i] much more rapidly than I is converted to P. This is described as a case of pre-equihbrium. At equihbrium, / y[A][S] = k [I]. Hence,... 

We now consider the solution of differential equations by means of Laplaee transforms. We have already solved one equation, namely, the first-order rate equation, but the technique is capable of more than this. It allows us to solve simultaneous differential equations. 

Complex reactions require the solution of simultaneous differential equations, and the Runge-Kutta procedure is applicable to these problems. To illustrate the method, Scheme XIV will be used. The rate equations are, in incremental form. 

The simultaneous differential equations are as follows - Parenchyma compartment... 

Thus, we have the / -electron wave function with separated spatial and spin parts only in the cases of two-electron singlet states and N-electron (N- - l)-plet states. The Hartree-Fock orbitals are defined as those functions t which make the wave functions (1.5), (1.6), and (1.7) best. The usual variation technique leads to the N(case A) or v(case B) simultaneous differential equations which have to be satisfied by... 

The evaluative fugacity model equations and levels have been presented earlier (1, 2, 3). The level I model gives distribution at equilibrium of a fixed amount of chemical. Level II gives the equilibrium distribution of a steady emission balanced by an equal reaction (and/or advection) rate and the average residence time or persistence. Level III gives the non-equilibrium steady state distribution in which emissions are into specified compartments and transfer rates between compartments may be restricted. Level IV is essentially the same as level III except that emissions vary with time and a set of simultaneous differential equations must be solved numerically (instead of algebraically). 

However, the complete solution of these three simultaneous differential equations is difficult to obtain and is no more instructive than the approximate solution that can be obtained by means of the steady-state approximation for intermediates. If one sets the time derivatives in equations 4.2.17 and 4.2.18 equal to zero and adds these equations, their sum is found to be... 

Vi A and B. This interaction is responsible for a bi-exponential evolution of their polarization which is accounted for by two simultaneous differential equations called Solomon equations... 

This problem is an extension of problems 7-10 and 7-11 on the dehydrogenation of ethane to produce ethylene. It can be treated as an open-ended, more realistic exercise in reaction mechanism investigation. The choice of reaction steps to include, and many aspects of elementary gas-phase reactions discussed in Chapter 6 (including energy transfer) are significant to this important industrial reaction. Solution of the problem requires access to a computer software package which can handle a moderately stiff set of simultaneous differential equations. E-Z Solve may be used for this purpose. 

Thus, for determination of time variation of q, and p the integration of 12 simultaneous differential equations are required. Further, reduction in number of dynamical equations by use of conservation of energy or total angular momentum is not worthwhile since the remaining equations become more complicated to solve. 

The mathematical problem posed is the solution of the simultaneous differential equations which arise from the mass-action treatment of the chemistry. For the homogeneous, well-mixed reactor, this becomes a set of ordinary, non-linear, first-order differential equations. For systems that are not... 

The processes of convection, axial diffusion, radial diffusion, and chemical reaction in the liquid and tissue layers all occur simultaneously. A rigorous approach requires solution of several simultaneous differential equations. To avoid this complexity in preliminary models, the transfer... 

Here, free protein E can react either with ligand S to form the complex ES, or react with free inhibitor I to form complex El. It follows that the overall rate of change in the concentrations of protein-ligand complexes [ S] and [ /] is described by the following simultaneous differential equations ... 

Note how much simpler this problem is to solve in the CSTR than in the PFTR (Figure 4—6), where we had to solve simultaneous differential equations. The CSTR involves only simultaneous algebraic equations so we just need to find roots of polynomials.]... 

These are two simultaneous differential equations with two initial conditions for a single reaction. For R simultaneous reactions we have to solve R + 1 simultaneous differential equations with R + 1 initial equations because there are R independent mass-balance equations and one temperature equatiorr... 

This expression must be solved numerically, but it is a single integral expression, which can be integrated directly rather than two simultaneous differential equations. 

The following simple BASIC program will integrate simultaneous differential equations to find Cj(t). Just type it into ary PC or Mac with BASIC, and it will draw a grid and find concentrations in A -> B C. Honest Use Print Screen to make a hard copy of your graphs. 

Let us emphasize that we have made no approximations yet. Equation (3.13) is a set of simultaneous differential equations for the coefficients cm that determine the state function (3.13) is fully equivalent to the time-dependent Schrodinger equation. [The column vector c(/) whose elements are the coefficients ck in (3.8) is the state vector in the representation that uses the tyj s as basis functions. Thus (3.13) is a matrix formulation of the time-dependent Schrodinger equation and can be written as the matrix equation ihdc/dt = Gc, where dc/dt has elements dcmf dt and G is the square matrix with elements exp(.iu>mkt)H mk. ... 

In many cases molecular displacements depend on other transition processes occurring previously in the neighborhood. Let us consider Fig. 27. The double kink may move from 1 to l Then segment 2 has the possibility to go from 2 to 2. Flowing units 4,5 and 6 may undergo similar processes. At last the molecule may reach the dashed structure. In every case the final position will be reached after some successive place exchange processes each determining the next step. We therefore get a system of simultaneous differential equations. 

In general, A 1 and this means that when A and B again are supplied in stoichiometric quantities—even if Ha is small—still ac will be rather large compared with b, at least in the direct vicinity of the phase boundary. Therefore, a solution as tried in Section II,B,1, where it was assumed that the concentration of A is constant throughout the cross section of the drops, cannot be obtained now since an equivalent assumption for the component B seems not to be justified. This means that for the real solution it is necessary to solve two simultaneous differential equations one for component A and one for component B. These equations can only be solved numerically by means of a computer. 

Numerical solution Solution of the simultaneous differential equations developed from Eqs. (2.5) and (2.6) without simplification. 

Eqs. (2.14), (2.30), and (2.31) with Eq. (2.9) can be solved simultaneously without simplification. Since the analytical solution of the preceding simultaneous differential equations are not possible, we need to solve them numerically by using a computer. Among many software packages that solve simultaneous differential equations, Advanced Continuous Simulation Language (ACSL, 1975) is very powerful and easy to use. 

Thus the mole fraction profile after the pressure increase is calculated by integrating the N simultaneous differential equations represented by Eq. (16) from t = 0 to t = Tp. 

PBPK models rely on a series of simultaneous differential equations that simulate chemical delivery to tissues via the arterial circulation and removal via the venous circulation. The models are run in time steps such that the entire course of chemical disposition can be presented for calculation of the area-under-the-curve (AUC) dose, often a key metric for chronic risk assessment. The physiologic parameters can be adapted for different species, sexes, age groups, and genetic variants to facilitate extrapolation from one type of receptor to another. 

The second group, i.e. at a slightly lower level, is the one where all factors that are part of an observed phenomenon are known, but we know or are only partly aware of their interrelationships, i.e. influences. This is usually the case when we are faced with a complex phenomenon consisting of numerous factors. Sometimes we can link these factors as a system of simultaneous differential equations but with no solutions to them. As an example we can cite the Navier-Stokes simultaneous system of differential equations, used to define the flow of an ideal fluid ... 

The exact solutions of Equations 10 and 11 can be obtained by Laplace transforms or the solution of simultaneous differential equations The resulting coefficients of f0 and fg are very complex functions of kt, k, kg, N and N. However, as seen in Figure 6, the reactivity o the p position is small relative to the a position. Thus, in the limiting condition where f is close to the initial value and neglecting exchange between thea and p positions,... 

In the last section we considered explicit expressions which predict the concentrations in elements at (t + At) from information at time t. An error is introduced due to asymmetry in relation to the simulation time. For this reason implicit methods, which predict what will be the next value and use this in the calculation, were developed. The version most used is the Crank-Nicholson method. Orthogonal collocation, which involves the resolution of a set of simultaneous differential equations, has also been employed. Accuracy is better, but computation time is greater, and the necessity of specifying the conditions can be difficult for a complex electrode mechanism. In this case the finite difference method is preferable7. 

Batch Reactors. One of the classic works in this area is by Gee and Melville (21), based on the PSSA for chain reaction with termination. Realistic mechanisms of termination, disproportionation, and combination, are treated with a variety of initiation kinetics, and analytical solutions are obtained. Liu and Amundson (37) solved the simultaneous differential equations for batch and transient stirred tank reactors by using digital computer without the PSSA. The degree of polymerization was limited to 100 the kinetic constants used were not typical and led to radical lifetimes of hours and to the conclusion that the PSSA is not accurate in the early stages of polymerization. In 1962 Liu and Amundson used the generating function approach and obtained a complex iterated integral which was later termed inconvenient for computation (37). The example treated was monomer termination. 

Beard(Z.) has developed a useful mathematical model of a convection dryer for studying the use of energy in a tenter frame. The model is based on a set of simultaneous differential equations which can be solved numerically to obtain fabric temperature and macroscopic moisture contents along the length of the dryer. The model considers the fabric as a moist layer of fabric sandwiched between two dry layers of fabric. Thermal energy is convected from the dryer to the external surface of the dry layer and then from the exterior of the fabric to the interface between the wet and dry layers. At the interface, the water is evaporated and diffuses as vapor through the dry layer to the surrounding hot make-up air. Assumptions in the model include ... 

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