Difference equations ordinary differential equation

By substituting this expression into (2) and (3), we ll obtain a set of amplitude equations, ordinary differential equations for the amplitudes, b, of the different harmonics or modes. But first we must also write the inflow as a Fourier... 

Rigorous error bounds are discussed for linear ordinary differential equations solved with the finite difference method by Isaacson and Keller (Ref. 107). Computer software exists to solve two-point boundary value problems. The IMSL routine DVCPR uses the finite difference method with a variable step size (Ref. 247). Finlayson (Ref. 106) gives FDRXN for reaction problems. 

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. 

Once the initial state x(f = 0) of the system is specified, future states, x(t), are uniquely defined for all times t . Moreover, the uniqueness theorem of the solutions of ordinary differential equations guarantees that trajectories originating from different initial points never intersect. 

Hyclic elimination method. We now focus the reader s attention on periodic solutions to difference schemes or systems of difference schemes being used in approximating partial and ordinary differential equations in spherical or cylindrical coordinates. A system of equations such as... 

Bagmut, G. (1969) Difference schemes of higher-order accuracy for an ordinary differential equations with singularity. Zh. Vychisl. Mat. i Mat. Fiz., 9, 221-226 (in Russian) English transl. in USSR Comput. Mathem. and Mathem. Physics. 

The basic notions of the theory of difference schemes are the error of approximation, stability, convergence, and accuracy of difference scheme. A more detailed exposition of these eoncepts will appear in Chapter 2. They are illustrated by considering a number of difference schemes for ordinary differential equations. In the same chapter we also outline the approach to the general formulations without regard to the particular form of the difference operator. 

Chapters 2-5 are concerned with concrete difference schemes for equations of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on homogeneous difference schemes for ordinary differential equations, by means of which we try to solve the canonical problem of the theory of difference schemes in which a primary family of difference schemes is specified (in such a case the availability of the family is provided by pattern functionals) and schemes of a desired quality should be selected within the primary family. This problem is solved in Chapter 3 using a particular form of the scheme and its solution leads us to conservative homogeneous schemes. 

The first and second terms contain functions of two different arguments and therefore each of them is constant. This allows us to replace the Laplace s equation by two ordinary differential equations, and they are... 

The best packages for stiff equations (see below) use Gear s backward difference formulas. The formulas of various orders are [Gear, G. W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N.J. (1971)]... 

This equation must be solved for yn +l. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL [Ascher, U. M., and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia (1998) and Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland Elsevier (1989)]. 

Packages to solve boundary value problems are available on the Internet. On the NIST web page http //gams.nist.gov/, choose problem decision tree and then differential and integral equations and then ordinary differential equations and multipoint boundary value problems. On the Netlibweb site http //www.netlib.org/, search on boundary value problem. Any spreadsheet that has an iteration capability can be used with the finite difference method. Some packages for partial differential equations also have a capability for solving one-dimensional boundary value problems [e.g. Comsol Multiphysics (formerly FEMLAB)]. 

For both the finite difference and weighted residual methods a set of coupled ordinary differential equations results which are integrated forward in time using the method of lines. Various software packages implementing Gear s method are popular. 

Occasionally, various methods for evaluating tracer data and for estimating the mixing parameter in the TIS model lead to different estimates for t and N In these cases, the accuracy of t and N must be verified by comparing the concentration-versus-time profiles predicted from the model with the experimental data. In general, the predicted profile can be determined by numerically integrating N simultaneous ordinary differential equations of the form ... 

Digital simulation is a powerful tool for solving the equations describing chemical engineering systems. The principal difficulties are two (1) solution of simultaneous nonlinear algebraic equations (usually done by some iterative method), and (2) numerical integration of ordinary differential equations (using discrete finite-difference equations to approximate continuous differential equations). 

As discussed in the introduction to this chapter, the solution of ordinary differential equations (ODEs) on a digital computer involves numerical integration. We will present several of the simplest and most popular numerical-integration algorithms. In Sec, 4.4.1 we will discuss explicit methods and in Sec. 4.4.2 we will briefly describe implicit algorithms. The differences between the two types and their advantages and disadvantages will be discussed. 

Method of lines. This method, which was introduced in the crystallization research by Tsuruoka and Randolph (JJ, transforms the population balance into a set of ordinary differential equations by discretization of the size axis in a fixed number of grid points. The differential 3 G L,t)n(L,t) /3L is then approximated by a finite difference scheme. As has been shown (X) a fourth-order accurate scheme using an equidistant grid (i- l.z) results in ... 

This complex system would be difficult to solve directly. However, the problem is separable by taking advantage of the widely different time scales of conversion and deactivation. For example, typical catalyst contact times for the conversion processes are on the order of seconds, whereas the time on stream for deactivation is on the order of days. [Note Catalyst contact time is defined as the volume of catalyst divided by the total volumetric flow in the reactor at unit conditions, PV/FRT. Catalyst volume here includes the voids and is defined as WJpp — e)]. Therefore, in the scale of catalyst contact time, a is constant and Eq. (1) becomes an ordinary differential equation ... 

This is a linear ordinary-differential-equation boundary-value problem that can be solved analytically (see Bird, Stewart, and Lightfoot, Transport Phenomena, Wiley, 1960). Here, however, proceed directly to numerical finite-difference solution, which can be implemented easily in a spreadsheet. Assuming a cone angle of a = 2° and a rotation rate of 2 = 30 rpm, determine f(0) — v /r. 

After cross differentiation, the difference of the two momentum equations produces the ordinary differential equation... 

The behavior of a mixture is determined by a system of ordinary differential equations, while the required state, either equilibrium or stationary, is determined by a time-independent system of algebraic equations. Therefore, at first glance one would not expect any qualitative difference between the equilibrium and stationary states. Ya.B. shows that in the equilibrium case, even for an ideal system, a variational principle exists which guarantees uniqueness. Such a principle cannot be formulated for the case of an open system with influx of matter and/or energy. 

Equation (88) was used to demonstrate [44] the differences in distributions of molecular sizes in pre-gel stages of an RA3 polymerization with substitution effects as calculated according to a statistical and a kinetics model. The moments of distribution and the gel points were calculated by the numerical solution of a few ordinary differential equations that were derived from Eq. (88) and compared with analogous quantities calculated from a statistical model. 

In the case of dynamic (unsteady) problems, even after the space discretization, we still have to solve a set of ordinary differential equations in time. Therefore, the second step is to discretize the temporal continuum. This is usually done by a finite difference approximation with the same properties of a FDM in space. Depending on the instant in which the information is taken, the time-discretization leads to ... 

To analyze this phenomenon further, 2D numerical simulations of (49) and (50) were performed using a central finite difference approximation of the spatial derivatives and a fourth order Runge-Kutta integration of the resulting ordinary differential equations in time. Details of the simulation technique can be found in [114, 119]. The material parameters of the polymer blend PDMS/PEMS were used and the spatial scale = (K/ b )ll2 and time scale r = 2/D were established from the experimental measurements of the structure factor evolution under a homogeneous temperature quench. 

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