Integration Initial value problems

This integral equation is a Volterra equation of the second land. Thus the initial-value problem is eqmvalent to a Volterra integral equation of the second kind. 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

Ordinaiy differential Eqs. (13-149) to (13-151) for rates of change of hquid-phase mole fractious are uouhuear because the coefficients of Xi j change with time. Therefore, numerical methods of integration with respect to time must be enmloyed. Furthermore, the equations may be difficult to integrate rapidly and accurately because they may constitute a so-called stiff system as considered by Gear Numerical Initial Value Problems in Ordinaiy Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1971). The choice of time... 

C (0). The analytieal solution to Equation 9-34 is rather eomplex for reaetion order n > 1, the (-r ) term is usually non-linear. Using numerieal methods, Equation 9-34 ean be treated as an initial value problem. Choose a value for = C (0) and integrate Equation 9-34. If C (A.) aehieves a steady state value, the eorreet value for C (0) was guessed. Onee Equation 9-34 has been solved subjeet to the appropriate boundary eonditions, the eonversion may be ealeulated from Caouc = Ca(0). 

In order to solve the differential equations, it is first necessary to initialise the integration routine. In the case of initial value problems, this is done by specifying the conditions of all the dependent variables, y, at initial time t = 0. If, however, only some of the initial values can be specified and other constant values apply at further values of the independent variable, the problem then becomes one of a split-boundary type. Split-boundary problems are inherently more difficult than the initial value problems, and although most of the examples in the book are of the initial value type, some split-boundary problems also occur. 

The Leibniz rule (see Integral Calculus ) can be used to show the equivalence of the initial-value problem consisting of the second-order differential equation d2y/cbd + A(x)(dy/dx) + B(x)y = fix) together with the prescribed initial conditions y(a) = y . y (a) = if, to the integral equation. 

An alternative (and much simpler) way of solving 20.4-6 is to covert it from a boundary- value problem to an initial-value problem, which may then be numerically integrated. 

But the major physical problem remained open Could one prove rigorously that the systems studied before 1979—that is, typically, systems of N interacting particles (with N very large)—are intrinsically stochastic systems In order to go around the major difficulty, Prigogine will take as a starting point another property of dynamical systems integrability. A dynamical system defined as the solution of a system of differential equations (such as the Hamilton equations of classical dynamics) is said to be integrable if the initial value problem of these equations admits a unique analytical solution, weekly sensitive to the initial condition. Such systems are mechanically stable. In order to... 

The initial value problem could be integrated using ODE software such as Vode [49]. However, for this simple problem, a fourth-order Runge-Kutta solution scheme [319,325] readily finds the solution and can be easily programmed or formed in a spreadsheet. 

Initial value problems, abbreviated by the acronym IVP, can be solved quite easily, since for these problems all initial conditions are specified at only one interval endpoint for the variable. More precisely, for IVPs the value of the dependent variable(s) are given for one specific value of the independent variable such as the initial condition at one location or at one time. Simple numerical integration techniques generally suffice to solve IVPs. This is so nowadays even for stiff differential equations, since good stiff DE solvers are widely available in software form and in MATLAB. 

For initial value problems such as the ones we have encountered in Chapter 4, there is only one possible direction of integration, namely from the initial value ujstart onwards. But for two-point BVPs we have function information at both ends ujstart < atend and we could integrate forwards from uj start to ujend, or backwards from uiend to uj start-Regardless of the direction of integration, each boundary value problem on [uJstart, uJend]... 

P.J. Van Der Houwen, Construction of Integration Formulas for Initial Value Problems, North-Holland, Amsterdam, 1977. 

Sincovec et al. [23] presented a very disquieting example of an initial value problem consisting of two ODEs. They showed that only one of the two state variables involved can be given an independent initial value. It was not hard to see why the problem occurs, but it was evident that such a problem could easily be hidden in a larger example. This work also proved that if an incorrect initial condition is specified and an implicit integration scheme is used, the solution will march directly to a solution that corresponds to one where a legitimate initial condition is used. The initial condition may not be one of interest, however. 

The initial value problem, Eqs. 1-3, can be integrated by any marching algorithm which is based on the Runge-Kutta or Adams-Moulton techniques. Based on the calculated space profiles of C,... 

Integration of this first-order, initial-value problem yields ... 

This first-order initial-value problem can easily be solved by the use of the integration factor method. That is. 

Integration of this initial-value problem with s... 

In ref 147 the authors apply Adomian decomposition method to develop an efficient algorithm of a special second-order ordinary initial value problems. The Adomian decomposition method is known that does not require discretization, so is computer time efficient. The authors are studied the Adomian decomposition method and the results obtained are compared with previously known results using the Quintic C -spline integration methods. 

In ref 148 the author obtain a new kind of trigonometrically fitted explicit Numerov-type method for the numerical integration of second-order initial value problems with... 

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