An initial-value problem

The solution may easily be obtained by the Laplace transform method. If the Laplace transform of the velocity is defined as 

By taking the inverse transform, we then find that the velocity is given by 

For x t (that is, for x ayoO sign of the real part of the argu- 

When x t, analogous reasoning shows that the contour of integration must be closed to the left, and the simple pole at s = 0 and branch points at s = — 1 and at s = — deolf foY will provide a nonzero contribution to V. Thus some disturbance always propagates with the velocity 

In order to find the shape of the wave front at large values of x and t, one may perform an asymptotic expansion of the integral in equation (119) for t and x approaching infinity with the ratio x /t fixed. By means of an interesting application of the method of steepest descents, the reader may show that 

Equation (120) shows that at large values of x and t, the main part of the signal travels with the equilibrium sound speed x/t = a Q, and the width of the signal broadens (as x and t increase) in proportion to l/y/t. Since f is the dimensionless time variable appearing in equation (119), the asymptotic shape given by equation (120) is attained earlier (at smaller values oft and x) for smaller values of the reaction time t. 

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The solution of an Initial value problem for these equations expresses... 

Adding time-dependent terms to the equations the simulation is treated as an initial value problem in which at a given reference time all stresses are zero the steady-state solution can be found iteratively. 

Modelling of steady-state free surface flow corresponds to the solution of a boundary value problem while moving boundary tracking is, in general, viewed as an initial value problem. Therefore, classification of existing methods on the basis of their suitability for boundary value or initial value problems has also been advocated. 

In mathematical language, the propagation problem is known as an initial-value problem (Fig. 3-2). Schematically, the problem is characterized by a differential equation plus an open region in which the equation holds. The solution of the differential equation must satisfy the initial conditions plus any side boundary conditions. 

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). 

Shooting methods attempt to convert a boundary value problem into an initial value problem. For example, given the preceding example restated as an initial value problem for which... 

U is unknown and must be chosen so that y(L) = 0. The equation may be solved as an initial value problem with predetermined step sizes so that will equal L at the end point. Since y(L) is a function of U, it will be denoted as y,(U) and an appropriate value of U sought so that... 

The extension to multiple reactions is done by writing Equation (3.1) (or the more complicated versions of Equation (3.1) that will soon be developed) for each of the N components. The component reaction rates are found from Equation (2.7) in exactly the same ways as in a batch reactor. The result is an initial value problem consisting of N simultaneous, first-order ODEs that can be solved using your favorite ODE solver. The same kind of problem was solved in Chapter 2, but the independent variable is now z rather than t. 

Oddson solved an Initial value problem that described the convective movement of an organic down from the soil surface for the following specific conditions ... 

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 boundary 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 value problem 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 ordinary differential equations become two-point boundary 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. 

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. 

Note that the numerical simulation of the turbulent reacting flow is now greatly simplified. Indeed, the only partial-differential equation (PDE) that must be solved is (5.100) for the mixture-fraction vector, which involves no chemical source term Moreover, (5.151) is an initial-value problem that depends only on the inlet and initial conditions and is parameterized by the mixture-fraction vector it can thus be solved independently of (5.100), e.g., in a pre(post)-processing stage of the flow calculation. For a given value of , the reacting scalars can then be stored in a chemical lookup table, as illustrated in Fig. 5.10. 

The evaluation of the semiclassical Van Vleck-Gutzwiller propagator (106) amounts to the solution of a boundary-value problem. That is, given a trajectory characterized by the position q(f) = q, and momentum p(f) = p, we need to hnd the roots of the equation = q floiPo)- To circumvent this cumbersome root search, one may rewrite the semiclassical expression for the transition amplitude (105) as an initial-value problem [104-111]... 

Here the problem is given as an initial value problem, although the concepts can easily be generalized to boundary value problems and even partial differential equations. Note also that both continuous variables, x (parameters), and functions of time, U(t) (control profiles), are included as decision variables. Constraints can also be enforced over the entire time domain and at final time. 

For the second step one establishes a solution method. The system under consideration may be static, dynamic, or both. Static cases require solving a boundary value problem, whereas dynamic cases involve an initial value problem. For the illustrative problem, we discuss the solution of a static Laplace (no sources) or Poisson (sources) equation such as... 

However, the situation is not so simple when the heat transport is coupled with stellar structure. Here, I would give another example. It is a problem of X-ray bursting neutron star. The X-ray burst proceeds in tens of second and it might be much shorter as compared with the time scale of heat transport in the envelope of the neutron star. During the burst the X-ray luminosity of the neutron star becomes very close to the Eddington luminosity and the outer layers of the envelope are pushed up by the radiation coming from the interior. Then the neutron star is puffed up and the time scale of heat transport becomes shorter and shorter. Finally, the envelope solution with steady mass flow in thermal equilibrium becomes a good approximation and such situation is also observation-ally confirmed. Before this has become understood, a specialist tried to calculate such expansion of the envelope all the way as an initial value problem by means of stellar evolution code, but it was found impracticable. 

One of the simplest algorithms to solve a boundary value problem is the "shooting method." In this method, we assume initial values needed to make a boundary value problem into an initial value problem. We repeat this process until the solution of the initial value problem satisfies the boundary conditions. Therefore, proper initial conditions for the solution of the preceding problem are... 

The system of DEs (6.144) describes an initial value problem in nine dimensions, once we have chosen the initial values for the dimensionless temperatures j/ (0) and the dimensionless concentrations XAi(0) and x-s/JO) of the components A and B, respectively, in each of the three tanks numbered = 1, 2, 3 at the dimensionless starting time r = 0. We study various sets initial conditions for the problem (6.144) that lead to different steady-state output concentrations xb3 of the desired component B in the three CSTR system. 

The numerical results of this section describe the dynamic behavior of a specific system that is modeled by an initial value problem in nine dimensions. The number of steady states of this system can be any odd number between 1 and 33 = 27, because if tank 1 has one steady state, there may be three in the second tank and nine in the third tank, since there may be three steady states in a subsequent tank for each steady state in the preceding tank. Therefore, if tank 1 has three steady states, the second tank can have nine and the third tank 27 steady states. The maximal number 27 of possible steady states in tank 3 is achieved if there are three steady states in tank 1, with each of these spawning three in the next tank, giving us maximally 9 steady states in tank 2. If each of these steady states in tank 2 gives us three steady states in tank 3, then there is a maximum of 27 steady states in tank 3. 

The dynamic model consists of the three differential equations (7.104), (7.110), and (7.120). These define an initial value problem with initial conditions at t = 0. The dynamic, unsteady state of this system is described by these highly nonlinear DEs, while the steady states are defined by nonlinear transcendental equations, obtained by setting all derivatives in the system of differential equations (7.104), (7.110), and (7.120) equal to zero. 

Equation (2.1) is an ODE system, and, since the values of the variables xi and X2 at t = 0 are provided, it is an initial value problem. By employing a small perturbation parameter 0 < e [Pg.12]

Initial Value Problem. If the conditions at the inlet of the reactor are known, the problem is classified as an initial value problem. One example is the simulation of an operating reactor (11). In this case, the temperature, the pressure, the feed composition at the reactor inlet, and the heat flux to each section of the reactor are known. The yield structure and fluid temperature at the end of the... 

Sometimes, instead of an initial value problem, the mathematical model of a chemical process is a boundary value problem in which values of the dependent variables are specified at different values of the independent variable t. The shooting technique consists of solving an initial value problem, but with an initial value vector a considered as a parameter to estimate (by optimization techniques) so that boundary conditions are satisfied. In this way, a boundary value problem is transformed into an initial value problem. 

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. 

Equations (2.4.10a) to (2.4.101) are six first order ODEs for the six unknown variables y to ye- Note that the order of system is increased from four to six in CMM, while the governing equation is transformed from a boundary value problem to an initial value problem. To solve these six equations, we therefore need to generate initial conditions for the unknowns. As we know the property of the fundamental solutions in the free stream, we can use that information to generate the initial conditions for yi to r/g. As at r/ —> 00 and (f>s, therefore we can estimate the... 

The formulation of the model as above has the advantage that mathematically it picturizes the bed as an initial value problem in contrast to the more complicated boundary value representation of the Fryer-Potter model. The implications of this reduced complexity become more evident (and considerably more important) when the reactions involved are nonlinear. 

The disadvantage of including axial dispersion is that an exit boundary condition must be specified, and in cases where an analytical solution is not available, a numerical boundary-value problem must be solved in the axial direction, rather than an initial-value problem. 

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