Initial value problem, solutions

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. 

Equations of the first land are very sensitive to solution errors so that they present severe numerical problems. Volterra equations are similar to initial value problems. 

NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS AS INITIAL VALUE PROBLEMS... 

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. 

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). 

The solution of boundary value problems depends to a great degree on the ability to solve initial value problems.) Any n -order initial value problem can be represented as a system of n coupled first-order ordinary differential equations, each with an initial condition. In general... 

Given two estimates of the root and Uj, two solutions of the initial value problem are calculated, yL(U,j, ) and yLCU,), a new estimate of U is obtained where... 

The above two examples were chosen so as to point out the similarity between a physical experiment and a simple numerical experiment (Initial Value Problem). In both cases, after the initial transients die out, we can only observe attractors (i.e. stable solutions). In both of the above examples however, a simple observation of the attractors does not provide information about the nature of the instabilities involved, or even about the nature of the observed solution. In both of these examples it is necessary to compute unstable solutions and their stable and/or unstable manifolds in order to track and analyze the hidden structure, and its implications for the observable system dynamics. 

The equations above describe how solutes in the soil will move in response to concentration or water potential gradients. Such gradients form when the rhizo-sphere is perturbed by the activities of the root including water and MN abstraction and carbon deposition. These activities need to be mathematically described and form one of the two boundary conditions required to solve the initial-value problem. 

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)]. 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements. One simply combines the methods for ordinary differential equations (see Ordinary Differential Equations—Boundary Value Problems ) with the methods for initial-value problems (see Numerical Solution of Ordinary Differential Equations as Initial Value Problems ). Fast Fourier transforms can also be used on regular grids (see Fast Fourier Transform ). 

Absorption columns can be modeled in a plate-to-plate fashion (even if it is a packed bed) or as a packed bed. The former model is a set of nonlinear algebraic equations, and the latter model is an ordinary differential equation. Since streams enter at both ends, the differential equation is a two-point boundary value problem, and numerical methods are used (see Numerical Solution of Ordinary Differential Equations as Initial-Value Problems ). 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

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]... 

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... 

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... 

Although one cannot obtain a general solution for solving the initial value problem... 

In both the Ito and Stratonovich formulations, the randomness in a set of SDEs is generated by an auxiliary set of statistically independent Wiener processes [12,16]. The solution of an SDE is defined by a hmiting process (which is different in different interpretations) that yields a unique solution to any stochastic initial value problem for each possible reahzation of this underlying set of Wiener processes. A Wiener process W t) is a Gaussian Markov diffusion process for which the change in value W t) — W(t ) between any two times t and t has a mean and variance... 

Clearly when t approaches , cA approaches (H)(cA+ + cA ), the average of the initial concentrations in the two half cells. Equation (136) is used in the analysis of closed-cell diffusion experiments when the time of diffusion is sufficiently short that the concentrations at the ends of the cell do not deviate from the initial values. The solution of this problem when Dab is a function of concentration has been studied by Stokes (S19), by Gillis and Kedem (G4), and by Fujita (F7, F8, F9). 

The problem posed by Eqs. (l)- 4) admits of solution in many forms. Montroll7 has treated the initial-value problem for an equation of the... 

Numerical Solution Equations 6.40 and 6.41 represent a nonlinear, coupled, boundary-value system. The system is coupled since u and V appear in both equations. The system is nonlinear since there are products of u and V. Numerical solutions can be accomplished with a straightforward finite-difference procedure. Note that Eq. 6.41 is a second-order boundary-value problem with values of V known at each boundary. Equation 6.40 is a first-order initial-value problem, with the initial value u known at z = 0. 

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. 

K.E. Brenan, S.L. Campbell, and L.R. Petzold. Numerical Solution of Initial-Value Problems in Differential Algebraic Equations. SIAM, Philadelphia, PA, second edition, 1996. 

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... 

Table 3.1 shows the ACSL program for the solution of Eq. (3. 34) when 0 = 2 and (3 = 5. This is the case when Rc = 0. Therefore, initially the substrate concentration is assumed to be zero, and it is checked that xs = 1 at r = 1 after solving the initial value problem. If it is not, a new value of xs is estimated using the equation ... 

Numerical Solution of the Resulting Initial Value Problem... 

Create the solution profiles of Figure 4.48 using fluidbedprof iles. m and interpret the behavior of the solution for the initial value problem trajectory of Figure 4.48. 

Due to these inner iterations via IVP solvers and due to the need to solve an associated nonlinear systems of equations to match the local solutions globally, boundary value problems are generally much harder to solve and take considerably more time than initial value problems. Typically there are between 30 and 120 I VPs to solve numerous times in each successful run of a numerical BVP solver. 

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