Initial value problems systems

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... 

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... 

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. 

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... 

It is interesting that in the soluble case studied in Section VI a close connection appears between initial value problems of dynamics and boundary value problems for dissipative structures. In discussion of initial value problems the concept of stability plays an important role. Only for simple dynamical systems such as separable systems do we find, in general, stability in the sense that trajectories originating from neighboring points remain close for all times. It would be very interesting to investigate along similar lines the stability of dissipative structures, and... 

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. 

It is instructive to study a much simpler mathematical equation that exhibits the essential features of boundary-layer behavior. There is a certain analogy between stiffness in initial-value problems and boundary-layer behavior in steady boundary-value problems. Stiffness occurs when a system of differential equations represents coupled phenomena with vastly different characteristic time scales. In the case of boundary layers, the governing equations involve multiple physical phenomena that occur on vastly different length scales. Consider, for example, the following contrived second-order, linear, boundary-value problem ... 

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. 

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. 

Moreover, the eigenvalues of the matrix A in the linearization of a nonlinear system determine the system s stiffness. We have encountered stiffness with DEs in Chapters 4 and 5 for both initial value problems (IVPs) and boundary value problems (BVPs). A formal definition of stiffness for systems of DEs is given in Section 5.1 on p. 276. 

Using difference quotients, the partial differential equations become ordinary differential equations. The boundary conditions (e.g., inlet and outlet gas temperature of the fluidized bed) can easily be implemented using difference quotient in the entire differential equation system. So, initial/boundary value problems are transferred into initial value problems. Now, the ordinary differential equations of order 1... 

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]

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... 

To avoid technical conditions, assume that / is such that solutions of initial value problems are unique and extend to [0,oo). Thus (D.l) generates a semidynamical system. Of course, one could assume (as has been done before) that solutions extend to all of K. However, checking the backward continuation of solutions presents a problem in one of the applications, so the results are stated for semidynamical systems. The form of the equations causes the positive cone to be invariant (Proposition B.7) and the coordinate axes and the bounding faces to be invariant (and represent lower-order dynamical systems). 

This appendix explains how to use DDAPLUS to solve nonlinear initial-value problems containing ordinary differential equations with or without algebraic equations, or to solve purely algebraic nonlinear equation systems by a damped Newton method. Three detailed examples are given. 

In this chapter we will consider only ordinary differential equations, that is, equations involving only derivatives of a single independent variable. As well, we will discuss only initial-value problems — differential equations in which information about the system is known at f = 0. Two approaches are common Euler s method and the Runge-Kutta (RK) methods. 

The shooting technique involves converting the given boundary value problem to a system of initial value problems. The unknown initial conditions are guessed. These unknown conditions are then updated using the known boundary condition at X = 1. In this technique, the unknown initial condition at x = 0 is estimated using an optimization procedure. This is best illustrated using the next example. 

In section 3.2.4, nonlinear boundary value problems were solved using shooting technique. The given boundary value problem was converted to a system of initial value problems. The unknown initial condition was obtained using an iteration and optimization procedure. This is a very robust technique and can be used to solve stiff boundary value problems. This technique is capable of predicting multiple steady states in a catalyst pellet. However, the number of iterations required for convergence can be prohibitively large for certain boundary value problems. 

Steady state mass or heat transfer in solids and current distribution in electrochemical systems involve solving elliptic partial differential equations. The method of lines has not been used for elliptic partial differential equations to our knowledge. Schiesser and Silebi (1997)[1] added a time derivative to the steady state elliptic partial differential equation and applied finite differences in both x and y directions and then arrived at the steady state solution by waiting for the process to reach steady state. [2] When finite differences are applied only in the x direction, we arrive at a system of second order ordinary differential equations in y. Unfortunately, this is a coupled system of boundary value problems in y (boundary conditions defined at y = 0 and y = 1) and, hence, initial value problem solvers cannot be used to solve these boundary value problems directly. In this chapter, we introduce two methods to solve this system of boundary value problems. Both linear and nonlinear elliptic partial differential equations will be discussed in this chapter. We will present semianalytical solutions for linear elliptic partial differential equations and numerical solutions for nonlinear elliptic partial differential equations based on method of lines. 

Irreversibility is an everyday phenomenon in nature but it remains one of the central issues in theoretical physics. In the early days of quantum physics, open system evolution was described using the Fermi Golden Rule, which leads inherently to an exponential decay of an excited quantum state. The underlying assumption is a continuum of final states that forces all the Poincare recurrences to infinity, and hence introduces irreversibility into the solution of an initial-value problem. While the Golden Rule yields the decay rate of the ex-... 

If the characteristics of the system of equations are found to be complex, the initial-value problem is said to be ill-posed [178]. A physical interpretation of this mathematical statement can be found by analyzing the flow instabilities predicted by this set of model equations. The instabilities predicted by a well-posed model system has some realistic physical meaning, while the instability always present in an ill-posed system is a mathematical mode having no physical origin indicating that the model is not treating small-scale phenomena correctly. 

For a closed, isolated system H is time independent time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem. If (to) is known, a formal solution to Eq. (1) is given by... 

The model (9.73)—(9.75) was presented as an initial value problem We were interested in the rate at which a system in state 0) decays into the continua L and R and have used the steady-state analysis as a trick. The same approach can be more directly applied to genuine steady state processes such as energy resolved (also referred to as continuous wave ) absorption and scattering. Consider, for example, the absorption lineshape problem defined by Fig. 9.4. We may identify state 0) as the photon-dressed ground state, state 1) as a zero-photon excited state and the continua R and L with the radiative and nonradiative decay channels, respectively. The interactions Fyo and correspond to radiative (e.g. dipole) coupling elements between the zero photon excited state 11 and the ground state (or other lower molecular states) dressed by one photon. The radiative quantum yield is given by the flux ratio Yr = Jq r/(Jq r Jq l) = Tis/(Fijj -F F1/,). 

Differential equations are equations that contain the derivatives of the unknown functions. They must be supplemented with auxiliary conditions to completely specify a problem. Auxiliary conditions must be prescribed at one or more points in the domain of the independent variables representing the boundary of the domain interface between different regions, and so on. Those equations with prescribed conditions at one point are called initial-value problems, and those with prescribed conditions on the boundary of the domain are appropriately called boundary-value problems. Initial-value problems generally govern the dynamics of the systems, while boundary-value problems describe the systems in steady state. 

An initial-value problem, consisting of a system of first-order ODEs and the corresponding set of initial conditions, can be expressed compactly in vector notation as... 

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