Differential equation Subject

The solution to this differential equation subject to the boundary condition that F(t) = 0 at t = 0 is ... 

Differential-Algebraic Systems Sometimes models involve ordinary differential equations subject to some algebraic constraints. For example, the equations governing one equilibrium stage (as in a distillation column) are... 

We next need to solve this second-order differential equation subject to the boundary conditions... 

Solve the following second order linear differential equation subject to the specified "boundary conditions" ... 

Thus the equations that we must solve are 12.196 and 12.197, which comprise a set of two coupled first-order differential equations, subject to the boundary conditions, Xj = 0.01395, and X2 = 0.00712 at z = 0 and Xj = X2 = 0 at z = Z, with the unknown fluxes Ni, N2 that must be found. This equation set could easily be solved as a two-point boundary-value problem using the spreadsheet-based iteration scheme discussed in Appendix D. However, for illustration purposes we choose to solve the equation set with a shooting method, mentioned in Section 6.3.4. We can solve the problem as an ordinary differential equation (ODE) initial-value problem, and iteratively vary Ni,N2 until the computed mole fractions X, X2 are both zero at z = Z. 

Such purely mathematical problems as the existence and uniqueness of solutions of parabolic partial differential equations subject to free boundary conditions will not be discussed. These questions have been fully answered in recent years by the contributions of Evans (E2), Friedman (Fo, F6, F7), Kyner (K8, K9), Miranker (M8), Miranker and Keller (M9), Rubinstein (R7, R8, R9), Sestini (S5), and others, principally by application of fixed-point theorems and Green s function techniques. Readers concerned with these aspects should consult these authors for further references. 

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the steady state distribution of heat or concentration across the slab or the material in which the experiment is performed. This steady state process involves solving second order ordinary differential equations subject to boundary conditions at two ends. Whenever the problem requires the specification of boundary conditions at two points, it is often called a two point boundary value problem. Both linear and nonlinear boundary value problems will be discussed in this chapter. We will present analytical solutions for linear boundary value problems and numerical solutions for nonlinear boundary value problems. 

The general solution to this differential equation subjected to the condition... 

We will now solve this differential equation subject to several sets of experimental boundary conditions. Extensional deformations will be used in these examples, but they could also be done equally well for shear. 

The solution of an optimal control problem requires the satisfaction of differential equations subject to initial as well as final conditions. Except when the equations are linear and the objective functional is simple enough, an analytical solution is impossible. This is the reality of most of the problems for which optimal controls can only be determined using numerical methods. 

In the previous chapters, the discussion was primarily about the initial value problems, for which one seeks a solution to a differential equation subject to conditions on the dependent variable and its derivatives specified at only one value of the independent variable. 

The rate laws that we have considered so far are differential rate laws they specify the time derivative of a concentration in terms of other concentrations. It is often more useful to work with an integrated rate law, in which the concentration is given as a function of the initial concentrations and the time. If the form of the differential rate law is not too complex, one can derive the integrated rate law from it by using the stoichiometric equation to express all of the concentrations in terms of a single concentration (say, [ 4]) and then integrating the resulting one-variable differential equation subject to the known initial conditions. 

Using scaling analysis and perturbation methods, we have been able to derive approximate expressions for the momentum and energy flux in dilute gases and liquids. These methods physically involve formal expansions about local equilibrium states, and the particular asymptotic restrictions have been formally obtained. The flux expressions now involve the dependent transport variables of mass or number density, velocity, and temperature, and they can be utilized to obtain a closed set of transport equations, which can be solved simultaneously for any particular physical system. The problem at this point becomes a purely mathematical problem of solving a set of coupled nonlinear partial differential equations subject to the particular boundary and initial conditions of the problem at hand. (Still not a simple matter see interlude 6.2.)... 

For a specific equation of state the differential equation (7.20) can be solved to yield an expression for the interfacial profile. The relationship between the interfacial profile and the surface tension takes the form (t = f m dp/dzf dz. Let /be the truncated Landau expansion given by eq 7.16. The equilibrium values of the liquid and vapour densities set the boundary conditions corresponding to the bulk phases. Taking the bulk liquid phase to be located at positive infinity, and solving the differential equation subject to this boundary condition, we find... 

Tbe solution of these two partial differential equations, subject to appropriate boundary and initial conditions, will yield the concentration profiles for solutes i =1,2 as a function of z and t. 

Solving the differential equation subject to conditions (12)-(14) yields an expression for the concentration C(x, 0 ... 

Assume that the rate constant ky is independent of position, for example, the catalyst is neither an eggshell nor an egg yolk design. Then the solution to this ordinary differential equation, subject to the boundary conditions of Eqns. (9-4a) and (9-4b), is... 

The solution of this differential equation, subject to the boundary conditions 16 75, gives for the asymptotic value of ci (00) the result... 

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