Time-dependent equations

Many transient flows of liquids may be analyzed by using the full time-dependent equations of motion for incompressible flow. However, there are some phenomena that are controlled by the small compressibility of liquids. These phenomena are generally called hydraulic transients. 

Since it wasnot within our ability to solve the time-dependent equation, we naturally solvedthesteady-stateproblemsothattheaccumulationtermwenttozero,whichleftonly thespatialderivative ... 

A) The first step in this problem is to set up asolution to these equations and agraphical display of the results. Using NDSolve, solve these time-dependent equations andby using Plotgraphtheconcentrationsasfunctionsoftimeforthefollowingsetofparameters ... 

Schrddinger s equations are usually written in a more succinct manner by invoking the Hamiltonian operator H, so for example the time-dependent equation for a single particle... 

Figure 3.17 illustrates the processes allowed in Point s model, and defines the rate constants to be used. The approach is very similar to that used in Sect. 3.5.2, and we shall use any results derived there which are applicable without repeating the calculation. The first stem can be of any length, l, and the number of such stems in an ensemble is N,. The net current between Nt and Nl+1 is S, which depends on the forward and backward rate constants for a segment, A and B. Subsequent stems are of the same length and the current between the kth and (k + l)lh stem of length / is Jlk and depends on the rate constants for a complete stem, Alk and Blk, and on the number of such stems, Mlk and Mlk+l. The time dependent equations are ... 

Difference Schemes for Time-Dependent Equations with... 

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

Homogeneous Difference Schemes for Time-Dependent Equations holds. Under this condition the relation... 

Equation (2.28) is now separable into two independent differential equations, one for each of the two independent variables x and t. The time-dependent equation is... 

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