Solution of differential equations

The following type of differential equation is encountered in the text, for example, in the analysis of the models for viscoelastic behaviour  

When a material is subjected to a uniaxial stress, there will be a strain in the direction of the stress and a strain of the opposite sense in the perpendicular directions. The latter is referred to as the Poisson s Ratio effect. 

The Z-direction is perpendicular to the page. For simplicity the material is assumed to be isotropic, ie same properties in all directions. However, in some cases for plastics and almost always for fibre composites, die properties will be anisotropic. Thus E and v will have different values in the x, y and z direction. Also, it should also be remembered that only at short times can E and v be assumed to be constants. They will both change with time and so for long-term loading, appropriate values should be used. 

If the material is subjected to biaxial stresses in both the x and y directions then the strains will be 

That is, the total strain will be the sum of the tensile strain due to and the negative strain due to the Poisson s ratio effect caused by Oy. 

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The finite element solution of differential equations requires function integration over element domains. Evaluation of integrals over elemental domains by analytical methods can be tedious and impractical and is not attempted in... 

The simplicity gained by choosing identical weight and shape functions has made the standard Galerkin method the most widely used technique in the finite element solution of differential equations. Because of the centrality of this technique in the development of practical schemes for polymer flow problems, the entire procedure of the Galerkin finite element solution of a field problem is further elucidated in the following worked example. 

Carnahan, B., and J. O. Wilkes. Numerical Solution of Differential Equations—An Overview in Foundations of Computei-Aided Chemical Fi ocess Design, AIChE, New York (1981). 

Integral-Transform Method A number of integral transforms are used in the solution of differential equations. Only one, the Laplace transform, will be discussed here [for others, see Integral Transforms (Operational Methods) ]. The one-sided Laplace transform indicated by L[f t)] is defined by the equation L[f t)] = /(O dt. It has... 

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

For packed towers, the continuous differential nature of the contact between gas and liqiiid leads to a design procedure involving the solution of differential equations, as described in the next subsection. 

The development of mathemafical models is described in several of the general references [Giiiochon et al., Rhee et al., Riithven, Riithven et al., Suzuki, Tien, Wankat, and Yang]. See also Finlayson [Numerical Methods for Problems with Moving Front.s, Ravenna Park, Washington, 1992 Holland and Liapis, Computer Methods for Solving Dynamic Separation Problems, McGraw-Hill, New York, 1982 Villadsen and Michelsen, Solution of Differential Equation Models by... 

An accurate indication is achieved by carrying out the calculations in small time steps, such as At = 0.004 s, and then by calculating the vaporization, humidity change, and corresponding temperature rise at each time step. This is the numerical solution of differential equations (4.326) and (4.328). The results of a calculation of this type are shown in Table 4.12. 

We now consider the solution of differential equations by means of Laplaee transforms. We have already solved one equation, namely, the first-order rate equation, but the technique is capable of more than this. It allows us to solve simultaneous differential equations. 

Solution of Sets of Simultaneous Linear Equations 71. Least Squares Curve Fitting 76. Numerical Integration 78. Numerical Solution of Differential Equations 83. 

Goldberg points out, however, that despite any apparent similarity between PFA supported solitons and the soliton solutions of differential equations such as the... 

For convenience, mass fraction units are used for [I] and [M] instead of moles per unit volume to eliminate density, which is assumed constant. With an appropriate variable transformation and series expansion, the analytical solution of differential Equation 10 can be derived. The solution is as follows. 

For a solution of differential equations (18.12) and (18.15) and for a quantitative calculation of the current distribution, we must know how the current density depends on polarization at constant reactant concentrations or on reactant concentrations at constant polarization. We must also formulate the boundary conditions. Examples of such calculations are reported below. 

Thus, (d/dcfe s= e and here the operator d/dx plays the role of the identity with respect to the function y e It will be employed in the solution of differential equations in Chapter 5. 

It should be evident that the expressions for the Laplace transforms of derivatives of functions can facilitate the solution of differential equations. A trivial example is that of the classical harmonic oscillator. Its equation of motion is given by Eq. (5-33), namely,... 

Note such simplification is allowed only when we use Equations (4) and (5) for solution of differential equations involving more stable products, for example, that for concentration of hydroperoxides, as follows ... 

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

The solution of differential equations is a specialized pursuit the precise method is often unique for a specific problem. A common equation with numerous applications will be solved by way of demonstration ... 

The solutions of differential equations often define series of related functions that can be obtained from simple generating functions or formulae. As an example consider the Legendre polynomials... 

For a more detailed analysis of measured transport restrictions and reaction kinetics, a more complex reactor simulation tool developed at Haldor Topsoe was used. The model used for sulphuric acid catalyst assumes plug flow and integrates differential mass and heat balances through the reactor length [16], The bulk effectiveness factor for the catalyst pellets is determined by solution of differential equations for catalytic reaction coupled with mass and heat transport through the porous catalyst pellet and with a film model for external transport restrictions. The model was used both for optimization of particle size and development of intrinsic rate expressions. Even more complex models including radial profiles or dynamic terms may also be used when appropriate. 

The Point at Infinity. In many problems we wish to find solutions of differential equations of the type (3.1) which are valid for large values of a . We seek solutions in... 

In developing series solutions of differential equations and in other formal calculations it is often convenient to make use of properties of gamma and beta functions. The integral... 

Villadsen J, Michelsen ML (1978) Solution of differential equation models by polynomial approximation. Prentice-Hall, Englewood Cliffs, N.J. 

Studying the dynamics of systems in the time domain involves direct solutions of differential equations. The computer simulation techniques of Part II are very general in the sense that they can give solutions to very complex nonlinear problems. However, they are also very specific in the sense that they provide a solution to only the particular numerical case fed into the computer. 

The authors applied this model to the situation of dissolving and deposited interfaces, involving chemically interacting species, and included rate kinetics to model mass transfer as a result of chemical reactions [60]. The use of a stochastic weighting function, based on solutions of differential equations for particle motion, may be a useful method to model stochastic processes at solid-liquid interfaces, especially where chemical interactions between the surface and the liquid are involved. 

Villadsen, J., and Michelsen, M. L., Solution of Differential Equation Models by Polynomial Approximation. Prentice-Hall Inc., New York, 1978. 

Figure 7. Form of solution of differential equations arising from decay...

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