Partial differential equation similarity transformations

Estimation of parameters present in partial differential equations is a very complex issue. Quite often by proper discretization of the spatial derivatives we transform the governing PDEs into a large number of ODEs. Hence, the problem can be transformed into one described by ODEs and be tackled with similar techniques. However, the fact that in such cases we have a system of high dimensionality requires particular attention. Parameter estimation for systems described by PDEs is examined in Chapter 11. 

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

Similarity and integral equation methods for solving the boundary layer equations have been discussed in the previous sections. In the similarity method, it will be recalled, the governing partial differential equations are reduced to a set of ordinary differential equations by means of a suitable transformation. Such solutions can only be obtained for a very limited range of problems. The integral equation method can, basically, be applied to any flow situation. However, the approximations inherent in the method give rise to errors of uncertain magnitude. Many attempts have been made to reduce these errors but this can only be done at the expense of a considerable increase in complexity, and, therefore, in the computational effort required to obtain the solution. 

A similarity transformation may be used when the solution to a parabolic partial differential equation, written in terms of two independent variables, can be expressed in terms of a new independent variable that is a combination of the original independent variables. The success of this transformation requires that ... 

Remember 2.3 Similarity transformations may be used for parabolic partial differential equations when the same condition applies when one independent variable is equal to zero and the other independent variable tends toward infinity. 

Solution The method for solving partial differential equations generally involves finding a method to express them as coupled ordinary differential equations. A similarity transformation is possible if c,- can be expressed as a function of only a new variable. This requirement implies that equation (2.48) can be expressed as a function of only the new variable, and that the three conditions (2.49) in time and position can collapse into two conditions in the new variable. 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a parabolic partial differential equation. For steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in semi-infinite domains using the Laplace transform technique, a similarity solution technique and Maple. In addition, we describe how numerical similarity solutions can be obtained for nonlinear partial differential equations in semi-infinite domains. 

Parabolic partial differential equations are solved using the similarity solution technique in this section. This method involves combining the two independent variables (x and t) as one (rj). For this purpose, the original initial and boundary conditions should become two boundary conditions in the new combined variable (rj). The methodology involves converting the governing equation (PDF) to an ordinary differential equation (ODE) in the combined variable (rj). This variable transformation is very difficult to do by hand. In this chapter, we will show how... 

Both the Laplace transform and the similarity solution techniques are powerful techniques for partial differential equations in semi-infinite domains. The Laplace transform technique can be used for all linear partial differential equations with all possible boundary conditions. The similarity solution can be used only if the independent variables can be combined and if the boundary conditions in x and t can be converted to boundary conditions in the combined variable. In addition, unlike the Laplace transform technique, the similarity solution technique cannot handle partial differential equations in which the dependent variable appears explicitly. The Laplace transform cannot handle elliptic or nonlinear partial differential equations. The similarity solution can be used for elliptic and for a few nonlinear partial differential equations as shown in section 4.6. There are thirteen examples in this chapter. 

Examination of Eqs. 5.1.7-5.1.10 shows that the similarity transformation reduces the original set of n — 1 coupled partial differential equations to a set of n — 1 uncoupled partial differential equations in the pseudocompositions. Equation (5.1.10) for the /th pseudocomponent is... 

The methods for solving a second-order partial differential equation are separation of variables, similarity variable, Laplace transform, Fourier transform, and Hankel transform. Each of the... 

One of the procedures found useful in solving partial differential equations is the so-called combination of variables method, or similarity transform. The strategy here is to reduce a partial differential equation to an ordinary one by judicious combination of independent variables. Considerable care must be given to changing independent variables. 

Another technique that is sometimes employed to reduce partial differential equations to ordinary differential equations is combination of variables or a similarity transformation. 

This case has also been considered by Kim and coworkers/ who derived a nonlinear partial differential equation of a form similar to that outlined in Eqn. 101. The problem was also considered by Hermans many years ago. The method of solution proposed by Bartlett and Gardner is perhaps the most comprehensive suggested to date, and it is based on the Neumann analysis of the phase transformation problem in a semiinfinite diffusion space described in the classic text written by Carlsaw and Jaeger. The reader is referred to the original paper for full details of the analysis. We note therefore that the process of dopant transport and reaction in a polymer matrix can be complex. 

Now all the information required for solving the Fick s law is known. We will use an operator approach. When a Laplace transform is applied to a linear differential equation, it gives an operator algebraic equation. Similarly, a linear differential equation is obtained from the partial differential equation (5.10) ... 

A number of analytical solutions have been developed since that of von Smoluchowski, all of which contain some assumptions and constraints. Friedlander [33] and Swift and Friedlander [34] developed an approach relaxing the above constraint of an initially monodisperse suspension. Using a continuous particle size distribution function, a nonlinear partial integro-differential equation (with no known solution) results from Eq. (5). Friedlander [35] demonstrated the utility of a similarity transformation for representation of experimental particle size distributions. Swift and Friedlander [34] employed this transformation to reduce the partial integro-differential equation to a total integro-differential equation, and dem-... 

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