Similarity Solution Technique for Nonlinear Partial Differential Equations

The current distribution at the electrode (y = O) is given by curr =subs(y=0,-diff(u,y))  

6 Similarity Solution Technique for Nonlinear Partial Differential Equations 

Nonlinear parabolic and elliptic partial differential equations are solved using the similarity solution technique in this section. The methods described in section 4.4 and sections 4.5 are valid for nonlinear partial differential equations, also. The methodology involves converting the governing equation (PDE) to an ordinary differential equation in the combined variable (Tj). This variable transformation is very difficult to do by hand. In this section, we will show how this variable transformation can be done using Maple. The original problem becomes a nonlinear boundary value problem (ODE) in the new combined variable (Tj). This is best illustrated with the following examples. 

4 Partial Differential Equations in Semi-infinite Domains 

Consider the transient diffusion in a rectangle in which the diffusivity varies linearly as a function of concentration.[10] The governing equation is  

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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. 

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. 

In section 4.4, the given linear parabolic partial differential equation in semi-infinite domain was solved by combining the independent variables (similarity solution). This technique is capable of providing special function solutions as shown in example 4.9. In section 4.5, elliptic partial differential equations were solved using the similarity solution technique. In section 4.6, similarity solution was extended for nonlinear parabolic and elliptic partial differential equations. 

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