Perturbation methods regular

Use limiting cases, which may be much more easily soluble, to box the problem. Perturbation methods, distinguishing between regular and singular, and asymptotics are often useful (see Perturbations and Asymptotics in Chapter 3). 

Two theoretical techniques worthy of serious review here, perturbation and Green function methods, can be considered complementary. Perturbation methods can be employed in systems which deviate only slightly from regular shape (mostly from planar geometry, but also from other geometries). However, they can be used to treat both linear and nonlinear PB problems. Green function methods on the other hand are applicable to systems of arbitrary irregularity but are limited to low surface potential surfaces for which the use of the linear PB equation is permitted. Both methods are discussed here with reference to surfactant solutions which are a potentially rich source of nonideal surfaces whether these be solid-liquid interfaces with adsorbed surfactants or whether surfactant self-assembly itself creates the interface. 

In the previous sections we have seen several examples of transport problems that are amenable to analysis by the method of regular perturbation theory. As we shall see later in this book, however, most transport problems require the use of singular-perturbation methods. The high-frequency limit of flow in a tube with a periodic pressure oscillation provided one example, which was illustrative of the most common type of singular-perturbation problem involving a boundary layer near the tube wall. Here we consider another example in which there is a boundary-layer structure that we can analyze by using the method of matched asymptotic expansions. 

This is a critical component of regular perturbation methods, since only the base case carries the primary boundary conditions, hence, by implication, we must have for the other solutions... 

The regular perturbation yields only one real root to the cubic equation. The other two roots are not found because of the simple fact that the cubic term was not retained in the solution for the zero-order coefficient, Zq, explaining why the regular perturbation method fails to locate the other two roots. 

IO3. Apply the regular perturbation method to solve the following first order ordinary differential equation... 

If X, is set to zero, the strained coordinates method will become the traditional regular perturbation method. 

Hopefully, these examples have demonstrated the importance of the regular perturbation method as a technique that can be used to reduce some nonlinear problems to sets of linear problems. And, as was stated earlier, we like to reduce new problems to old ones that we know how to solve. 

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