Method of Frobenius

In solving Transport Phenomena problems, cylindrical and spherical coordinate 

We shall assume that the functions Q(x) and Pix) are convergent around the point a q = 0 with radius of convergence R 

Under these conditions, the equation can be solved by the series Method of Frobenius. Such series will also be convergent for Ul R. 

for the general case, we start with the expansion 

The first stage of the analysis is to find suitable values for c through the indicial relation. The second stage is to find the relations for a from the recurrence relation. This second stage has many twists and turns and can best be learned by example. We consider these relations in the next section. 

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For regular singular points, a series solution of the differential equation can be obtained by the method of Frobenius. This is based on the following generalization of the power series expansion ... 

Linear homogeneous equations containing nonconstant coefficients were not treated, except for the elementary Euler-Equidimensional equation, which was reduced to a constant coefficient situation by letting x = exp(r). In the next chapter, we deal extensively with the nonconstant coefficient case, starting with the premise that all continuous solutions are in fact representable by an infinite series of terms, for example ejqj (jc) = 1 + x + x /2 + x /3 + . This leads to a formalized procedure, called the Method of Frobenius, to find all the linearly independent solutions of homogeneous equations, even if coefficients are nonconstant. 

Use the method of Frobenius to find the two linearly independent solutions when a = p = 2. Introducing... 

The method of Frobenius can be applied to Eq. 3.143 (Bessel s equation) to yield two linearly independent solutions, which are widely tabulated as... 

The reader can see, had we applied the method of Frobenius, the second solution would take the form... 

Apply the method of Frobenius and show that the only solution that is finite at jc = 0 has the first few terms as... 

We have seen in Chapter 3 that finite difference equations also arise in Power Series solutions of ODEs by the Method of Frobenius the recurrence relations obtained there are in fact finite-difference equations. In Chapters 7 and 8, we show how finite-difference equations also arise naturally in the numerical solutions of differential equations. 

Another exceptional case occurs when repeated roots arise. This can be treated in a manner analogous to the case for repeated roots in the Method of Frobenius (CASE II). Thus, the second solution is obtained by taking the limit... 

Attempting to solve this differential equation by the method of Frobenius gives c = 0 and c = 2 as roots of the indicial equation. Then, for c = 2 we get... 

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