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TAM310hw06 - x ±ind the Frst 4 nonzero terms 2b...

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TAM 310 Advanced Engineering Analysis I Spring 2008 Prof.R.Rand Homework No.6 (Due Tuesday March 4) 1. We have seen in class that the equation xy primeprime + y prime + xy = 0 (1) has the general solution y = c 1 J 0 ( x ) + c 2 Y 0 ( x ) (2) where J 0 ( x ) is the Bessel function of order zero of the first kind, and where Y 0 ( x ) is the Bessel function of order zero of the second kind. See for example my online notes, eq.(130), or Edwards and Penney, section 8.4, example 3. These Bessel functions can be generated in MATLAB using the notation besselj(0,x) and bessely(0,x). (Type “help bessel” in MATLAB.) 1a. In MATLAB, plot J 0 ( x ) and Y 0 ( x ) from x=0 to x=30. Hand in a copy of your plot. 1b. Use MATLAB to find the first zero of J 0 ( x ) to 8 significant figures. 2. A more general version of Bessel’s equation (1) is: x 2 y primeprime + xy prime + ( x 2 - p 2 ) y = 0 (3) See Edwards and Penney, section 8.5. Eq.(3) has a solution J p ( x ) which reduces to J 0 ( x ) when p = 0. 2a. Use the method of Frobenius to obtain a series solution for
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Unformatted text preview: x ). ±ind the Frst 4 nonzero terms. 2b. Di²erentiate your result of 2a and show that d dx ( xJ 1 ( x )) = xJ ( x ) (4) 2c. Similarly di²erentiate the series for J ( x ) and show that d dx ( J ( x )) =-J 1 ( x ) (5) 2d. In MATLAB, plot J ( x ) and J 1 ( x ) from x=0 to x=30 on the same plot. Hand in a copy of your plot. 3. Using the search engine GOOGLE, search for an application of Bessel functions to a topic that you are interested in. ±or example, googling “blood ³ow Bessel” comes up with a number of references of mathematical models of blood ³ow which involve Bessel functions. ±rom the various hits google delivers, choose one and write a paragraph describing the application and how Bessel functions enter into the analysis. Print out a sample page of the application and include it with your homework....
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