Bessel Functions / Eigenvalues / Heat Equation
- From: phioder@xxxxxxxxx
- Date: Fri, 14 Dec 2007 09:24:10 -0800 (PST)
Hello
Trying to calculate and simulate with Matlab the Steady State
Temperature in the circular cylinder I came to the book of Dennis G.
Zill Differential Equations with Boundary-Value Problems 4th edition
pages 521 and 522
The temperature in the cylinder is given in cylindrical coordinates
by:
u(r,z)= u_0 [Sum from n=1 to Infinite] of:
sinh( lambda_n*z ) * J_0( lambda_n * r )
________________________________________________
lambda_n * sinh(4 * lambda_n) * J_1(2 *lambda_n}
My problems:
-I don't understand very well the Bessel Function either the
Eigenvalues and need a bit of help
-PDE Knowledge and simulations is basic
Information:
With the separation of variables method in cylindrical coordinates and
having U as temperature the equations are defined as follows:
Initial Conditions:
u(2,z)=0 0<z<4
u(r,0)=0 0<r<2
Boundary Condition:
u(r,4)=u_0 0<r<2
u=R(r)Z(z)
r*R'' + R' + ((lambda)^2)*r*R = 0 Cauchy-Euler equation
Z'' + 0 - ((lambda)^2) * Z = 0
With solutions:
R = c_1 * J_0 ( lambda * r ) + c_2 * Y_0 (lambda * r)
Z = c_3 * cosh( lambda * z ) + c_4 * sinh(lambda * z)
The book states "the assumption that the function u is bounded at r =
0 demands that c_2 = 0"
The condition u(2,z) = 0 implies that R(2) = 0
The positive eingenvalues lambda_n of the problem are defined by:
J_0(2*lambda)=0
Now I come to my questions:
1.- What is meant by "the function u is bounded at r = 0"?
Is it right to understand that c_2 = 0 because the Bessel Function of
the Second Kind of Order Zero (Y_0) tends to minus infinite while
aproaching to r=0 from the right side, what is meant by bounded at
r=0?
2.- I did some research on the Bessel Functions of the First and
Second Kinds, solved the Bessel equation step by step and "more or
less" understood it. My problem is that I don't understand neither how
to calculate the eigenvalues lambda_n of the steady state temperature
in a circular cylinder.
Does the equation J_0(2*lambda) = 0 means that:
2*lambda_{1} = 2.4048
2*lambda_{2} = 5.5201
2*lambda_{3} = 8.6537
..
..
..
lambda_{1} = 2.4048 / 2 ???
Or in words said: The eigenvalues are defined by the division by two
of the x value where J_0 is a zero or a root?
3.- If we go back to the final solution there are two terms a
J_0(lambda_n*r) and a J_1(2*lambda_n) and my goal is to implement this
terms on Matlab to understand better the temperature U.
So my approach would continue trying to define in Matlab a vector for
J_0(lambda_n * r), is it right to think that having two vectors of the
same size lambda_n and r, being r defined from 0 to 2, find out which
is the value of the bessel function J_0 at say J_0( (2.4048/2)*r )?
Unfortunately I can't write my post in a less complex way, hope it is
understood, any help, hint or tip would be kindly appreciated.
Best Regards
.
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