10.34 – Fall 2006
Homework #1
Due Date:
Wednesday, Sept. 13
th
, 2006 – 9 AM
Problem 1: Bessel Functions
Bessel functions are commonly encountered in heat and mass transfer problem, where the
geometry is cylindrical. Bessel functions of the first kind (
) and second kind
(
) are the two linearly independent solutions of the differential equation
( )
Jx
ν
(
)
Yx
2
22
2
()
dy
d
y
xx
x
y
dx
dx
++
−
=
2
0
The solution of the equation is exactly
for boundary condition
and
( )
(0)
1
y
=
(0)
0
dy
dx
=
.
This second order differential equation can be converted into a system of two coupled
first order differential equations by defining new variable
u
1
and
u
2
as follows:
1
uy
=
2
dy
u
dx
=
The two differential equations thus obtained are
1
2
du
u
dx
=
2
1
2
1
du
u
u
dx
x
x
⎛⎞
=−
−
−
⎜⎟
⎝⎠
(1)
A matlab code is presented below which solves the above problem with boundary
conditions.
1
(0)
1
u
=
and
2
(0)
0
u
=
(Notice that at x =0,
2
du
dx
in equation 1 becomes singular. To write the matlab code we
have used the property of Bessel functions
' (0)
0.5
J
=
, for
1
≥
).
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to
Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu),
Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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% problem 1, HW set 1
% Solution of bessel's equation function using ode23s.
function
plot_bessel_using_ode(x_end)
%u_init is the intial condition at t=0
u_init = [1 0];
%solve the differential equation using ode23s to generate vectors for x
and
%u.
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 Fall '04
 ClarkColton
 Bessel Functions, Numerical Methods Applied

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