problem_set1 - 10.34 Fall 2006 Homework #1 Due Date:...

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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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problem_set1 - 10.34 Fall 2006 Homework #1 Due Date:...

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