Characteristic Equation:

Cartesian Coordinates

M.M. Yovanovich

CECART.MWS

Characteristic equation in cartesian coordinates.

The characteristic equation which arises from the application of the Separation of Variables Method to the one-dimensional diffusion and wave equations and the two-dimensional Laplace equation is written as where the dimensionless parameter ranges from to . The characteristic equation

has infinitely many positive roots for any value of . Maple can be used to find the roots which lie in the

general interval: for .

> restart:

Definition of characteristic equation.

> ce:= Bi->delta*sin(delta) - Bi*cos(delta)=0;

Maple procedure to find the nth root for particular value of Bi number.

> rootce:= (n,Bi)->fsolve(ce(Bi), delta= (n-1)*Pi..(2*n-1)*Pi/2);

Calculate the first root for three values of Bi = 0.1, 1,10.

> rootce(1,0.1); rootce(1,1); rootce(1,10);

Calculate a list of values of the first root for values of Bi=10^j for j=-3 to 3 by 1.

> rts1:= [seq(rootce(1,10^j), j = -3..3)];

Calculate a list of values of the second root for values of Bi=10^j for j=-6 to 6 by 1.

> rts2:= [seq(rootce(2,10^j), j = -6..6)];

Calculation of the first 10 roots for Bi = 1.

> rts10:= [seq(rootce(n,1.), n = 1..9)];

The difference between two consecutive roots approach .

> difference:= (n,Bi)->rootce(n+1,Bi) - rootce(n,Bi);

> evalf(difference(400,1)/Pi);

>