# A Boundary Value Problem Requiring Quarterwave Symmetry

1. Apr 10, 2016

### TheJfactors

I can't seem to find an explicit or analytical solution to a boundary value problem and thought I might ask those more knowledgeable on the subject than me. If t is an independent variable and m(t) and n(t) are two dependent variables with the following 8 constraints:

a) m' =0 @T=0 and t=pi/2 b) n' =0 @T=0 and t=pi/2
c) m' ' =0 @T=0 and t=pi/2 d) n' ' =0 @T=0 and t=pi/2

I need to solve the following differential equation:

(((1+1/4)+((pi/2-t)+m)/pi)∙(1+m^' )+((3-1/4)-((pi/2-t)-n)/pi)∙(1-n^' )) + (((1+1/4)+(t+m)/pi)∙(1+m^' )+((3-1/4)-(t-n)/pi)∙(1-n^' )) = 1

for m(t) and n(t), you can assume a profile for one of them as long as it is satisfies the conditions listed above. Thus far I've had no luck finding a profile for both that can satisfy those boundary conditions.

2. Apr 11, 2016

### TheJfactors

Here's a slightly easier to read version, I didn't see the formatting errors when I copied it over from equation editor:
$(((1+\frac{1}{4})+\frac{(\frac{pi}{2}-t)+m}{pi})∙(1+m' )+((3-\frac{1}{4})-\frac{(\frac{pi}{2}-t)-n}{pi})∙(1-n' )) + (((1+\frac{1}{4})+\frac{t+m}{pi})∙(1+m' )+((3-\frac{1}{4})-\frac{t-n}{pi})∙(1-n' )) = 1$