Approximate Current on a Wire – A Differential Equation Method Adam Schreiber, Yuriy Goykhman, Chalmers Butler

Outline Derivation Solution Method – Solve DE and Iterate Sample Data Discussion Conclusion

Integral Equation Reference Integral Equation Method Break K(z-z’) into it’s real and imaginary parts  K R (z-z’) resembles the delta function  K I (z-z’) resembles sin(x)/x

Justification of K R Approximation

Approximation For 0.4 & 0.8 λ

Differential Equation Second Order Differential Equation I(-h)=I(h)=0

Pulse Test Evaluate the Differential Equation at m points Creates N equations in I(z) Changes intervals from (-h, h) to (z m - Δ/2, z m + Δ/2)

Triangle Expansion Replaces I(z) with N unknowns We now have N equations with N unknowns

MatrixEquation

Solution Method Generate tri-diagonal matrix Find I 0 with J 0 = 0 Generate a new right hand side Find I p Repeat the above 2 steps till convergence Compare results with integral equation data

0.4 Wavelength

0.8 Wavelength

1.0 Wavelength

Error:

Lengths without Convergence Region around 0.5 λ Region around 1.5 λ Odd multiples of 0.5 λ

Reasons for Error Initial delta function approximation Accuracy of evaluating Almost singular matrices at non- convergent lengths

Future Work Increase efficiency/speed Extend algorithm to bent/curved wires Improve numerical integration methods

Conclusion Provides good approximations for current on wires, although not near lengths equal to odd multiples of 0.5 λ Can be adjusted to improve speed and increase accuracy Perhaps, method can be fixed for lengths near 0.5 λ

Acknowledgements Clemson University NSF Dr.Butler Dr. Noneaker & Dr. Xu

Psi R