DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

NUMERICAL SOLUTION OF A TRANSMISSION LINE
PROBLEM OF ELECTRODYNAMICS IN A CLASS
OF DISCONTINUOUS FUNCTIONS
TURHAN KARAGULER AND MAHIR RASULOV
A special numerical method for the solution of first-order partial differential equation
which represents the transmission line problem in a class of discontinuous functions is
described. For this, first, an auxiliary problem having some advantages over the main
problem is introduced. Since the differentiable property of the solution of the auxiliary
problem is one order higher than the differentiability of the solution of the main problem,
the application of classical methods to the auxiliary problem can easily be performed.
Some economical algorithms are proposed for obtaining a numerical solution of the auxiliary
problem, from which the numerical solution of the main problem can be obtained.
In order to show the effectiveness of the suggested algorithms, some comparisons between
the exact solution and the numerical solution are carried out.
Copyright © 2006 T. Karaguler and M. Rasulov. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. The Cauchy problem
It is known from the electromagnetic field and the circuit theories that the equations for
a current and potential in a transmission line have the following form [1, 3, 4]:
L ∂i(x,t)
∂t
C ∂v(x,t)
∂t
+ ∂v(x,t)
∂x
+ ∂i(x,t)
∂x
+ Ri(x,t) = 0, (1.1)
+ Gv(x,t) = 0. (1.2)
Here v(x,t) andi(x,t) are potential and current at any points x and t, R is resistance
per unit length, L is inductance per unit length, C is capacitance per unit length, and G is
conductance per unit length. These line parameters are taken constant since the medium
is assumed as linear and homogeneous.
The initial condition for (1.1), (1.2)are
i(x,0)= i 0 (x), (1.3)
v(x,0)= v 0 (x), (1.4)
where i 0 (x)andv 0 (x) are given as continuous or piecewise continuous functions.
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 501–508