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The main purpose of the present book is to propose a method for
solving the mixed problem for transmission line systems reducing it
to a neutral equation (or system) on the boundary. Arising
non-linearities in the neutral systems are caused by non-linear
characteristics of the RGCL-loads. In view of the applications we
consider mainly periodic and oscillatory problems for loss-less
transmission lines. We point out, however, that here we propose an
extended procedure for reducing the mixed problem for lossless and
lossy transmission lines. We introduce also an extension of
Heaviside condition and this way we can consider the case of
time-varying specific parameters-per-unit length resistance,
conductance, inductance and capacitance. We find a solution of the
obtained neutral equations by discovering operators whose fixed
points in suitable function spaces are periodic or oscillatory
solutions of the formulating problems. Using fixed point theorems
for contractive mappings in uniform and metric spaces (proved by
the author in the previous papers) we prove existence -- uniqueness
results for periodic and oscillatory problems. We obtain also
successive approximations of the solution with respect to a
suitable family of pseudo-metrics and give an estimate of the rate
of convergence. Although the question of finding the initial
approximation is not trivial. We show that one can begin with a
simple harmonic initial approximation. The rate of convergence
depends on the parameters of the transmission lines and
characteristics of the non-linear RCL-loads. Our conditions are
applicable even in some cases to non-uniform transmission lines.
Numerical examples demonstrate the applicability of the main
results to design of circuits. It is easy to verify a system of
inequalities between basic parameters without examining the proofs
of the theorems.
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