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methods for solving ODEs. But the digital computing has also its limitations. Thus, recently
some researchers are exploring ways to return to the use of the analog computing method
again [111], especially in cases where ultrafast speed of the solving process is needed (see
realtime simulation needs for example). There are many reasons behind this decision. The
first and most important issue is the processing speed. Much of the physical phenomena in
the real world are measured/expressed by calculus; therefore with analog computing we
can simulate these models very fast. Due to the inherent process and computing
parallelization, analog computers are capable of producing complex solutions in real time.
In the early days of the analog computing age (more than 40 years ago) they were facing a
series of limitations due amongst others to the use of discrete electronic components of
this analog computing paradigm: imperfect connections between elements, limitation in
the voltage scaling, variability of elements characteristics during the process due to the
temperature, etc [32], [108]. Also the precision of the component characteristics values is a
serious technological non-scalable limitation. In a real analog computer, there is a
limitation of the voltage scaling. The voltage is limited between the noise level and the high
voltage level. Because of this physical limitation one cannot reach solutions that are out of
this boundary (interval). Since the 1980’s up to now, many researchers have been trying to
implement analog computation systems for specific problems in VLSI chips [112, 113]. In
VLSI chips, one can rescale the voltages within CMOS or TTL ranges. One particular
weakness of this approach (i.e., analog VLSI implementation) is that the circuit is not re-
configurable and for solving a new problem we have to design another chip, what is a very
expensive issue. Another problem is that we are not able to re-scale the time in this case. In
some cases, for coupling system components, time synchronization becomes necessary.
Due to the limitations of the VLSI approach, we have started thinking of an alternative that
consists in the essence of an emulation concept of the analog computer on top of
reconfigurable and scalable digital platforms, especially FPGA chips [111, 112, 114, 115].
This research shows that this emulation has been successful and that we are capable of
modeling and solving any type of higher order even nonlinear ODE at an ultra-fast speed
on this fully digital structure.
