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Generalized Dynamic Systems Solution by Decomposed Physical Reactions

Shantanu Das

Abstract


Mathematical modeling of many engineering and physics problem leads to extraordinary differential equations (Non-linear, Delayed, Fractional Order). We call them Generalized Dynamic System. An effective method is required to analyze the mathematical model which provides solutions conforming to physical reality. For instant a Fractional Differential Equation (FDE), where the leading differential operator is Reiman-Liouvelli (RL) type requires fractional order initial states which are sometimes hard to physically relate. Therefore, we must be able to solve these dynamic systems, in space, time, frequency, area, volume, with physical reality conserved. The usual procedures, like Runga-Kutta, Grunwarld-Letnikov Discretization with short memory principle etc, necessarily change the actual problems in essential ways in order to make it mathematically tractable by conventional methods. Unfortunately, these changes necessarily change the solution; therefore, they can deviate, sometimes seriously, from the actual physical behavior. The avoidance of these limitations so that physically correct solutions can be obtained would add in an important way to our insight into natural behavior of physical systems and would offer a potential for advances in science and technology. Adomian Decomposition Method (ADM) is applied here in this paper by physical process description; where a process reacts to external forcing function. This reactions-chain generates internal modes from zero mode reaction to first mode second mode and to infinite modes; instantaneously in parallel time or space-scales; at the origin and the sum of all these modes gives entire system reaction. By this approach formulation of Fractional Differential Equation (FDE) by RL method it is found that there is no need to worry about the fractional initial states; instead one can use integer order initial states (the conventional ones) to arrive at solution of FDE. This new finding is highlighted in this paper, which eases out solving for the generalized dynamic systems

Keywords


Riemann-Liouvelli Derivative, Caputo Derivative, Integer Order Initial States, Fractional Order Initial States, Adomian Decomposition Method (ADM), Adomian-Polynomial, Modal-Reactions

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