Deterministic analysis of distributed order systems using operational matrix

Abstract

a b s t r a c tRecently, distributed order systems as a generalized concept of fractional order have been a major focus in science and engineering areas, and have rapidly extended application across a wide range of disciplines. However, only a few numerical methods are available for analyzing the distributed order systems. This paper proposes a novel numerical scheme to analyze the behavior of single input single output linear systems in the time domain with a single distributed order differentiator/integrator by using operational matrix technique. The proposed method reduces different analysis problems to a system of algebraic equations by using block pulse functions, which makes it easy to handle an arbitrary input. Numerical examples were used to illustrate the accuracy and computational efficiency of the proposed method. The proposed method was found to be an efficient tool for analyzing linear distributed order systems.

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

“…These tables and Figure 3 show that we obtain more accurate results with lower numbers of bases than the methods in [50,53,58]. Moreover, for comparison of our results with the method reported in [57], we plot the exact and approximate solutions for t ∈ [0, 30) in Figure 4 which is stated that by using 100 numbers of bases we obtain more accurate results than the method reported in [57] that used 3000 BPFs to solve this problem.…”

“…Consider the following equations describing the motion of the fractional distributed order oscillator [50,51,53,57,58,75]:…”

“…Consider the following initial value problem for the fractional relaxation equation of distributed order with the given initial condition [9,57,76]:…”

“…Therefore, the development of effective and easy-to-use numerical schemes for solving such equations acquires an increasing interest. While several numerical techniques have been proposed to solve many different problems (see, for instance , and references therein), there have been few research studies that developed numerical methods to solve DOFDEs (see [50][51][52][53][54][55][56][57][58]). The development, however, for efficient numerical methods to solve DOFDEs is still an important issue [51].…”

On an operational matrix method based on generalized Bernoulli polynomials of level m

“…The so-called operational matrix method is one of the numerical methods available to solve a wide class of differential equations. This technique turns the differential equation problem into a system of algebraic equations by means of a finite set of orthogonal basis functions, which simplifies the problem (see [1,[3][4][5][7][8][9]). Furthermore, the requirement of orthogonality for the basis can be skipped when formulating those algebraic equations and, therefore, the problem is significantly simplified (cf.…”