Exact Solution of Linear Fractional Distributed Order Systems With Exponential Order Weight Functions

The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus

“…For instance assuming F (s) = exp (s/4) − 1 and a = 0, the Laplace inverse L −1 s→t F 1=s ð Þ f gis simply calculated from (20) as…”

“…Distributed order operators have recently found important applications in signal processing, control systems, electrical circuits, etc. Therefore, the exact impulse response of distributed order operators is of utmost importance when designing circuits containing distributed order elements for instance.…”

“…Applications of the results are therefore presented subsequently, including analytic solutions of fractional distributed order relaxation processes and time domain impulse response of fractional distributed order operators in new simple series expansions. Due to the physical implications and importance of these two problems as well as omnipresence of fractional (distributed order) differential equations, numerical schemes for these kinds of equations and analytical exact and approximate expressions of their solutions that are simple and easy to compute are interesting to scientists . In addition to that, new alternative series expansions for some special functions are obtained in this paper.…”

“…Due to the physical implications and importance of these two problems [9][10][11][12][13] as well as omnipresence of fractional (distributed order) differential equations, 14 numerical schemes for these kinds of equations 15 and analytical exact and approximate expressions of their solutions that are simple and easy to compute are interesting to scientists. [16][17][18][19][20] In addition to that, new alternative series expansions for some special functions are obtained in this paper. The provided numerical simulations suggest that the obtained solutions and expressions are convergent and valid.…”

Asymptotic analysis of fundamental solution of multi-dimensional distributed-order time-fractional diffusion equation with unit density function

“…is a modification of sub-diffusion and has attracted a lot attention by many researchers in the last two decades. In the literature, this equation relates to the initial works [13,14,[20][21][22][23][24], and provides new mathematical and theoretical aspects for studying of the physical and natural sciences in different cases of b(σ) and fractional operators using the analytical and numerical methods [6,8,9,18,25,27,34,35,44,45,47,49,50].…”