On infinite series concerning zeros of Bessel functions of the first kind

Abstract

Abstract. A relevant result independently obtained by Rayleigh and Sneddon on an identity on series involving the zeros of Bessel functions of the first kind is derived by an alternative method based on Laplace transforms. Our method leads to a Bernstein function of time, expressed by Dirichlet series, that allows us to recover the RayleighSneddon sum. We also consider another method arriving at the same result based on a relevant formula by Calogero. Moreover, we also provide an electrical example in which this sum results to be extremely useful in order to recover the analytical expression for the response of the system to a certain external input.

Prabhakar-like fractional viscoelasticity

“…If we define a n pk; tq " p´aq´n t βn pγnq k Γpα k`βn`1q pωt α q k k! " p´aq´n Γpγn`kq Γpγnq Γpα k`βn`1q 15) and recalling the asymptotic behaviour of the rate of gamma functions (see [8; 10]) Γ pz`aq Γ pz`bq " z a´b , for |z| Ñ 8, |Argpzq| ď π´ , |Argpz`aq| ď π´ , 0 ă ă π, it is easy to see thaťˇˇˇa n`1 pk; tq a n pk; tqˇˇˇˇ"ˇˇˇˇt β β a 1 nˇˇˇˇ, n Ñ 8 , @t ą 0, @k P N Y t0u , (2.16) hence lim nÑ8ˇa n`1 pk; tq a n pk; tqˇˇˇˇ" 0 , @t ą 0, @k P N Y t0u , (2.17) that concludes our proof of the absolute convergence of (2.12).…”

Dispersion relations for the time-fractional Cattaneo-Maxwell heat equation

“…where τ is the so called relaxation time, and it leads to a modified version of the heat equation, In the last few years, the quest for potential applications of fractional calculus [11][12][13][14] in biology [15][16][17][18], thermodynamics [19][20][21][22], viscoelasticity [24][25][26][27] has been attracting much attention in the mathematical community. Nevertheless, despite this growing interest, not much work has been done in the study of dispersion relations for fractional models for wave propagation.…”

Storage and Dissipation of Energy in Prabhakar Viscoelasticity

“…Now, it is very well known that a viscoelastic system can usually be mapped onto a class of electrical ladder networks and vice versa; see, for example, [32,33].…”