Fractional Calculus and Waves in Linear Viscoelasticity

10.1142/p926 ◽  
2021 ◽  
Author(s):  
Francesco Mainardi
Keyword(s):  
Fractional Calculus ◽  
Linear Viscoelasticity

scholarly journals On the asymptotics of wright functions of the second kind

2021 ◽  
Vol 24 (1) ◽  
pp. 54-72
Author(s):  
Richard B. Paris ◽  
Armando Consiglio ◽  
Francesco Mainardi
Keyword(s):  
Fractional Calculus ◽  
Asymptotic Expansions ◽  
Linear Viscoelasticity ◽  
Delta Function ◽  
Numerical Results ◽  
Dirac Delta Function ◽  
Dirac Delta ◽  
Sigma 1

Abstract The asymptotic expansions of the Wright functions of the second kind, introduced by Mainardi [see Appendix F of his book Fractional Calculus and Waves in Linear Viscoelasticity (2010)], F σ ( x ) = ∑ n = 0 ∞ ( − x ) n n ! Γ ( − n σ )   ,   M σ ( x ) = ∑ n = 0 ∞ ( − x ) n n ! Γ ( − n σ + 1 − σ )   ( 0 < σ < 1 ) $$F_\sigma(x)=\sum\limits_{n=0}^\infty \frac{(-x)^n}{n! {\mathrm{\Gamma}}(-n\sigma)}~,\quad M_\sigma(x)=\sum\limits_{n=0}^\infty \frac{(-x)^n}{n! {\mathrm{\Gamma}}(-n\sigma+1-\sigma)}\quad(0 \lt \sigma \lt 1) $$ for x → ± ∞ are presented. The situation corresponding to the limit σ → 1− is considered, where M σ (x) approaches the Dirac delta function δ(x − 1). Numerical results are given to demonstrate the accuracy of the expansions derived in the paper, together with graphical illustrations that reveal the transition to a Dirac delta function as σ → 1−.

scholarly journals An historical perspective on fractional calculus in linear viscoelasticity

2012 ◽  
Vol 15 (4) ◽  
Author(s):  
Francesco Mainardi
Keyword(s):  
Fractional Calculus ◽  
Linear Viscoelasticity ◽  
Historical Perspective ◽  
Imperial College ◽  
Historical Survey

AbstractThe article provides an historical survey of the early contributions on the applications of fractional calculus in linear viscoelasticty. The period under examination covers four decades, since 1930’s up to 1970’s, and authors are from both Western and Eastern countries. References to more recent contributions may be found in the bibliography of the author’s book.This paper reproduces, with Publisher’s permission, Section 3.5 of the book: F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press-London and World Scienific-Singapore, 2010.

A note on fractional integral operator associated with multiindex Mittag-Leffler functions

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1931-1939 ◽  
Author(s):  
Junesang Choi ◽  
Praveen Agarwal
Keyword(s):  
Integral Operator ◽  
Fractional Calculus ◽  
Fractional Integral ◽  
Fractional Integration ◽  
Fractional Integral Operator ◽  
Integration Formula ◽  
Special Cases

Recently Kiryakova and several other ones have investigated so-called multiindex Mittag-Leffler functions associated with fractional calculus. Here, in this paper, we aim at establishing a new fractional integration formula (of pathway type) involving the generalized multiindex Mittag-Leffler function E?,k[(?j,?j)m;z]. Some interesting special cases of our main result are also considered and shown to be connected with certain known ones.

scholarly journals A remark on local fractional calculus and ordinary derivatives

2016 ◽  
Vol 14 (1) ◽  
pp. 1122-1124 ◽  
Author(s):  
Ricardo Almeida ◽  
Małgorzata Guzowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Short Note ◽  
General Definition ◽  
Local Fractional Derivative ◽  
Fundamental Properties ◽  
Local Fractional Calculus ◽  
Definition Of

AbstractIn this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

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