Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models

By Francesco Mainardi. 2010

2,173 citations recorded by the source corpus.

Subfields: Biomedical Engineering, Fluid Flow and Transfer Processes, Modeling and Simulation

How researchers see this paper

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In Biomedical Engineering

1. G. Em Karniadakis
2. Xiao-jun Yang
3. C. Sin
4. Kexue Li
5. Lu Lu
6. Rodrigo Ponce
7. Sverre Holm
8. Sven Peter Näsholm

In Fluid Flow and Transfer Processes

1. Xiao-jun Yang
2. G. H. McKinley
3. C. Sin
4. Kexue Li
5. T. Faber
6. A. Jaishankar
7. Rodrigo Ponce
8. Sverre Holm

In Modeling and Simulation

1. Dumitru Băleanu
2. Roberto Garrappa
3. G. Em Karniadakis
4. Yong Zhou
5. Xiao-jun Yang
6. Bashir Ahmad
7. Ali H. Bhrawy
8. Juan Trujillo

Citations in the corpus

• fPINNs: Fractional Physics-Informed Neural Networks (2018, background): “Among these are solute transport in fractured and porous media [2, 3], acoustic wave propagation with frequency-dependent dissipation [4, 5, 6, 7], laminar and turbulent flows [8, 9, 10], viscoelastic constitutive laws [11], just to mention a few.”
• Fractional calculus: A survey of useful formulas (2013, neutral): “Beta function B(x, y) = B(y, x) = Γ(x)Γ(y) Γ(x+y) (10) Digamma function ψ(x) = d log Γ(x) dx = 1 Γ(x) dΓ(x) dx (11) Error function erf (z) = 2 √ π ∫ z 0 e−t 2 dt, z ∈ C (12) Complementary error function erfc (z) = 1− erf (z) , z ∈ C (13)”
• A unified Petrov-Galerkin spectral method for fractional PDEs (2015, background): “operators appear as attractive and potentially powerful modeling tools in many areas such as viscoelastic materials [3,4], porous or fractured media [5], fluid mechanics [6–8], bioengineering [9], and anomalous diffusion (nonMarkovian) processes [10,11].”
• Nonlocal Fractional Evolution Inclusions of Order α ∈ (1,2) (2019, background): “More recently, time-fractional diffusion and wave equations have been attracting the widespread attention of many fields of science and engineering [7,8].”
• Modified Galerkin algorithm for solving multitype fractional differential equations (2019, methodology): “By differentiating the relation (80) of order to with respect to x and using the orthogonality relation (5), we get ( D(2)xφr(x), ρ (γ,δ) i (x) ) w = r ∑”
• Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems (2016, methodology): “The FD of constant order The FD of the Caputo type with respect to the singular power-law kernel is given by [2, 5, 10]: ( ) ( ) ( ) ( ) 1 1 d ( ) D d 1 d x”
• Hölder continuous solutions for fractional differential equations and maximal regularity (2013, background): “See, for instance [1, 26, 31] and [34] for further details.”
• Science metrics on fractional calculus development since 1966 (2013, background): “Figure 2 (left) depicts the evolution of Da [19, 12, 56, 75, 57, 99, 60, 87, 100, 100, 77, 68, 85, 95, 69, 70, 71, 72, 78, 23, 102, 13, 96, 64, 114, 36, 79, 73, 91, 63, 84, 80, 105, 109, 4, 111, 82, 22, 27, 28, 110, 81, 67, 32, 29, 86, 52, 33, 117, 106, 55, 58, 112, 113, 108, 6, 15, 2, 41, 59, 10, 4”