Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives

By Nicole Heymans, Igor Podlubny. 2005

667 citations recorded by the source corpus.

On a series of examples from the field of viscoelasticity we demonstrate that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann–Liouville fractional derivatives, and that it is possible to obtain initial values for such initial conditions by appropriate measurements or observations.

Subfields: Fluid Flow and Transfer Processes, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/68ea9e762de0a87e29bc58a71573e94551dfd645

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Fluid Flow and Transfer Processes

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

EXISTENCE AND UNIQUENESS OF SOLUTIONS TO IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS (2009, background): “For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann-Liouville and Caputo types see [27, 40].”
Existence results for boundary value problems with non-linear fractional differential equations (2008, background): “For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann–Liouville and Caputo types see [35,36].”
Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations (2013, background): “(2.2) The Laplace transform of the Caputo derivative is given by Podlubny [32], L { dα dtα f (t); s } = sα f̃ (s) − n−1∑ k=0 sα−k−1f (k)(0), n − 1 < α ≤ n, (2.3) where f̃ (s) =L{f (t); s}.”
A finite element solution for the fractional advection–dispersion equation (2008, background): “[20], Heymans and Podlubny [17], Li and Deng [21], and Zhang et al.”
Boundary Value Problems for Differential Equations Involving Riemann-Liouville Fractional Derivative on the Half-Line (2011, background): “For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann-Liouville and Caputo types see [14, 22].”
Creep, relaxation and viscosity properties for basic fractional models in rheology (2011, methodology): “We refer the reader to the paper by Heymans and Podubny [16] for the physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville derivatives, especially in viscoelasticity.”
Time-fractional derivatives in relaxation processes: a tutorial survey (2008, methodology): “We refer the reader to the paper by Heymans and Podubny [65] for the physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville derivatives, especially in viscoelasticity.”
Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives ✩ (2012, background): “However, on some examples from the field of viscoelasticity, Heymans and Podlubny [7] demonstrated that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann–Liouville fractional derivatives or integrals, and it is possible to obtain initial values for such”
Economic interpretation of fractional derivatives (2017, background): “There are different interpretations of the fractional derivatives and integrals [21]-[36] such as probabilistic interpretations [24]-[26], informatic interpretation [27], geometric interpretations [28]-[30] and [31]-[32], physical interpretations [31]-[32] and [33]-[36].”
Pontryagin maximum principle for fractional ordinary optimal control problems (2014, background): “Several physical interpretations of this condition are demonstrated in [12].”