Fractional Differential Equations

By Tadeusz Kaczorek, K. Rogowski. 2015

10,198 citations recorded by the source corpus.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/c1c7dc1aac34f8ac46c75f718d919838a1b9acfc

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In Numerical Analysis

Citations in the corpus

A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions (2018, methodology): “These new fractional-order models are more adequate than the previously used integer-order models, because fractional-order derivatives and integrals enable the description of the memory and hereditary properties of different quantities [8].”
Numerical solution of time fractional diffusion systems (2017, background): “Some fundamental notions on fractional calculus may be found in [35].”
Global uniqueness in an inverse problem for time fractional diffusion equations (2016, background): “See [37, 47, 53, 57] regarding fractional calculus, and [3, 26, 45], and references therein, for studies of partial differential equations with time fractional derivatives.”
Review of Some Promising Fractional Physical Models (2013, background): “It should be noted the books on fractional differential equation by Podlubny [7] and an introduction to fractional calculus for physicists by Herrmann [8].”
Approximated Fractional Order Chebyshev Lowpass Filters (2015, methodology): “One definition of a fractional derivative of order α is given by the Caputo derivative [16] as”
Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation (2013, neutral): “Before defining the boundary-value problem, we start with some preliminary definitions of fractional calculus [4].”
Fractional differential equations with dependence on the Caputo-Katugampola derivative (2016, background): “Since 17th century, when fractional calculus was born, several different derivatives have been introduced: Riemann-Liouville, Hadamard, Grunwald–Letnikov, Caputo, just to mention a few [13, 14, 17], each of them with its own advantages and disadvantages.”
Theoretical prospects of fractional order weakly singular Volterra Integro differential equations and their approximations with convergence analysis (2021, neutral): “where the last inequality follows from (5).”
The Use of Fractional Order Derivative to Predict the Groundwater Flow (2013, methodology): “(52) Now applying the inverse Laplace transformoperator on both sides in the previous equation, we obtain the following in terms of the Bessel function first kind [24]:”
Describing the firmness, springiness and rubberiness of food gels using fractional calculus. Part I: Theoretical framework (2017, positive): “211 The fractional constitutive framework builds on ideas from fractional calculus, that incor212 porate integrals and derivatives of arbitrary order (Podlubny, 1999; Mainardi, 2010; Jaishankar 213 and McKinley, 2014).”
Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions (2018, background): “Fractional differential equations describe many phenomena in several fields of engineering and scientific disciplines such as physics, biophysics, chemistry, biology (for example, the primary infection with HIV), economics, control theory, signal and image processing, thermoelasticity, aerodynamics,”
A Lyapunov-Type Inequality for a Fractional Differential Equation under a Robin Boundary Condition (2015, neutral): “Γ (kα + β) , α > 0, β > 0, z ∈ C (47) is analytic in the whole complex plane; it will be referred to [12–14] as the Mittag-Leffler function with parameters (α, β).”
A finite difference scheme for the nonlinear time‐fractional partial integro‐differential equation (2020, methodology): “Using (7), Lemma 8, and the Cauchy-Schwarz inequality, we can get”
The -Transform Method and Delta Type Fractional Difference Operators (2015, background): “The particular interpretation of the fractional derivative was presented, for example, in [1] or [2].”
A numerical approach for solving fractional optimal control problems with mittag-leffler kernel (2021, background): “…and widespread fields of physics and engineering, such as rheology, viscoelasticity, electromagnetic theory, diffusive transport, fluid flow, and electrical networks (Ali et al., 2020; Aydogan et al., 2020; Kilbas et al., 2006; Mainardi, 2010; Podlubny, 1999; Samko et al., 1993; Zaslavsky, 2005).”
Existence results of Hilfer integro-differential equations with fractional order (2020, background): “1.[16, 32] Fractional equation of order q for f : [a,∞)→ R is I a+f(ω) = 1 Γq ω ∫”