Fractional calculus ≈ Fractional calculus
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New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model Open
In this manuscript we proposed a new fractional derivative with non-local and\n no-singular kernel. We presented some useful properties of the new derivative\n and applied it to solve the fractional heat transfer model.
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Fractional Calculus: Models and Numerical Methods Open
Survey of Numerical Methods to Solve Ordinary and Partial Fractional Differential Equations Specific and Efficient Methods to Solve Ordinary and Partial Fractional Differential Equations Fractional Variational Principles Continuous-Time Ra…
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A class of second order difference approximations for solving space fractional diffusion equations Open
A class of second order approximations, called the weighted and shifted Gr\"{u}nwald difference operators, are proposed for Riemann-Liouville fractional derivatives, with their effective applications to numerically solving space fractional…
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Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial Open
Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable and efficient way is much more difficult than in the standard integer-order case; moreover, the majority of the computational tools do not prov…
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On a new class of fractional operators Open
This manuscript is based on the standard fractional calculus iteration procedure on conformable derivatives. We introduce new fractional integration and differentiation operators. We define spaces and present some theorems related to these…
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Fractional viscoelastic models for power-law materials Open
Power law materials exhibit a rich range of behaviours interpolating continuously from the linear elastic to the linear viscous responses.
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Fractional differential equations with a Caputo derivative with respect to a Kernel function and their applications Open
This paper is devoted to the study of the initial value problem of nonlinear fractional differential equations involving a Caputo‐type fractional derivative with respect to another function. Existence and uniqueness results for the problem…
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A Discrete Grönwall Inequality with Applications to Numerical Schemes for Subdiffusion Problems Open
We consider a class of numerical approximations to the Caputo fractional\nderivative. Our assumptions permit the use of nonuniform time steps, such as is\nappropriate for accurately resolving the behavior of a solution whose\nderivatives a…
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Two-scale mathematics and fractional calculus for thermodynamics Open
A three dimensional problem can be approximated by either a two-dimensional or one-dimensional case, but some information will be lost. To reveal the lost information due to the lower dimensional approach, two-scale mathematics is needed. …
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Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal Open
In this work, a theoretical study of diffusion of neumatic liquid crystals was done using the concept of fractional order derivative. This version of fractional derivative is very easy to handle and obey to almost all the properties satisf…
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On the generalized fractional derivatives and their Caputo modification Open
In this manuscript, we define the generalized fractional derivative onWe present some of the properties of generalized fractional derivatives of these functions and then we define their Caputo version.
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Applications of variable-order fractional operators: a review Open
Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modellin…
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Transient heat diffusion with a non-singular fading memory: From the Cattaneo constitutive equation with Jeffrey’s Kernel to the Caputo-Fabrizio time-fractional derivative Open
Starting from the Cattaneo constitutive relation with a Jeffrey’s kernel the\n derivation of a transient heat diffusion equation with relaxation term\n expressed through the Caputo-Fabrizio time fractional derivative has been\n developed. …
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On a Fractional Operator Combining Proportional and Classical Differintegrals Open
The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f ( t ) , by a fractional integral operator ap…
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Analysis of the Keller–Segel Model with a Fractional Derivative without Singular Kernel Open
Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Capu…
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Analysis of the model of HIV-1 infection of $CD4^{+}$ T-cell with a new approach of fractional derivative Open
By using the fractional Caputo–Fabrizio derivative, we investigate a new version for the mathematical model of HIV. In this way, we review the existence and uniqueness of the solution for the model by using fixed point theory. We solve the…
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NUMERICAL SOLUTION OF TRAVELING WAVES IN CHEMICAL KINETICS: TIME-FRACTIONAL FISHERS EQUATIONS Open
This paper addresses the numerical solution of nonlinear time-fractional Fisher equations via local meshless method combined with explicit difference scheme. This procedure uses radial basis functions to compute space derivatives while Cap…
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A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow Open
In this article we propose a new fractional derivative without singular\n kernel. We consider the potential application for modeling the steady\n heat-conduction problem. The analytical solution of the fractional-order heat\n flow is also …
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Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels Open
In this manuscript we propose the discrete versions for the recently introduced fractional derivatives with nonsingular Mittag-Leffler function. The properties of such fractional differences are studied and the discrete integration by part…
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A fractional Laplace equation: Regularity of solutions and finite element approximations Open
This paper deals with the integral version of the Dirichlet homogeneous fractional Laplace equation. For this problem weighted and fractional Sobolev a priori estimates are provided in terms of the Holder regularity of the data. By relying…
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On History of Mathematical Economics: Application of Fractional Calculus Open
Modern economics was born in the Marginal revolution and the Keynesian revolution. These revolutions led to the emergence of fundamental concepts and methods in economic theory, which allow the use of differential and integral calculus to …
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Comparison and analysis of the Atangana–Baleanu and Caputo–Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction Open
Atangana and Baleanu (AB) in their recent work introduced a new version of fractional derivatives which uses the generalized Mittag-Leffler function as the non-singular and non-local kernel and accepts all properties of fractional derivati…
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On Fractional Operators and Their Classifications Open
Fractional calculus dates its inception to a correspondence between Leibniz and L’Hopital in 1695, when Leibniz described “paradoxes” and predicted that “one day useful consequences will be drawn” from them. In today’s world, the study of …
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The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators Open
In engineering, a fast estimation of the periodic property of a nonlinear oscillator is much needed. This paper reviews some simplest methods for nonlinear oscillators, including He’s frequency formulation, the max-min approach and the hom…
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A fractional differential equation model for the COVID-19 transmission by using the Caputo–Fabrizio derivative Open
We present a fractional-order model for the COVID-19 transmission with Caputo–Fabrizio derivative. Using the homotopy analysis transform method (HATM), which combines the method of homotopy analysis and Laplace transform, we solve the prob…
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EXACT TRAVELING-WAVE SOLUTION FOR LOCAL FRACTIONAL BOUSSINESQ EQUATION IN FRACTAL DOMAIN Open
The new Boussinesq-type model in a fractal domain is derived based on the formulation of the local fractional derivative. The novel traveling wave transform of the non-differentiable type is adopted to convert the local fractional Boussine…
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APPLICATION OF THE CAPUTO-FABRIZIO FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL TO KORTEWEG-DE VRIES-BURGERS EQUATION∗ Open
In order to bring a broader outlook on some unusual irregularities observed in wave motions and liquids’ movements, we explore the possibility of extending the analysis of Korteweg–de Vries–Burgers equation with two perturbation’s levels t…
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Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel Open
The numerical approximation of the Caputo–Fabrizio fractional derivative with fractional order between 1 and 2 is proposed in this work. Using the transition from ordinary derivative to fractional derivative, we modified the RLC circuit mo…
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Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo–Fabrizio derivatives Open
\n This paper presents a Caputo–Fabrizio fractional derivatives approach to the thermal analysis of a second grade fluid over an infinite oscillating vertical flat plate. Together with an oscillating boundary motion, the heat transfer is c…
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A conformable fractional calculus on arbitrary time scales Open
A conformable time-scale fractional calculus of order $\\alpha \\in ]0,1]$ is\nintroduced. The basic tools for fractional differentiation and fractional\nintegration are then developed. The Hilger time-scale calculus is obtained as a\npart…