scholarly journals Spectral Approximation of Fractional PDEs in Image Processing and Phase Field Modeling

2017 ◽  
Vol 17 (4) ◽  
pp. 661-678 ◽  
Author(s):  
Harbir Antil ◽  
Sören Bartels
Keyword(s):  
Image Processing ◽  
Phase Field ◽  
Differential Operators ◽  
Mathematical Tool ◽  
Phase Field Models ◽  
Field Modeling ◽  
Regularity Properties ◽  
Fractional Pdes ◽  
Fractional Differential ◽  
Fractional Differential Operators

AbstractFractional differential operators provide an attractive mathematical tool to model effects with limited regularity properties. Particular examples are image processing and phase field models in which jumps across lower dimensional subsets and sharp transitions across interfaces are of interest. The numerical solution of corresponding model problems via a spectral method is analyzed. Its efficiency and features of the model problems are illustrated by numerical experiments.

scholarly journals Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Ming-Sheng Hu ◽  
Ravi P. Agarwal ◽  
Xiao-Jun Yang
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Series Solution ◽  
Differential Operators ◽  
Fractional Wave Equation ◽  
Local Fractional Calculus ◽  
Vibrating String ◽  
Fractional Differential ◽  
Fractional Differential Operators

We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag-Leffler function.

scholarly journals The Extension of the Physical and Stochastic Problems to Space-Time-Fractional Differential Equations

2021 ◽  
Vol 2090 (1) ◽  
pp. 012031
Author(s):  
E.A. Abdel-Rehim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Differential Operators ◽  
Space Time ◽  
Partial Differential ◽  
Stochastic Problems ◽  
Fractional Differential ◽  
Fractional Differential Operators ◽  
Time Fractional Differential Equations

Abstract The fractional calculus gains wide applications nowadays in all fields. The implementation of the fractional differential operators on the partial differential equations make it more reality. The space-time-fractional differential equations mathematically model physical, biological, medical, etc., and their solutions explain the real life problems more than the classical partial differential equations. Some new published papers on this field made many treatments and approximations to the fractional differential operators making them loose their physical and mathematical meanings. In this paper, I answer the question: why do we need the fractional operators?. I give brief notes on some important fractional differential operators and their Grünwald-Letnikov schemes. I implement the Caputo time fractional operator and the Riesz-Feller operator on some physical and stochastic problems. I give some numerical results to some physical models to show the efficiency of the Grünwald-Letnikov scheme and its shifted formulae. MSC 2010: Primary 26A33, Secondary 45K05, 60J60, 44A10, 42A38, 60G50, 65N06, 47G30,80-99

Dynamical behavior of fractionalized simply supported beam: An application of fractional operators to Bernoulli-Euler theory

2021 ◽  
Vol 10 (1) ◽  
pp. 231-239
Author(s):  
Kashif Ali Abro ◽  
Abdon Atangana ◽  
Ali Raza Khoso
Keyword(s):  
Differential Operators ◽  
Special Functions ◽  
Dynamical Behavior ◽  
Analytic Solutions ◽  
Simply Supported Beam ◽  
Simply Supported ◽  
Parametric Resonances ◽  
Fractional Differential ◽  
Fractional Differential Operators ◽  
Graphical Illustration

Abstract The complex structures usually depend upon unconstrained and constrained simply supported beams because the passive damping is applied to control vibrations or dissipate acoustic energies involved in aerospace and automotive industries. This manuscript aims to present an analytic study of a simply supported beam based on the modern fractional approaches namely Caputo-Fabrizio and Atanagna-Baleanu fractional differential operators. The governing equation of motion is fractionalized for knowing the vivid effects of principal parametric resonances. The powerful techniques of Laplace and Fourier sine transforms are invoked for investigating the exact solutions with fractional and non-fractional approaches. The analytic solutions are presented in terms of elementary as well as special functions and depicted for graphical illustration based on embedded parameters. Finally, effects of the amplitude of vibrations and the natural frequency are discussed based on the sensitivities of dynamic characteristics of simply supported beam.

scholarly journals Phase-field modeling of geologic fracture incorporating pressure-dependence and frictional contact

2020 ◽  
Vol 205 ◽  
pp. 03004
Author(s):  
Jinhyun Choo ◽  
Fan Fei
Keyword(s):  
Pressure Dependence ◽  
Phase Field ◽  
Frictional Contact ◽  
Induced Earthquakes ◽  
Phase Field Modeling ◽  
Phase Field Models ◽  
Fracture Simulation ◽  
Field Modeling ◽  
Geologic Fractures ◽  
New Phase

Geologic fractures such as joints and faults are central to many problems in energy geotechnics. Notable examples include hydraulic fracturing, injection-induced earthquakes, and geologic carbon storage. Nevertheless, our current capabilities for simulating the development and evolution of geologic fractures in these problems are still insufficient in terms of efficiency and accuracy. Recently, phase-field modeling has emerged as an efficient numerical method for fracture simulation which does not require any algorithm for tracking the geometry of fracture. However, existing phase-field models of fracture neglected two distinct characteristics of geologic fractures, namely, the pressure-dependence and frictional contact. To overcome these limitations, new phase-field models have been developed and described in this paper. The new phase-field models are demonstrably capable of simulating pressure-dependent, frictional fractures propagating in arbitrary directions, which is a notoriously challenging task.

scholarly journals Fractional Differential Operators: Eigenfunctions

2018 ◽  
Author(s):  
Rubens De Figueiredo Camargo ◽  
Eliana Contharteze Grigoletto ◽  
Edmundo Capelas De Oliveira
Keyword(s):  
Differential Operators ◽  
Fractional Differential ◽  
Fractional Differential Operators

scholarly journals On multi-order fractional differential operators in the unit disk

Filomat ◽  
2016 ◽  
Vol 30 (1) ◽  
pp. 73-81 ◽  
Author(s):  
Rabha Ibrahim ◽  
Cenap Ozel
Keyword(s):  
Unit Disk ◽  
Differential Operators ◽  
Analytic Formula ◽  
Fractional Operators ◽  
Multi Index ◽  
Owa Operators ◽  
Generalized Wright Function ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Fractional Differential Operators

In this article, we generalize fractional operators (differential and integral) in the unit disk. These operators are generalized the Srivastava-Owa operators. Geometric properties are studied and the advantages of these operators are discussed. As an application, we impose a method, involving a memory formalism of the Beer-Lambert equation based on a new generalized fractional differential operator. We give solutions in terms of the multi-index Mittag-Leffler function. In addition, we sanctify the out come from a stochastic standpoint. We utilize the generalized Wright function to obtain the analytic formula of solutions.

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