scholarly journals Numerical Simulation for Fractional Percolation Equation

2021 ◽  
Vol 8 (3) ◽  
pp. 425-430
Author(s):  
Iman I. Gorial
Keyword(s):  
Exact Solution ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Stability And Convergence ◽  
Initial Condition ◽  
One Dimensional ◽  
Grid Points ◽  
Explicit Finite Difference ◽  
Mixed Fractional Derivatives ◽  
The Given

The aims of this paper are to propose approach of explicit finite difference mathod (EFDM), clarify the problem the mixed fractional derivative in one-dimensional fractional percolation equation (O-DFPE), and the study of consistency, stability, and convergence methods. Use of estimated Grunwald estimation in the analysis of mixed fractional derivatives. However, the given method is successfully applied to the mixed fractional derivative classes with the initial condition (IC) and derivative boundary conditions (DBC). To illustrate the efficiency and validity of the proposed algorithm, examples are given and the results are compared with the exact solution. From the figures shown for the examples in this work, the approximate solution values given by the EFDM for the various grid points are equivalent to the exact solution values with high-precision approximation. To show the effectiveness of the proposed method, where the error between the EFDM and the exact method is zero, the fractional derivative was used with various and random values. Using the package MATLAB and MathCAD 12 Figures were introduced.

Fractional derivatives of multidimensional Colombeau generalized stochastic processes

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Danijela Rajter-Ćirić ◽  
Mirjana Stojanović
Keyword(s):  
Stochastic Process ◽  
Fractional Derivative ◽  
Stochastic Processes ◽  
Compact Support ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
One Dimensional ◽  
Compactly Supported ◽  
Derivatives Of ◽  
Do So

AbstractWe consider fractional derivatives of a Colombeau generalized stochastic process G defined on ℝn. We first introduce the Caputo fractional derivative of a one-dimensional Colombeau generalized stochastic process and then generalize the procedure to the Caputo partial fractional derivatives of a multidimensional Colombeau generalized stochastic process. To do so, the Colombeau generalized stochastic process G has to have a compact support. We prove that an arbitrary Caputo partial fractional derivative of a compactly supported Colombeau generalized stochastic process is a Colombeau generalized stochastic process itself, but not necessarily with a compact support.

scholarly journals Explicit Finite-Difference Scheme for the Numerical Solution of the Model Equation of Nonlinear Hereditary Oscillator with Variable-Order Fractional Derivatives

2016 ◽  
Vol 26 (3) ◽  
pp. 429-435 ◽  
Author(s):  
Roman I. Parovik
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Fractional Derivatives ◽  
Stability And Convergence ◽  
Variable Order ◽  
The Difference ◽  
Explicit Finite Difference ◽  
The Stability ◽  
Variable Order Fractional Derivatives

Abstract The paper deals with the model of variable-order nonlinear hereditary oscillator based on a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate the stability and convergence of the difference scheme. It is argued that the approximation, stability and convergence are of the first order, while the scheme is stable and converges to the exact solution.

scholarly journals Davey-Stewartson Equation with Fractional Coordinate Derivatives

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
H. Jafari ◽  
K. Sayevand ◽  
Yasir Khan ◽  
M. Nazari
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Stability And Convergence ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Good Agreement

We have used the homotopy analysis method (HAM) to obtain solution of Davey-Stewartson equations of fractional order. The fractional derivative is described in the Caputo sense. The results obtained by this method have been compared with the exact solutions. Stability and convergence of the proposed approach is investigated. The effects of fractional derivatives for the systems under consideration are discussed. Furthermore, comparisons indicate that there is a very good agreement between the solutions of homotopy analysis method and the exact solutions in terms of accuracy.

A Second-Order Finite Difference Method for Two-Dimensional Fractional Percolation Equations

2016 ◽  
Vol 19 (3) ◽  
pp. 733-757 ◽  
Author(s):  
Boling Guo ◽  
Qiang Xu ◽  
Ailing Zhu
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Fractional Derivatives ◽  
Second Order ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Mixed Fractional Derivatives ◽  
The Fourier Transform ◽  
Nicolson Scheme

AbstractA finite difference method which is second-order accurate in time and in space is proposed for two-dimensional fractional percolation equations. Using the Fourier transform, a general approximation for the mixed fractional derivatives is analyzed. An approach based on the classical Crank-Nicolson scheme combined with the Richardson extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Consistency, stability and convergence of the method are established. Numerical experiments illustrating the effectiveness of the theoretical analysis are provided.

scholarly journals A Discrete Method Based on the CE-SE Formulation for the Fractional Advection-Dispersion Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Silvia Jerez ◽  
Ivan Dzib
Keyword(s):  
Numerical Simulation ◽  
Analytical Solution ◽  
Fractional Derivative ◽  
Dispersion Equation ◽  
Numerical Algorithm ◽  
Initial Condition ◽  
Discrete Method ◽  
One Dimensional ◽  
Advection Dispersion ◽  
The One

We obtain a numerical algorithm by using the space-time conservation element and solution element (CE-SE) method for the fractional advection-dispersion equation. The fractional derivative is defined by the Riemann-Liouville formula. We prove that the CE-SE approximation is conditionally stable under mild requirements. A numerical simulation is performed for the one-dimensional case by considering a benchmark with a discontinuous initial condition in order to compare the results with the analytical solution.

scholarly journals Efficient and Stable Numerical Methods for Multi-Term Time Fractional Sub-Diffusion Equations

2014 ◽  
Vol 4 (3) ◽  
pp. 242-266 ◽  
Author(s):  
Jincheng Ren ◽  
Zhi-zhong Sun
Keyword(s):  
Fractional Derivatives ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Caputo Fractional Derivatives ◽  
Numerical Examples ◽  
One Dimensional ◽  
Compact Difference ◽  
Spatial Discretisation ◽  
The Stability

AbstractSome efficient numerical schemes are proposed for solving one-dimensional (1D) and two-dimensional (2D) multi-term time fractional sub-diffusion equations, combining the compact difference approach for the spatial discretisation and L1 approximation for the multi-term time Caputo fractional derivatives. The stability and convergence of these difference schemes are theoretically established. Several numerical examples are implemented, testifying to their efficiency and confirming their convergence order.

scholarly journals New Series Solution of the Caputo Fractional Ambartsumian Delay Differential Equationation by Mittag-Leffler Functions

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 157
Author(s):  
Weam Alharbi ◽  
Snezhana Hristova
Keyword(s):  
Exact Solution ◽  
Fractional Derivative ◽  
Decomposition Method ◽  
Series Solution ◽  
Fractional Derivatives ◽  
Adomian Decomposition Method ◽  
Delay Equation ◽  
Adomian Decomposition ◽  
The Laplace Transform ◽  
Delay Differential

The fractional generalization of the Ambartsumian delay equation with Caputo’s fractional derivative is considered. The Ambartsumian delay equation is very difficult to be solved neither in the case of ordinary derivatives nor in the case of fractional derivatives. In this paper we combine the Laplace transform with the Adomian decomposition method to solve the studied equation. The exact solution is obtained as a series which terms are expressed by the Mittag-Leffler functions. The advantage of the present approach over the known in the literature ones is discussed.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Fractional Derivative ◽  
Singular Integral ◽  
Fractional Derivatives ◽  
Continuous Functions ◽  
Finite Part ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Fundamental Properties ◽  
Logarithmic Series ◽  
The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

scholarly journals Fractional-Order Colour Image Processing

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 457
Author(s):  
Manuel Henriques ◽  
Duarte Valério ◽  
Paulo Gordo ◽  
Rui Melicio
Keyword(s):  
Image Processing ◽  
Edge Detection ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Grey Scale ◽  
Colour Image ◽  
Colour Image Processing ◽  
Image Processing Algorithms ◽  
Processing Algorithms

Many image processing algorithms make use of derivatives. In such cases, fractional derivatives allow an extra degree of freedom, which can be used to obtain better results in applications such as edge detection. Published literature concentrates on grey-scale images; in this paper, algorithms of six fractional detectors for colour images are implemented, and their performance is illustrated. The algorithms are: Canny, Sobel, Roberts, Laplacian of Gaussian, CRONE, and fractional derivative.

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