Numerical treatment for studying the blood ethanol concentration systems with different forms of fractional derivatives

2020 ◽  
Vol 31 (03) ◽  
pp. 2050044 ◽  
Author(s):  
M. M. Khader ◽  
Khaled. M. Saad
Keyword(s):  
Fractional Derivatives ◽  
Ethanol Concentration ◽  
Algebraic Equations ◽  
Efficient Tool ◽  
Blood Ethanol Concentration ◽  
Fractional Models ◽  
Fractional Differential ◽  
Relative Errors ◽  
The Given ◽  
First Time

The purpose of this paper is to implement an approximate method for obtaining the solution of a physical model called the blood ethanol concentration system. This model can be expressed by a system of fractional differential equations (FDEs). Here, we will consider two forms of the fractional derivative namely, Caputo (with singular kernel) and Atangana–Baleanu–Caputo (ABC) (with nonsingular kernel). In this work, we use the spectral collocation method based on Chebyshev approximations of the third-kind. This procedure converts the given model to a system of algebraic equations. The implementation of the proposed method to solve fractional models in ABC-sense is the first time. We satisfy the efficiency and the accuracy of the given procedure by evaluating the relative errors. The results show that the implemented technique is an easy and efficient tool to simulate the solution of such models.

scholarly journals Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 96 ◽  
Author(s):  
İbrahim Avcı ◽  
Nazim I. Mahmudov
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Main Idea ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Fractional Order Differential Equations ◽  
Fractional Differential ◽  
The Given

In this article, we propose a numerical method based on the fractional Taylor vector for solving multi-term fractional differential equations. The main idea of this method is to reduce the given problems to a set of algebraic equations by utilizing the fractional Taylor operational matrix of fractional integration. This system of equations can be solved efficiently. Some numerical examples are given to demonstrate the accuracy and applicability. The results show that the presented method is efficient and applicable.

scholarly journals A multiple-step adaptive pseudospectral method for solving multi-order fractional differential equations

2019 ◽  
Vol 8 (1) ◽  
pp. 702-718
Author(s):  
Mahmoud Mashali-Firouzi ◽  
Mohammad Maleki
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Pseudospectral Method ◽  
Algebraic Equations ◽  
Computational Accuracy ◽  
Step Size ◽  
Collocation Points ◽  
Fractional Differential ◽  
Multiple Step

Abstract The nonlocal nature of the fractional derivative makes the numerical treatment of fractional differential equations expensive in terms of computational accuracy in large domains. This paper presents a new multiple-step adaptive pseudospectral method for solving nonlinear multi-order fractional initial value problems (FIVPs), based on piecewise Legendre–Gauss interpolation. The fractional derivatives are described in the Caputo sense. We derive an adaptive pseudospectral scheme for approximating the fractional derivatives at the shifted Legendre–Gauss collocation points. By choosing a step-size, the original FIVP is replaced with a sequence of FIVPs in subintervals. Then the obtained FIVPs are consecutively reduced to systems of algebraic equations using collocation. Some error estimates are investigated. It is shown that in the present multiple-step pseudospectral method the accuracy of the solution can be improved either by decreasing the step-size or by increasing the number of collocation points within subintervals. The main advantage of the present method is its superior accuracy and suitability for large-domain calculations. Numerical examples are given to demonstrate the validity and high accuracy of the proposed technique.

scholarly journals Numerical solutions of multi-order fractional differential equations by Boubaker polynomials

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 226-230 ◽  
Author(s):  
A. Bolandtalat ◽  
E. Babolian ◽  
H. Jafari
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Boubaker Polynomials ◽  
Fractional Differential ◽  
The Given

AbstractIn this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on Boubaker polynomials. Using this operational matrix, the given problem is converted into a set of algebraic equations. Illustrative examples are are given to demonstrate the efficiency and simplicity of this technique.

Highly accurate technique for studying some chaotic models described by ABC-fractional differential equations of variable-order

2020 ◽  
pp. 2150018
Author(s):  
M. M. Khader ◽  
Ibrahim Al-Dayel
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Polynomial Interpolation ◽  
Numerical Technique ◽  
Variable Order ◽  
Lagrange Polynomial ◽  
Efficient Tool ◽  
Fractional Differential ◽  
The Given

The propose of this paper is to introduce and investigate a highly accurate technique for solving the fractional Logistic and Ricatti differential equations of variable-order. We consider these models with the most common nonsingular Atangana–Baleanu–Caputo (ABC) fractional derivative which depends on the Mittag–Leffler kernel. The proposed numerical technique is based upon the fundamental theorem of the fractional calculus as well as the Lagrange polynomial interpolation. We satisfy the efficiency and the accuracy of the given procedure; and study the effect of the variation of the fractional-order [Formula: see text] on the behavior of the solutions due to the presence of ABC-operator by evaluating the solution with different values of [Formula: see text]. The results show that the given procedure is an easy and efficient tool to investigate the solution for such models. We compare the numerical solutions with the exact solution, thereby showing excellent agreement which we have found by applying the ABC-derivatives. We observe the chaotic solutions with some fractional-variable-order functions.

Numerical study of the nanofluid thin film flow past an unsteady stretching sheet with fractional derivatives using the spectral collocation Chebyshev approximation

2020 ◽  
pp. 2150026
Author(s):  
M. M. Khader
Keyword(s):  
Thin Film ◽  
Fractional Derivatives ◽  
Volume Fraction ◽  
Numerical Study ◽  
Chebyshev Approximation ◽  
Algebraic Equations ◽  
Thin Film Flow ◽  
Spectral Collocation ◽  
Unsteady Stretching Surface ◽  
Fractional Differential

In this work, a mathematical model of fractional-order in fluid will be analyzed numerically to describe and study the influence of thermal radiation on the magnetohydrodynamic flow of nanofluid thin film which moves due to the unsteady stretching surface with viscous dissipation. The set of nonlinear fractional differential equations in the form of velocity, temperature and concentration which describe our proposed problem are tackled through the spectral collocation method based on Chebyshev polynomials of the third-kind. This method reduces the presented model to a system of algebraic equations. The effect of the influence parameters which governs the process of flow and mass heat transfer is discussed. The numerical values of the dimensionless velocity, temperature and concentration are depicted graphically. Also, computations of the values of skin-friction, Nusselt number and Sherwood number have been carried out and presented in the same figures. Finally, our numerical analysis shows that both the magnetic and the unsteadiness parameters can enhance the free surface temperature and nanoparticle volume fraction.

Numerical solution of fractional differential equations via a Volterra integral equation approach

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Shahrokh Esmaeili ◽  
Mostafa Shamsi ◽  
Mehdi Dehghan
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Volterra Integral Equation ◽  
Equation Approach ◽  
Algebraic Equations ◽  
Integral Equation Approach ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
The Given

AbstractThe main focus of this paper is to present a numerical method for the solution of fractional differential equations. In this method, the properties of the Caputo derivative are used to reduce the given fractional differential equation into a Volterra integral equation. The entire domain is divided into several small domains, and by collocating the integral equation at two adjacent points a system of two algebraic equations in two unknowns is obtained. The method is applied to solve linear and nonlinear fractional differential equations. Also the error analysis is presented. Some examples are given and the numerical simulations are also provided to illustrate the effectiveness of the new method.

On the operational solutions of fuzzy fractional differential equations

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Djurdjica Takači ◽  
Arpad Takači ◽  
Aleksandar Takači
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Software Package ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Fractional Differential ◽  
Definition Of ◽  
The Given ◽  
Fuzzy Fractional Differential Equations

AbstractFuzzy fractional differential equations with fuzzy coefficients are analyzed in the frame of Mikusiński operators. Systems of fuzzy operational algebraic equations are obtained, in view of the definition of fuzzy derivatives. Their exact and approximate solutions are constructed and their characters are analyzed, considering them as the corresponding solutions of the given problem. The described procedure of the construction of solutions is illustrated on an example and the obtained approximate solutions of the considered problems are visualized by using the GeoGebra software package.

scholarly journals Genocchi Wavelet-like Operational Matrix and its Application for Solving Non-linear Fractional Differential Equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 463-472 ◽  
Author(s):  
Abdulnasir Isah ◽  
Chang Phang
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Fractional Differential ◽  
First Time ◽  
Linear Fractional Differential Equations

AbstractIn this work, we propose a new operational method based on a Genocchi wavelet-like basis to obtain the numerical solutions of non-linear fractional order differential equations (NFDEs). To the best of our knowledge this is the first time a Genocchi wavelet-like basis is presented. The Genocchi wavelet-like operational matrix of a fractional derivative is derived through waveletpolynomial transformation. These operational matrices are used together with the collocation method to turn the NFDEs into a system of non-linear algebraic equations. Error estimates are shown and some illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed technique.

scholarly journals Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1979
Author(s):  
Ravi P. Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan ◽  
Ricardo Almeida
Keyword(s):  
Fractional Derivatives ◽  
Finite Interval ◽  
Iterative Scheme ◽  
Mild Solutions ◽  
Initial Value ◽  
Monotone Iterative ◽  
Fractional Differential ◽  
The One ◽  
The Right ◽  
The Given

The main aim of the paper is to present an algorithm to solve approximately initial value problems for a scalar non-linear fractional differential equation with generalized proportional fractional derivative on a finite interval. The main condition is connected with the one sided Lipschitz condition of the right hand side part of the given equation. An iterative scheme, based on appropriately defined mild lower and mild upper solutions, is provided. Two monotone sequences, increasing and decreasing ones, are constructed and their convergence to mild solutions of the given problem is established. In the case of uniqueness, both limits coincide with the unique solution of the given problem. The approximate method is based on the application of the method of lower and upper solutions combined with the monotone-iterative technique.

scholarly journals A collocation method based on cubic trigonometric B-splines for the numerical simulation of the time-fractional diffusion equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Muhammad Yaseen ◽  
Muhammad Abbas ◽  
Muhammad Bilal Riaz
Keyword(s):  
Collocation Method ◽  
Fractional Derivatives ◽  
Fractional Diffusion ◽  
Biological Models ◽  
B Splines ◽  
Fractional Diffusion Equations ◽  
Numerical Tests ◽  
Symmetry Properties ◽  
Fractional Differential ◽  
The Given

AbstractFractional differential equations sufficiently depict the nature in view of the symmetry properties, which portray physical and biological models. In this paper, we present a proficient collocation method based on cubic trigonometric B-Splines (CuTBSs) for time-fractional diffusion equations (TFDEs). The methodology involves discretization of the Caputo time-fractional derivatives using the typical finite difference scheme with space derivatives approximated using CuTBSs. A stability analysis is performed to establish that the errors do not magnify. A convergence analysis is also performed The numerical solution is obtained as a piecewise sufficiently smooth continuous curve, so that the solution can be approximated at any point in the given domain. Numerical tests are efficiently performed to ensure the correctness and viability of the scheme, and the results contrast with those of some current numerical procedures. The comparison uncovers that the proposed scheme is very precise and successful.

Sign in / Sign up

Export Citation Format

Share Document