scholarly journals Shifted Jacobi polynomials for nonlinear singular variable-order time fractional Emden–Fowler equation generated by derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
M. H. Heydari ◽  
Z. Avazzadeh ◽  
A. Atangana
Keyword(s):  
Jacobi Polynomials ◽  
Operational Matrix ◽  
High Accuracy ◽  
Computational Method ◽  
Singular Kernel ◽  
Variable Order ◽  
Fractional Differentiation ◽  
Established Method ◽  
Shifted Jacobi Polynomials ◽  
Fowler Equation

AbstractIn this work, a nonlinear singular variable-order fractional Emden–Fowler equation involved with derivative with non-singular kernel (in the Atangana–Baleanu–Caputo type) is introduced and a computational method is proposed for its numerical solution. The desired method is established upon the shifted Jacobi polynomials and their operational matrix of variable-order fractional differentiation (which is extracted in the present study) together with the spectral collocation method. The presented method transforms obtaining the solution of the main problem into obtaining the solution of an algebraic system of equations. Several numerical examples are examined to show the validity and the high accuracy of the established method.

Modified wavelet method for solving fractional variational problems

2020 ◽  
pp. 107754632093202
Author(s):  
Haniye Dehestani ◽  
Yadollah Ordokhani ◽  
Mohsen Razzaghi
Keyword(s):  
Error Analysis ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variational Problems ◽  
High Accuracy ◽  
Construction Method ◽  
Computational Method ◽  
Wavelet Method ◽  
Fractional Variational Problems ◽  
Operational Matrix Of Integration

In this article, a newly modified Bessel wavelet method for solving fractional variational problems is considered. The modified operational matrix of integration based on Bessel wavelet functions is proposed for solving the problems. In the process of computing this matrix, we have tried to provide a high-accuracy operational matrix. We also introduce the pseudo-operational matrix of derivative and the dual operational matrix with the coefficient. Also, we investigate the error analysis of the computational method. In the examples section, the behavior of the approximate solutions with respect to various parameters involved in the construction method is tested to illustrate the efficiency and accuracy of the proposed method.

Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Eid Doha ◽  
Ali Bhrawy ◽  
Samer Ezz-Eldien
Keyword(s):  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Diffusion ◽  
High Accuracy ◽  
Variable Coefficients ◽  
Diffusion Equations ◽  
Algebraic Equations ◽  
Fractional Differentiation ◽  
Fractional Diffusion Equations ◽  
General Method

AbstractIn this paper, a class of fractional diffusion equations with variable coefficients is considered. An accurate and efficient spectral tau technique for solving the fractional diffusion equations numerically is proposed. This method is based upon Chebyshev tau approximation together with Chebyshev operational matrix of Caputo fractional differentiation. Such approach has the advantage of reducing the problem to the solution of a system of algebraic equations, which may then be solved by any standard numerical technique. We apply this general method to solve four specific examples. In each of the examples considered, the numerical results show that the proposed method is of high accuracy and is efficient for solving the time-dependent fractional diffusion equations.

scholarly journals Shifted Fractional-Order Jacobi Collocation Method for Solving Variable-Order Fractional Integro-Differential Equation with Weakly Singular Kernel

2021 ◽  
Vol 6 (1) ◽  
pp. 19
Author(s):  
Mohamed A. Abdelkawy ◽  
Ahmed Z. M. Amin ◽  
António M. Lopes ◽  
Ishak Hashim ◽  
Mohammed M. Babatin
Keyword(s):  
Collocation Method ◽  
Fractional Order ◽  
Jacobi Polynomials ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
Singular Kernel ◽  
Variable Order ◽  
Algebraic Equations ◽  
Weakly Singular Kernel ◽  
Weakly Singular

We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fractional integral and derivative and fractional-order shifted Jacobi polynomials, the approximate solutions of VO-FIDE-WSK are derived by solving systems of algebraic equations. The superior accuracy of the method is illustrated through several numerical examples.

A Robust Algorithm for Nonlinear Variable-Order Fractional Control Systems with Delay

2018 ◽  
Vol 19 (3-4) ◽  
pp. 231-238 ◽  
Author(s):  
José António Tenreiro Machado ◽  
Behrouz Parsa Moghaddam
Keyword(s):  
Control Systems ◽  
High Accuracy ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Variable Order ◽  
The Novel ◽  
Fractional Differentiation ◽  
Nonlinear Feedback ◽  
The Stability ◽  
Fractional Control

AbstractIn this paper, we propose a high-accuracy linear B-spline finite-difference approximation for variable-order (VO) derivative. We consider VO fractional differentiation as a control parameter for improving the stability in systems exhibiting vibrations. The method is applied to nonlinear feedback with VO fractional derivative. The results demonstrate the efficiency and high accuracy of the novel algorithm.

scholarly journals Numerical Solution of Variable-Order Fractional Differential Equations Using Bernoulli Polynomials

2021 ◽  
Vol 5 (4) ◽  
pp. 219
Author(s):  
Somayeh Nemati ◽  
Pedro M. Lima ◽  
Delfim F. M. Torres
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Unknown Function ◽  
Operational Matrix ◽  
Bernoulli Polynomials ◽  
High Accuracy ◽  
Basis Functions ◽  
Variable Order ◽  
Algebraic Equations ◽  
Fractional Differential

We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the Riemann–Liouville integral operator was used to give approximations for the unknown function and its variable-order derivatives. An operational matrix of variable-order fractional integration was introduced for the Bernoulli functions. By assuming that the solution of the problem is sufficiently smooth, we approximated a given order of its derivative using Bernoulli polynomials. Then, we used the introduced operational matrix to find some approximations for the unknown function and its derivatives. Using these approximations and some collocation points, the problem was reduced to the solution of a system of nonlinear algebraic equations. An error estimate is given for the approximate solution obtained by the proposed method. Finally, five illustrative examples were considered to demonstrate the applicability and high accuracy of the proposed technique, comparing our results with the ones obtained by existing methods in the literature and making clear the novelty of the work. The numerical results showed that the new method is efficient, giving high-accuracy approximate solutions even with a small number of basis functions and when the solution to the problem is not infinitely differentiable, providing better results and a smaller number of basis functions when compared to state-of-the-art methods.

scholarly journals Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 269-280 ◽  
Author(s):  
M.A. Abdelkawy ◽  
Engy A. Ahmed ◽  
Rubayyi T. Alqahtani
Keyword(s):  
Numerical Algorithm ◽  
Jacobi Polynomials ◽  
High Accuracy ◽  
Basis Functions ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
One Dimensional ◽  
Numerical Tests ◽  
Shifted Jacobi Polynomials ◽  
Derivatives Of

AbstractWe introduce a new numerical algorithm for solving one-dimensional time-fractional Tricomi-type equations (T-FTTEs). We used the shifted Jacobi polynomials as basis functions and the derivatives of fractional is evaluated by the Caputo definition. The shifted Jacobi Gauss-Lobatt algorithm is used for the spatial discretization, while the shifted Jacobi Gauss-Radau algorithmis applied for temporal approximation. Substituting these approximations in the problem leads to a system of algebraic equations that greatly simplifies the problem. The proposed algorithm is successfully extended to solve the two-dimensional T-FTTEs. Extensive numerical tests illustrate the capability and high accuracy of the proposed methodologies.

scholarly journals New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1573
Author(s):  
Waleed Mohamed Abd-Elhameed ◽  
Badah Mohamed Badah
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Tau Method ◽  
Riccati Differential Equation ◽  
Shifted Jacobi Polynomials ◽  
Linearization Problem ◽  
Linearization Coefficients ◽  
New Formula ◽  
Connection Formulas

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

New Operational Matrix for Solving Multiterm Variable Order Fractional Differential Equations

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
A. M. Nagy ◽  
N. H. Sweilam ◽  
Adel A. El-Sayed
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Algebraic System ◽  
Order Differential Equation ◽  
Fundamental Problem ◽  
Operational Matrix ◽  
Variable Order ◽  
Engineering Problems ◽  
Fractional Differential ◽  
Fourth Kind

The multiterm fractional variable-order differential equation has a massive application in physics and engineering problems. Therefore, a numerical method is presented to solve a class of variable order fractional differential equations (FDEs) based on an operational matrix of shifted Chebyshev polynomials of the fourth kind. Utilizing the constructed operational matrix, the fundamental problem is reduced to an algebraic system of equations which can be solved numerically. The error estimate of the proposed method is studied. Finally, the accuracy, applicability, and validity of the suggested method are illustrated through several examples.

An Efficient Operational Matrix Technique for Multidimensional Variable-Order Time Fractional Diffusion Equations

2016 ◽  
Vol 11 (6) ◽  
Author(s):  
M. A. Zaky ◽  
S. S. Ezz-Eldien ◽  
E. H. Doha ◽  
J. A. Tenreiro Machado ◽  
A. H. Bhrawy
Keyword(s):  
Present Method ◽  
Operational Matrix ◽  
Solution Process ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Variable Order ◽  
Algebraic Equations ◽  
Global Approximation ◽  
Fractional Diffusion Equations ◽  
Accurate Numerical Algorithm

This paper derives a new operational matrix of the variable-order (VO) time fractional partial derivative involved in anomalous diffusion for shifted Chebyshev polynomials. We then develop an accurate numerical algorithm to solve the 1 + 1 and 2 + 1 VO and constant-order fractional diffusion equation with Dirichlet conditions. The contraction of the present method is based on shifted Chebyshev collocation procedure in combination with the derived shifted Chebyshev operational matrix. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations, and it reduces such problems to those of solving a system of algebraic equations, which greatly simplifies the solution process. In addition, we analyze the convergence of the present method graphically. Finally, comparisons between the algorithm derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones.

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