Analytical numerical solutions of the fractional multi-pantograph system: Two attractive methods and comparisons

El-Ajou, Ahmad; Oqielat, Moa’ath N.; Al–Zhour, Zeyad; Momani, Shaher

doi:10.1016/j.rinp.2019.102500

Cited by 37 publications

(19 citation statements)

References 24 publications

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“…This section extends the fractional power series that discussed in the references [14,[16][17][18] to the matrix case, as general, and establish some new nice results related to the convergent and radii of convergence for MFPS in Caputo sense. In addition, we study some important definitions and theorems which are very useful to investigate our results in the approximation of the matrix fractional derivative in Caputo sense and fractional integral in Riemann-Liouville sense and also in the solutions of some MFDEs as in Sections 3 and 4.…”

Section: Matrix Fractional Power Series (Mfps)mentioning

confidence: 62%

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Taylor's expansion for fractional matrix functions: theory and applications

El-Ajou¹

2020

J. Math. Computer Sci.

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In this paper, several aims and tasks have been accomplished that can be summarized in the following points. Firstly, we recover some nice results related to the convergence and radii of convergence for the matrix fractional power series formula. Secondly, the Frobenius norm approximations for the matrix fractional derivatives in Caputo sense and fractional integrals in Riemann-Liouville sense are presented. Thirdly, we present the general exact and numerical solutions of four important and interesting matrix fractional differential equations and a new computational technique is also applied for getting the general solutions of the non-linear case in Caputo sense. Finally, some illustrated examples and special cases are also given and considered to show our new approach.

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Section: Matrix Fractional Power Series (Mfps)mentioning

confidence: 62%

“…Many of the properties of the previous definitions exist in the references [6,9,14,15,18,31,[33][34][35]37]. The most important of these properties that we will need during this work can be summarized in the following lemma.…”

Section: Introductionmentioning

confidence: 99%

Taylor's expansion for fractional matrix functions: theory and applications

El-Ajou¹

2020

J. Math. Computer Sci.

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“…In the last five years, the residual power series method (RPSM) has achieved an advanced rank among the methods used to find ASs for many fractional differential and integral equations. It has been used in determining ESs and ASs for many equations such as homogeneous and non-homogeneous time-and space-fractional telegraph equation [23], timefractional Boussinesq-type and space-fractional Klein-Gordon-type equations [24], fractional multipantograph system [25], space-and time-fractional linear and nonlinear KdV-Burgers equation [26], multi-energy groups of neutron diffusion equations [27], and other equations. The RPSM is characterized by its ease and speed in finding solutions for equations in the form of a power series.…”

Section: Introductionmentioning

confidence: 99%

A Vector Series Solution for a Class of Hyperbolic System of Caputo Time-Fractional Partial Differential Equations With Variable Coefficients

El-Ajou

Al–Zhour

2021

Front. Phys.

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In this paper, we introduce a series solution to a class of hyperbolic system of time-fractional partial differential equations with variable coefficients. The fractional derivative has been considered by the concept of Caputo. Two expansions of matrix functions are proposed and used to create series solutions for the target problem. The first one is a fractional Laurent series, and the second is a fractional power series. A new approach, via the residual power series method and the Laplace transform, is also used to find the coefficients of the series solution. In order to test our proposed method, we discuss four interesting and important applications. Numerical results are given to authenticate the efficiency and accuracy of our method and to test the validity of our obtained results. Moreover, solution surface graphs are plotted to illustrate the effect of fractional derivative arrangement on the behavior of the solution.

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“…Recently, due to the importance of MDEs, the solving of these equations have been frequently considered in the literature and many numerical and analytical methods have been presented [ 6 ]. For example, in References [ 7 , 8 , 9 ], a homotopy approach and power series are developed for solving linear MDEs and coinciding of the estimated solution with the exact solution is investigated. In Reference [ 10 ], the spectral tau method is studied and the convergence of the presented approach is investigated by

norm.…”

Section: Introductionmentioning

confidence: 99%

Machine Learning for Modeling the Singular Multi-Pantograph Equations

Mosavi

Shokri

Mansor

et al. 2020

Entropy

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In this study, a new approach to basis of intelligent systems and machine learning algorithms is introduced for solving singular multi-pantograph differential equations (SMDEs). For the first time, a type-2 fuzzy logic based approach is formulated to find an approximated solution. The rules of the suggested type-2 fuzzy logic system (T2-FLS) are optimized by the square root cubature Kalman filter (SCKF) such that the proposed fineness function to be minimized. Furthermore, the stability and boundedness of the estimation error is proved by novel approach on basis of Lyapunov theorem. The accuracy and robustness of the suggested algorithm is verified by several statistical examinations. It is shown that the suggested method results in an accurate solution with rapid convergence and a lower computational cost.

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Analytical numerical solutions of the fractional multi-pantograph system: Two attractive methods and comparisons

Cited by 37 publications

References 24 publications

Taylor's expansion for fractional matrix functions: theory and applications

Taylor's expansion for fractional matrix functions: theory and applications

A Vector Series Solution for a Class of Hyperbolic System of Caputo Time-Fractional Partial Differential Equations With Variable Coefficients

Machine Learning for Modeling the Singular Multi-Pantograph Equations

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