Fractional-order Euler functions for solving fractional integro-differential equations with weakly singular kernel

Wang, Yanxin; Zhu, Li; Zhi, Wang

doi:10.1186/s13662-018-1699-3

Cited by 17 publications

(10 citation statements)

References 33 publications

(41 reference statements)

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“…In order to compare with the second-kind Chebyshev polynomials method (SKCPM) in [8] and the fractional order Euler functions method (FEFsM) in [9], we computed the L 2 errors of the present scheme with various values of M and list the results in Table 2, where N denotes the maximal degree of the polynomials in the space spanned by all polynomials for SKCPM and FEFsM, and M indicates the truncation in the present scheme. The degree of all polynomials was no more than three when m = 4 in the present scheme.…”

Section: Examplementioning

confidence: 99%

“…There have been plenty of works, especially those considering real-world physical systems, where the Caputo-type fractional derivative and the Riemann-Liouville fractional integral have been used to describe complex dynamical systems. As most fractional order equations cannot be resolved analytically, numerical methods have been developed to give their numerical solutions [7][8][9][10]. In recent years, multi-term fractional order equations have received increasing attention due to their wider applicability.…”

Section: Introductionmentioning

confidence: 99%

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Quasilinearized Semi-Orthogonal B-Spline Wavelet Method for Solving Multi-Term Non-Linear Fractional Order Equations

Liu

Zhang

2020

Mathematics

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In the present article, we implement a new numerical scheme, the quasilinearized semi-orthogonal B-spline wavelet method, combining the semi-orthogonal B-spline wavelet collocation method with the quasilinearization method, for a class of multi-term non-linear fractional order equations that contain both the Riemann–Liouville fractional integral operator and the Caputo fractional differential operator. The quasilinearization method is utilized to convert the multi-term non-linear fractional order equation into a multi-term linear fractional order equation which, subsequently, is solved by means of semi-orthogonal B-spline wavelets. Herein, we investigate the operational matrix and the convergence of the proposed scheme. Several numerical results are delivered to confirm the accuracy and efficiency of our scheme.

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Section: Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quasilinearized Semi-Orthogonal B-Spline Wavelet Method for Solving Multi-Term Non-Linear Fractional Order Equations

Liu

Zhang

2020

Mathematics

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“…Euler polynomials : Here we recall definition and some properties of Euler polynomials [1]. The Euler polynomials of degree m are constructed from the following relation.…”

Section: Preliminariesmentioning

confidence: 99%

“…In 2016 a fractional extension of Euler polynomials is constructed for the solution of generalized Pantograph equations [10] and the same is used for the solution of the class of space fractional diffusion equation [12]. Recently in [1] Yanxin Wang used fractional order Euler functions(FEFs) for solving fractional integro-differential equations with weakly singular kernel.…”

Section: Introductionmentioning

confidence: 99%

Logarithmic fractional order Euler functions for solving Caputo type modified Hadamard fractional integro-differential equations

Shameema¹,

Ranjini²

2019

Malaya J. Mat.

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In this paper, a new set of logarithmic fractional order Euler functions(LFEFs) is constructed to obtain a numerical method for solving fractional integro-differential equation with kernel of the form log(x t). To do this an operational matrix of fractional order integration in the Hadamard sense for the LFEFs is derived. The properties and operational matrix of LFEFs is used to transform the problem of fractional integro-differential equation into a system of linear algebraic equations. The error analysis of approximation in terns of LFEFs is also given.

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“…Many researchers have attempted to solve IVP quickly and accurately using a variety of methods including the Euler method. To solve the fractional integro-differential equation with weakly singular kernel, a FO Euler function based on Euler polynomials has developed (Wang et al ., 2018). The Euler and Runge-kutta methods have been found to be compatible with solving third-order FrDEs (Mechee and H Aidi, 2022).…”

Section: Introductionmentioning

confidence: 99%

A robust study on fractional order HIV/AIDS model by using numerical methods

Roshan,

Ghosh,

Chauhan

et al. 2023

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PurposeThe fractional order HIV model has an important role in biological science. To study the HIV model in a better way, the model is presented with the help of Atangana- Baleanu operator which is in Caputo sense. Also, the characteristics of the solutions are described briefly with the help of the advance numerical techniques for the different values of fractional order derivatives. This paper aims to discuss the aforementioned objectives.Design/methodology/approachIn this work, Adams-Bashforth method and Euler method are used to get the solution of the HIV model. These are the important numerical methods. The comparison results also are described with the physical meaning of the solutions of the model.FindingsHIV model is analyzed under the view of fractional and AB derivative in Atangana-Baleanu-Caputo sense. The uniqueness of the solution is proved by using Banach Fixed point. The solution is derived with the help of Sumudu transform. Further, the authors employed fractional Adam-Bashforth method and Euler method to enumerate numerical results. The authors have used several values of fractional orders to present the outcomes graphically. The above calculations have been done with the help of MATLAB (R2016a). The numerical scheme used in the proposed study is valid and fruitful, and the same can be used to explore other real issues.Research limitations/implicationsThis investigation can be done for the real data sets.Practical implicationsThis paper aims to express the solution of the HIV model in a better way with the effect of non-locality, this work is very useful.Originality/valueIn this work, HIV model is developed with the help of Atangana- Baleanu operator in Caputo sense. By using Banach Fixed point, the authors proved that the solution is unique. Also, the solution is presented with the help of Sumudu transform. The behaviors of the solutions are checked for different values of fractional order derivatives with the physical meaning with help of the Adam-Bashforth method and the Euler method.

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Fractional-order Euler functions for solving fractional integro-differential equations with weakly singular kernel

Cited by 17 publications

References 33 publications

Quasilinearized Semi-Orthogonal B-Spline Wavelet Method for Solving Multi-Term Non-Linear Fractional Order Equations

Quasilinearized Semi-Orthogonal B-Spline Wavelet Method for Solving Multi-Term Non-Linear Fractional Order Equations

Logarithmic fractional order Euler functions for solving Caputo type modified Hadamard fractional integro-differential equations

A robust study on fractional order HIV/AIDS model by using numerical methods

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