Numerical solution of fractional Mathieu equations by using block-pulse wavelets

Pirmohabbati, P.; Sheikhani, A. H. Refahi; Najafi, H. Saberi; Ziabari, Ali Abdolahzadeh

doi:10.1016/j.joes.2019.05.005

Cited by 13 publications

(11 citation statements)

References 11 publications

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“…, which is superior to that of BPFs approximation [14]. This is also true for the upper error bound given in Eq.…”

Section: Remark 53 Eq 51 Shows That the Upper Error Bound Between mentioning

confidence: 60%

“…Orthogonal functions or polynomials, such as Haar wavelets [13], block-pulse functions (BPFs) [14], shifted Legendre polynomials [15] and Genocchi polynomials [16] have been widely adopted to handle different kinds of FDEs. Their foundation is to convert the FDE to a system of algebraic equations through an operational matrix, thus simplifying the problem.…”

Section: Introductionmentioning

confidence: 99%

“…Among these functions or polynomials, BPFs are piecewise constants and orthogonal in the interval [0, T ), and can approximate any square integrable functions in [0, T ) with arbitrary precision. BPFs have been utilized to solve fractional partial differential equation [17], nonlinear system of fractional integral-differential equations [18] and fractional Mathieu equation [14]. However, BPFs are not smooth enough.…”

Section: Introductionmentioning

confidence: 99%

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Solving fractional differential equations using collocation method based onhybrid of block-pulse functions and Taylor polynomials

Lü¹,

Tang²

2021

Turk J Math

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In this paper, a novel approach is proposed to solve fractional differential equations (FDEs) based on hybrid functions. The hybrid functions consist of block-pulse functions and Taylor polynomials. The exact formula for the Riemann-Liouville fractional integral of the hybrid functions is derived via Laplace transform. The FDE under consideration is converted into an algebraic equation with undetermined coefficients by using this formula. A set of linear or nonlinear equations are obtained through collocating the algebraic equation at Newton-Cotes nodes. The numerical solution of the FDE is achieved by solving the linear or nonlinear equations. Error analysis is performed on the proposed method. Several numerical examples are given, and the results have proven that the proposed method is effective.

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“…, which is superior to that of BPFs approximation [14]. This is also true for the upper error bound given in Eq.…”

Section: Remark 53 Eq 51 Shows That the Upper Error Bound Between mentioning

confidence: 60%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Solving fractional differential equations using collocation method based onhybrid of block-pulse functions and Taylor polynomials

Lü¹,

Tang²

2021

Turk J Math

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“…Wavelets have been used for solving linear and nonlinear ordinary and partial differential equations since the 1980s. The interesting features in this method are the possibility to find‐out singularities, irregular structure, and transient phenomena exhibited by the analysed equations such as wavelets method for solving systems of nonlinear singular fractional Volterra integro‐differential equations (see, eg, Wang & Fan, Heydari et al, Pirmohabbati et al, and Rehman & Khan ).…”

Section: Introductionmentioning

confidence: 99%

A study of fractional Lotka‐Volterra population model using Haar wavelet and Adams‐Bashforth‐Moulton methods

Kumar

Agarwal

et al. 2020

Math Methods in App Sciences

285

123

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The Lotka-Volterra (LV) system is an interesting mathematical model because of its significant and wide applications in biological sciences and ecology. A fractional LV model in the Caputo sense is investigated in this paper. Namely, we provide a comparative study of the considered model using Haar wavelet and Adams-Bashforth-Moulton methods. For the first method, the Haar wavelet operational matrix of the fractional order integration is derived and used to solve the fractional LV model. The main characteristic of the operational method is to convert the considered model into an algebraic equation which is easy to solve.To demonstrate the efficiency and accuracy of the proposed methods, some numerical tests are provided. KEYWORDS Adams-Bashforth-Moulton method, fractional LV model, Haar wavelet method, operational matrix MSC CLASSIFICATION 26A33; 34A08; 34A34 Math Meth Appl Sci. 2020;43:5564-5578. wileyonlinelibrary.com/journal/mma

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“…The dynamics of Mathieu equation with two kinds of van der Pol (VDP) fractional-order terms is investigated by Wen et al [19]. Recently the Block-Pulse wavelets approximation method [20] was used by Pirmohabbati et al for numerical solution of fractional Mathieu equation.…”

Section: Introductionmentioning

confidence: 99%

Numerical Study of Fractional Mathieu Differential Equation Using Radial Basis Functions

Ghorbani¹,

Mahmoudi²,

Saei³

2020

MMEP

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In this paper, we introduce a method based on Radial Basis Functions (RBFs) for the numerical approximation of Mathieu differential equation with two fractional derivatives in the Caputo sense. For this, we suggest a numerical integration method for approximating the improper integrals with a singularity point at the right end of the integration domain, which appear in the fractional computations. We study numerically the affects of characteristic parameters and damping factor on the behavior of solution for fractional Mathieu differential equation. Some examples are presented to illustrate applicability and accuracy of the proposed method. The fractional derivatives order and the parameters of the Mathieu equation are changed to study the convergency of the numerical solutions.

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Numerical solution of fractional Mathieu equations by using block-pulse wavelets

Cited by 13 publications

References 11 publications

Solving fractional differential equations using collocation method based onhybrid of block-pulse functions and Taylor polynomials

Solving fractional differential equations using collocation method based onhybrid of block-pulse functions and Taylor polynomials

A study of fractional Lotka‐Volterra population model using Haar wavelet and Adams‐Bashforth‐Moulton methods

Numerical Study of Fractional Mathieu Differential Equation Using Radial Basis Functions

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