An effective method for solving nonlinear fractional differential equations

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ORIGINAL ARTICLE

An effective method for solving nonlinear fractional differential equations Hoa T. B. Ngo1 · Thieu N. Vo1 · Mohsen Razzaghi2  Received: 30 April 2020 / Accepted: 8 August 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020

Abstract A new technique based on beta functions is applied to compute the exact formula for the Riemann–Liouville fractional integral of the fractional-order generalized Chelyshkov wavelets. An approximation method based on the wavelets is proposed to effectively solve nonlinear fractional differential equations. Illustrative examples show that the proposed method gives solutions with less errors in comparison with the previous methods. Keywords  Fractional differential equation · Fractional order · Chelyshkov wavelet · Regularized beta function

1 Introduction Fractional calculus (FC) and fractional differential equations (FDEs) have gone through a long history. Starting from the seventeenth century by famous mathematicians, such as G.W Leibniz (1695) and L. Euler (1730), FC and FDEs flourished through time and they still remain a focus of attention nowadays. Survey of the history of FC can be found in [1–5]. In recent decades, various fields of applied sciences, industrial and engineering problems have been investigated by means of FC and FDEs, for instance, in physics [6–8] chemistry [9], economics[10], biology [11], medicine[12] and optimal control [13, 14]. It is difficult, sometimes impossible, to calculate the exact solutions for FDEs. Therefore, it is essential to find numerical methods to approximate the solutions. Various methods for approximating the solutions of FDEs have been proposed. In recent decades, a collection of the numerical * Mohsen Razzaghi [email protected] Hoa T. B. Ngo [email protected] Thieu N. Vo [email protected] 1



Fractional Calculus, Optimization and Algebra Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam



Department of Mathematics and Statistics, Mississippi State University, Starkville, USA

2

methods used orthogonal functions and wavelets were introduced to solve different problems in fractional calculus, such as Legendre wavelets [15, 16], Chebyshev wavelets [17], CAS wavelets [18], Haar wavelets [19], and Chelyshkov wavelets [20–22]. In these methods, for solving the fractional calculus problems, the operational matrices, say P𝛽 , were used in the following approximation I 𝛽 Ψ(t) ≃ P𝛽 Ψ(t) , where I 𝛽 is the Riemann–Liouville fractional integral operator (RLFIO) and Ψ(t) is the used wavelet. To improve the efficiency of the method, the authors in [23, 24] determine the exact formula for the RLFIO of the hybrid of blockpulse function and Bernoulli polynomials and then used the formula to solve certain types of FDEs. The exact formulas for the RLFIO of the Taylor wavelets were also obtained in [25, 26]. Although classical integer-order polynomials and wavelet bases have good convergence behavior for the numerical solutions of differential and

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