Solving the Nonlinear Variable Order Fractional Differential Equations by Using Euler Wavelets

Abstract

An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper. The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented. Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations. And the convergence of the Euler wavelets basis is given. The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.Many phenomena in fluid mechanics, chemistry, physics, finance and other sciences can be described successfully by models using mathematical tools from fractional calculus, i.e. the theory of derivatives and integrals of fractional order [Miller and Ross (1993)]. Due to the fractional order exponents in differential operators, analytical solutions of fractional equations are usually difficult to obtain. Consequently, different methods have been developed to give numerical solutions for fractional equations, including fractional differential transform method [Wei and Chen (2014)], Adomian decomposition method [Song and Wang (2013)], Chebyshev pseudo-spectral method [Khader and Sweilam (2013)], Homotopy perturbation method [Abdulaziz, Hashim and Momani (2008)], Homotopy analysis method [Dehghan, Manafian and Saadatmandi (2010)], and wavelet method [Li and Zhao (2010); Wang and Fan (2012); Wang and Zhu (2016)]. Recently, the concepts of fractional derivatives of variable order have been introduced and some research works of the relative practical applications have arisen [Sun, Chen and Chen (2009); Samko (2013)]. Several numerical approximation methods are proposed to solve the variable order fractional differential equation [Chen, Liu, Li et al.

A Wavelet-Based Numerical Algorithm for Some Fractional Models

“…Using (20), (21) in (19), we have 𝐶𝜒(𝑡) − 𝜌(𝐶𝑃 𝜇 𝜒(𝑡) + 66 − 32) = 0. (22) We can attain the coefficient vector 𝐶 by solving the equation (22) at 𝑡 𝑖 = (2𝑖−1)𝑇…”

“…Recently, orthogonal wavelets have become more popular numerical techniques for solving differential and integral equations due to their excellent properties. Many researchers have employed the operational matrices of fractional integrations of Chebyshev wavelets [3], Haar wavelets [9], Müntz-Legendre wavelets [13], Bernoulli wavelets [12], Legendre wavelets [14,15,20], Taylor wavelets [19], Second kind Chebyshev wavelets [21] and Euler wavelets [22] to find the approximate solutions of arbitrary order differential and integral equations.…”

Variable-Order Conformable Fractional Derivatives using 2-stage Runge-Kutta and Euler Methods

“…As a result, it is necessitating the use of numerical approaches to comprehend the results of solutions to linear and nonlinear problems. There are a number of numerical approaches that have been presented to date, like: "Polynomial methods" [8][9][10], "Lagrange multipliers technique" [11], "collocation method" [12], "Galerkin finite element method" [13], "wavelet methods" [14], and another method using "artificial neural network" [15].…”

Wavelet methods for fractional electrical circuit equations

“…However, the same complex modelling feature renders finding the analytical solutions to those FDEs very difficult, and therefore it is crucial to have accurate numerical solutions. In recent years, certain numerical methods have been used for FDEs, such as spectral method [15], Galerkin method [16], Homotopy analysis method [17], Jacobi Tau method [18], waveform relaxation method [19], fractional finite difference method [20], Adomian decomposition method [21,22], Fourier transform [23], power series method [24], B-spline collocation method [25], block-by-block method [26], variational iteration method [27], polynomial methods [28][29][30][31][32] and wavelet methods [33][34][35][36][37][38][39]. Wavelets are mathematical functions that can be used to decompose data into various time-frequency components.…”