Numerical Solution of Fractional Differential Equations Using Haar Wavelet Operational Matrix Method

Green–Haar wavelets method for generalized fractional differential equations

“…By solving the linear system in (27), we can determine the vector C t , and putting this vector into equation 23, u (n) (t) can be calculated for each t ∈ [a, b]. Now assuming x i ∈ [a, b] and u (n) (x i ) are calculated for i = 1, 2, 3, .…”

“…Wavelets are being used for analyzing signals, for representation of waveform and segmentation, optimal control, numerical analysis, fast algorithm for easy implementation, and time-frequency analysis [26]. There are many kinds of wavelets, for example, Haar [27][28][29][30], Daubechies [31], B-spline [32], Battle-Lemarie [33], Legender [34], as well as Green-CAS [30]. A naive form of orthonormal wavelets which employ compact support has been used by many researchers and is called the Haar wavelet.…”

Generalized wavelet quasilinearization method for solving population growth model of fractional order

“…During the last decade or so, the operational matrices of integration based on Haar wavelets, Legendre wavelets, Chebyshev wavelets, CAS wavelets, Bernoulli wavelets, Gegenbauer wavelets, fractional wavelets, wavelet frames, and the spline wavelets have been developed in order to solve a wide variety of differential, integral, and integro-differential equations. [14][15][16][17][18] The Haar wavelets are a specific kind of compactly supported wavelets generated by the combined action of dyadic dilations and integer translations of a rectangular pulse wave, and these wavelets have gained prominence among researchers due to their simple and lucid structure. Moreover, these wavelets can be integrated analytically in arbitrary times and permit straight inclusion of the different types of boundary conditions in the numerical algorithms.…”

An approach based on Haar wavelet for the approximation of fractional calculus with application to initial and boundary value problems

“…Heydari et al 24 derived the general formulation for the Legendre and Chebyshev wavelet operational matrices of fractional‐order integration to solve the multi‐order fractional differential equations, see also Singh and Mehra 25 . Shah et al 26 derived the Haar wavelet operational matrices of fractional‐order integration without using block pulse functions and used it for the numerical solution of fractional differential equations.…”

“…Motivated by above works, in this paper, we propose the numerical approximation of fractional initial and boundary value problems using Haar wavelet. The proposed work has following advantages in comparison to the abovementioned works using the Haar wavelet: Unlike above approaches, 22,26 where fractional derivative of the function was expressed in terms of Haar wavelet basis functions, we approximate the function and its classical derivatives, that is, f ( n ) ( x ) by the Haar wavelet basis function and use it to find an approximation to the fractional derivative which then leads to the approximation of fractional differential equations. In the solution process, we do not require the inverse of the Haar matrix for the approximation of function which effectively results in reduced computation costs of our process. The error bounds in the approximation of fractional integrals and fractional derivatives are derived, which depend on the index J of the approximation space V J and on the fractional order α . …”