Green–Haar wavelets method for generalized fractional differential equations

Abstract

The objective of this paper is to present two numerical techniques for solving generalized fractional differential equations. We develop Haar wavelets operational matrices to approximate the solution of generalized Caputo–Katugampola fractional differential equations. Moreover, we introduce Green–Haar approach for a family of generalized fractional boundary value problems and compare the method with the classical Haar wavelets technique. In the context of error analysis, an upper bound for error is established to show the convergence of the method. Results of numerical experiments have been documented in a tabular and graphical format to elaborate the accuracy and efficiency of addressed methods. Further, we conclude that accuracy-wise Green–Haar approach is better than the conventional Haar wavelets approach as it takes less computational time compared to the Haar wavelet method.

Green-Haar method for fractional partial differential equations

“…In this work, a method depended on the two-dimensional Haar wavelet is proposed, called Green-Haar technique. This method extends the Green-Haar method developed in Rehman et al (2019) for the numerical solutions of fractional ordinary differential equations. Green-Haar technique is used for solving fractional partial differential equations subject to the initial and boundary conditions.…”

The existence of nonnegative solutions for a nonlinear fractional q-differential problem via a different numerical approach

“…In 2019, Ren and Zhai [ 27 ] discussed the existence of a unique solution and multiple positive solutions for the fractional q -differential equation for each with nonlocal boundary conditions and where is the standard Riemann–Liouville fractional q -derivative of order α such that and , , is nonnegative, is a linear functional given by involving the Stieltjes integral with respect to a nondecreasing function such that is right-continuous on , left-continuous at , , and is a positive Stieltjes measure. Rehman et al [ 28 ] developed Haar wavelets operational matrices to approximate the solution of generalized Caputo–Katugampola fractional differential equations. They introduced the Green–Haar approach for a family of generalized fractional boundary value problems and compared the method with the classical Haar wavelets technique.…”

“…where D α q is the standard Riemann-Liouville fractional q-derivative of order α such that 2 < α ≤ 3 and α -1β > 0, q ∈ (0, 1), φ ∈ L 1 [0, 1] is nonnegative, μ[x] is a linear functional given by μ[x] = 1 0 x(t) dN(t) involving the Stieltjes integral with respect to a nondecreasing function N : [0, 1] → R such that N(t) is right-continuous on [0, 1), leftcontinuous at t = 1, N(0) = 0, and dN is a positive Stieltjes measure. Rehman et al [28] developed Haar wavelets operational matrices to approximate the solution of generalized Caputo-Katugampola fractional differential equations. They introduced the Green-Haar approach for a family of generalized fractional boundary value problems and compared the method with the classical Haar wavelets technique.…”

Fast and Efficient Numerical Finite Difference Method for Multiphase Image Segmentation

“…e authors in [20] proposed a parametric snake model for image segmentation. Image segmentation methods based on fractional calculus are popular emerging methods [21,22]. Recently, deep learning-based segmentation methods have been actively studied in the image segmentation [23][24][25][26].…”