Green-Haar method for fractional partial differential equations

• Published in: Engineering computations Vol. 37; no. 4; pp. 1473 - 1490
• Main Authors: Ismail, Muhammad, Rehman, Mujeeb ur, Saeed, Umer
• Format: Journal Article
• Language: English
• Published: Bradford Emerald Publishing Limited 16.04.2020; Emerald Group Publishing Limited
• Subjects: Approximation; Boundary conditions; Calculus; Error analysis; Green's functions; Ordinary differential equations; Partial differential equations; Wavelet analysis; Green-Haar method; Fractional partial differential equations; Caputo derivative; Haar wavelets; Picard technique
• ISSN: 0264-4401; 1758-7077
• DOI: 10.1108/EC-05-2019-0234

Purpose The purpose of this study is to obtain the numerical scheme of finding the numerical solutions of arbitrary order partial differential equations subject to the initial and boundary conditions. Design/methodology/approach The authors present a novel Green-Haar approach for the family of fractional partial differential equations. The method comprises a combination of Haar wavelet method with the Green function. To handle the nonlinear fractional partial differential equations the authors use Picard technique along with Green-Haar method. Findings The results for some numerical examples are documented in tabular and graphical form to elaborate on the efficiency and precision of the suggested method. The obtained results by proposed method are compared with the Haar wavelet method. The method is better than the conventional Haar wavelet method, for the tested problems, in terms of accuracy. Moreover, for the convergence of the proposed technique, inequality is derived in the context of error analysis. Practical implications The authors present numerical solutions for nonlinear Burger’s partial differential equations and two-term partial differential equations. Originality/value Engineers and applied scientists may use the present method for solving fractional models appearing in applications.