Available online at www.isr-publications.com/jmcs J. Math. Computer Sci., 21 (2020), 164–175 Research Article Online: ISSN 2008-949X Journal Homepage: www.isr-publications.com/jmcs On averaging methods for general parabolic partial differen- tial equation Mahmoud M. El-Borai a,* , Hamed Kamal Awad b , Randa Hamdy M. Ali b a Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria, Egypt. b Department of Mathematics, Faculty of Science, Damanhour University, Behera, Egypt. Abstract The averaging method of the quantitative and the qualitative analysis of the parabolic partial differential equations appears as an exciting field of the investigation. The aim of this paper is to generalize some known results due to Krol on the averaging methods and use them to solve the fractional parabolic partial differential equations and a special case of these equations is studied. We treat some different cases related to the averaging method. Keywords: Averaging method, fractional parabolic partial differential equation, Existence and uniqueness of solutions. 2020 MSC: 34A07, 34A60, 35A05, 03E72, 34C29, 34K05. c 2020 All rights reserved. 1. Introduction The averaging method is an important computational technique. The investigation in the field of the averaging method of the qualitative and the quantitative analysis of the parabolic partial differential equa- tions is more exciting field to be studied. We study the fractional parabolic partial differential equation in this paper using the technique of the averaging method of the linear operator. In Section 2, we discuss the averaging of the linear operator where we generalize some known results due to Krol [13]. We will consider the following fractional parabolic partial differential equation in the form ∂ α ∂t α [ ∂u(x, t) ∂t -  |q|=2m a q (x)D q u(x, t)] = ε  |q|<2m b q (x, t)D q u(x, t), (1.1) u(x,0)= ϕ(x), ∂u(x,0) ∂t = ψ(x), (1.2) * Corresponding author Email addresses: m_m_elborai@yahoo.com (Mahmoud M. El-Borai), hamedk66@sci.dmu.edu.eg (Hamed Kamal Awad), rhamdy1989@gmail.com, rali6565@sci.dmu.edu.eg (Randa Hamdy M. Ali) doi: 10.22436/jmcs.021.02.08 Received: 2019-06-27 Revised: 2020-02-29 Accepted: 2020-03-03