Academic Journal of Applied Mathematical Sciences ISSN(e): 2415-2188, ISSN(p): 2415-5225 Vol. 6, Issue. 1, pp: 1-4, 2020 URL: https://arpgweb.com/journal/journal/17 DOI: https://doi.org/10.32861/ajams.61.1.4 Academic Research Publishing Group *Corresponding Author 1 Original Research Open Access Method of Averaging for Some Parabolic Partial Differential Equations Mahmoud M. El-Borai * Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria, Egypt Hamed Kamal Awad Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt Randa Hamdy M. Ali Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt Abstract Quantitative and qualitative analysis of the Averaging methods for the parabolic partial differential equation appears as an exciting field of the investigation. In this paper, we generalize some known results due to Krol on the averaging methods and use them to solve the parabolic partial differential equation. Keywords: Averaging; Averaging method; Partial differential equation; Parabolic partial differential equation. CC BY: Creative Commons Attribution License 4.0 1. Introduction The investigation in the field of the qualitative and quantitative analysis of the Averaging methods for the parabolic partial differential equation is more exciting field to be studied. We study the parabolic partial differential equation in this paper using the technique of the averaging method of the linear operator. In section 2, we study the averaging of the linear operator where we generalize some known results due to Krol [1]. We consider the following parabolic partial differential equation in the form:               where     ∑      ||            is the dimensional Euclidean space,        is an dimensional multi index, ||                     The coefficients     and  are bounded continuous with bounded derivatives and     are bounded on         In section 3, we discuss a special case for the problem (1), (2). Compare [2-11]. 2. Averaging a Linear Operator By averaging the coefficients     over , we can average the operator         ∫         for all       producing the averaged operator     and all the coefficients     ||   are bounded continuous with bounded derivatives on   .        ∫ ∑      ||    like as an approximating problem for (1), (2), we take                    another straightforward analysis display the existence and uniqueness of the solutions of problems (1), (2) and (5), (6) on the time-scale    We consider the domain    [ ] The norm ‖ ‖  is defined by the supremum norm on  and denoted by ‖ ‖     | |