On averaging methods for general parabolic partial differential equation

Volume 21, Issue 2, pp 164--175 http://dx.doi.org/10.22436/jmcs.021.02.08

Publication Date: April 17, 2020 Submission Date: June 27, 2019 Revision Date: February 29, 2020 Accteptance Date: March 03, 2020

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Authors

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, Behera, Egypt. Randa Hamdy M. Ali - 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.

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Keywords

• Averaging method
• fractional parabolic partial differential equation
• Existence and uniqueness of solutions

MSC

•  34A07
•  34A60
•  35A05
•  03E72
•  34C29
•  34K05

References

• [1] A. Ben Lemlih, J. A. Ellison, Method of averaging and the quantum anharmonic oscillator, Phys. Rev. Lett., 55 (1985), 1950--1953
  • View Article
  • MathSciNet
  • Google Scholar


• [2] S. D. Èĭdelʹman, On fundamental solutions of parabolic systems, Math. Sobornik, 38 (1956), 51--92
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] M. M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Soliton Fractals, 14 (2002), 433--440
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] M. M. El-Borai, Evolution equations without semi groups, J. Appl. Math. Comp., 149 (2004), 815--821
  • View Article
  • Google Scholar


• [5] M. M. El-Borai, H. K. Awad, R. Hamdy, M. Ali, A parabolic transform and averaging methods for general partial differential equations, J. Adv. Math., 17 (2019), 352--361
  • Google Scholar


• [6] M. M. El-Borai, K. El-Said El-Nadi, A parabolic transform and some stochastic Ill-posed problem, British J. Math. Comput. Sci., 9 (2015), 418--426
  • View Article
  • Google Scholar


• [7] M. M. El-Borai, K. El-Said El-Nadi, On the solutions of Ill-posed Cauchy problems for some singular integro-partial differential equations, Global J. Math., 13 (2019), 899--905
  • View Article
  • Google Scholar


• [8] M. M. El-Borai, K. El-Said El-Nadi, E. G. El-Akabawy, On some fractional evolution equations, Comput. Math. Appl., 59 (2010), 1352--1355
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] M. M. El-Borai, K. El-Said El-Nadi, H. A. Foad, On some fractional stochastic delay differential equations, Comput. Math. Appl., 59 (2010), 1165--1170
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] M. M. El-Borai, W. G. El-Sayed, F. N. Ghaffoori, On the Cauchy problem for some parabolic fractional partial differential equations with time delays, J. Math. Syst. Sci., 6 (2016), 194--199
  • View Article
  • Google Scholar


• [11] M. M. El-Borai, O. L. Moustafa, F. H. Michael, On the correct formulation of a nonlinear differential equations in Banach space, Int. J. Math. Math. Sci., 26 (2001), 8 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [12] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs (1964)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] M. S. Krol, On the averaging method in nearly time-periodic advection-diffusion problems, SIAM J. Appl. Math., 51 (1991), 1622--1637
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] M. H. Protter, H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs (1967)
  • View Article
  • MathSciNet
  • Google Scholar


• [15] F. Verhulst, On averaging methods for partial differential equations, In: Symmetry and perturbation theory, 1999 (1999), 79--95
  • View Article
  • MathSciNet
  • Google Scholar