A Parabolic Transform and Averaging Methods for General Partial Differential Equations
Averaging method of the fractional general partial differential equations and a special case of these equations are studied, without any restrictions on the characteristic forms of the partial differential operators. We use the parabolic transform, existence and stability results can be obtained.
A. Ben Lemlih and J.A. Ellison, Method of averaging and the quantum anharmonic oscillator, Phys. Rev. Lett. 55 (1985), 1950-1953.
A. Friedman, Partial Di_erential Equations of Parabolic Type, Prentice-Hall, Englewood Cli_s, NJ (1964).
Krol, M.S., On the averaging method in nearly time-periodic advection-di_usion problems, SIAM J. Appl. Math. 51 (1991), 1622-1637.
Mahmoud M. El-Borai, Evolution equations without semi groups, J. of Appl. Math. And Comp., 149 (2004), 815-821.
Mahmoud M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Soliton and Fractals 14 (2002), 433-440.
Mahmoud M. El-Borai, O. L. Moustafa, F. H. Michael, On the correct formulation of a nonlinear differential equations in Banach space, Int. J. Math. 22(1) (1999).
Mahmoud M. El-Borai, Khairia El-Said El-Nadi, A parabolic transform and some stochastic Ill-posed problem, British Journal of Mathematics and Computer Science, 9(5) (2015), 418-426.
Mahmoud M. El-Borai, Khairia El-Said El-Nadi, On the solutions of Ill-posed Cauchy problems for some singular integro-partial di_erential equations, Global Journal of Mathematics, 13(2) (2019), 899-905.
Mahmoud M. El-Borai, Khairia El-Said El-Nadi and Eman G. El-Akabawy, On some fractional evolution equations, Computers and Mathematics with Applications, 59 (2010), 1352-1355.
Mahmoud M. El-Borai, Khairia El-Said El-Nadi and Hoda A. Foad, On some fractional stochastic delay differential equations, Computers and Mathematics with Applications, 59 (2010), 1165-1170.
Mahmoud M. El-Borai, Wagdy G. El-Sayed, Faez N. Gha_oori, On the Cauchy problem for some parabolic fractional partial differential equations with time delays, Journal of Mathematics and System Science, 6 (2016), 194-199.
Protter, M.H., and Weinberger, H.F., Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cli_s, NJ (1967).
Verhulst, F., On averaging methods for partial differential equations, SPT98-Symmetry and Perturbation Theory II, A.Degasperis and G.Gaeta eds., World Scientifc (1999), 79-95.
Copyright (c) 2019 Mahmoud Mohammed Mostafa El-Borai, Hamed Kamal Awad Awad, Randa Hamdy. M. Ali Ali
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors retain the copyright of their manuscripts, and all Open Access articles are distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided that the original work is properly cited.