Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, University of Technical and Vocational, Tabriz, Iran.

2 Department of Mathematics, Maragheh Branch, Islamic Azad University, Maragheh, Iran.

Abstract

‎In this paper‎, ‎for a singular perturbation problem consist of the Cauchy-Euler equation with local and non-local boundary conditions‎. ‎We investigate the condition of the self-adjoint and the non-self-adjoint‎, ‎also look for the formation or non-formation of boundary layers for local boundary conditions using the Frequent uniform limit method‎. ‎Also‎, ‎for the state of non-local conditions‎, ‎we convert the non-local boundary conditions into local conditions by finding the fundamental solution and then obtaining the necessary conditions with the help of the 4-step method‎. ‎Finally‎, ‎we determine the formation or non-formation of a boundary layer for non-local conditions such as local conditions‎.

Keywords


[1] E.P. Doolan, J.J. Miller and W.H. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980.
[2] M. Khakshour and G. Aghamollaei, Some results on polynomial numerical hulls of perturbed matrices, Sahand Commun. Math. Anal., 14 (1), (2019), pp. 147-158.
[3] A.Najati, B. Noori and M.B. Moghimi, On approximation of some mixed functional equations, Sahand Commun. Math. Anal., 18 (1), (2021), pp. 35-46.
[4] J.R.E. O'mally, Introduction to Singular Perturbation, Academic Press, New York, 1974.
[5] J.R.E. O'mally, Singular Perturbation Methods For O.D.E's, Springer Verlag, New York, 1991.
[6] A.R. Sarakhsi and M. Jahanshahi, Investigation of boundary layers of singular perturbation problem including second order linear differential equation with non-local boundary conditions, J. Science. Kharazmi., 13 (3), (2013), pp. 809-818.
[7] A.R. Sarakhsi and M. Jahanshahi, Asymptotic solution of singular perturbation problem for second order linear O. D. E with local boundary conditions, in: proc. 40th Annual Iranian Mathematics Conference, Tehran, Iran, 2009.
[8] A.R. Sarakhsi and M. Jahanshahi, Investigation of boundary layers in singular perturbation problems with general linear non-local boundary conditions, in: proc. IV Congress of the Turkic World Mathematical Society, Baku, Azarbayjan, 2011.
[9] A.R. Sarakhsi and M. Jahanshahi, Asymptotics Solution of problem of singular perturbation of second-order linear with constant coefficients with dirichlet condition, J. Science. Kharazmi., 10 (1), (2012).
[10] A.R. Sarakhsi, M. Jahanshahi, S. Ashrafi and M. Sarakhsi, Investigation of boundary layers in some singular perturbation problems including fourth order ordinary differential equation, World Applied Sciences. J., 22 (12), (2012), pp. 1695-1701.
[11] A.R. Sarakhsi and M. Jahanshahi, Boundary layer problem for system of first order of ordinary differential equations with linear non-local boundary conditions, I. J. Science and Technology., 37A3, (2013), pp. 389-396.
[12] A.R. Sarakhsi and M. Jahanshahi, Detecting the location of the boundary layers in singular perturbation problems with general linear non-local boundary conditions, Int. J. Industrial Mathematics., 7 (4), (2015), pp. 321-326.
[13] A.R. Sarakhsi, M. Jahanshahi and M. Sarakhsi, Investigation of approximate solution of mathematical model of singular perturbation problem of including second order linear equation with variable coefficients and Dirichlet boundary conditions, JAMM, J. Adv. Math. Model., 2 (2), (2012), pp. 49-70.
[14] A.R. Sarakhsi and M. Jahanshahi, Asymptotic expansinns for singular problem of 2-dimensional dynamical system, in: proc. 6th Iranian Seminar of Geometry and Topology, 6SGT., Bonab, Iran, 2011, pp. 47-52.
[15] A.R. Sarakhsi and M. Jahanshahi, Investigation of boundary layers in some singular perturbation problems including fourth order ordinary differential equation, in: proc. 9th Iranian Seminar of Differential Equations and Dynamical Systems, Tabriz, Iran, 2012, pp. 259-262.