On A Boundary Value Problem for A Singularly Perturbed Differential Equation of Non-Classical Type
Mahir M Sabzaliev1* and Mahbuba E Kerimova2
1Department of Mathematics, Azerbaijan State University of Oil and Industry, Azerbaijan
2Department of Mathematics, Baku Business University, Azerbaijan
Submission: February 27, 2018; Published: April 04, 2018
*Corresponding author: Mahir M Sabzaliev, Department of Mathematics, Azerbaijan State University of Oil and Industry, Azerbaijan; Email: sabzalievm@mail.ru
How to cite this article: Mahir M S, Mahbuba E K. On A Boundary Value Problem for A Singularly Perturbed Differential Equation of Non-Classical Type. Biostat Biometrics Open Acc J. 2018; 6(1): 555676. DOI: 10.19080/BBOAJ.2018.06.555676.
Abstract
In a semi-infinite strip we consider a boundary value problem for a non-classical type equation of third order degenerating into a hyperbolic equation. The asymptotic expansion of the problem under consideration is construction is constructed in a small parameter to within any positive degree of a small parameter.
Introduction
Boundary value problems for non-classical singularly perturbed differential equations were not studied enough. We can show the papers devoted to construction of asymptotic solutions to some boundary value problems for non-classical type differential equations [1-3].
In this note in the infinite semi-strip we consider the following boundary value problem
Where &>0 is a small parameter, is the given function.
The goal of the work is to construct the complete asymptotics in a small parameter of the solution of problem (1)-(3). When constructing the asymptotics we follow the M.I. Vishik LA & Lusternik [4] technique.
The following theorem is proved.
Theorem
where the functions Wi are determined by the first iterative process, Vj are the boundary layer type functions near the boundary x=1 determined by the second iterative process, en+1 Z is a remainder term, and for z we have the estimation
Where c1>0, c2>0 are the constants independent of e. References
References
- Mahir MS, Mahbuba EK (2014) Asymptotics of the Solution of Boundary Value Problem for One-Characteristic Differential Equation Degenerating into a Parbolic Equation in an infinite Strip. Nonl Analysis and Differential Equations 2(3): 125-133.
- Mahir MS, Mahbuba EK (2014) Asymptotics of the solution a rectangle of a boundary value problem for one-characteristic differential equation degenerating into a parabolic equation. Transactions of NAS of Azerbaijan iss math mech 34(4): 97-106.
- Mahir MS, Ilhama MS (2017) Asymptotics of Solution of a Boundary Value Problem for Quasi linear Non-Classical Type Differential Equation of Arbitrary Odd Order. British Journal of Mathematics Computer Science 22(4): 1-19.
- Vishik MI, Lusternik LA (1957) Regular degeneration and a boundary layer for linear differential equations with a small parameter. Uspekhi mathematicheskikh nauk 12: 5(77): 3-122.