scholarly journals Asymptotic Estimates of the Solution of a Restoration Problem with an Initial Jump

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Duisebek Nurgabyl
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Problem Solution ◽  
Asymptotic Estimates ◽  
Cauchy Function ◽  
Unperturbed Problem ◽  
Boundary Functions ◽  
Restoration Problem ◽  
Initial Jump

The asymptotic behavior of the solution of the singularly perturbed boundary value problemLεy=htλ,Liy+σiλ=ai,i=1,n+1̅is examined. The derivations prove that a unique pair(ỹt,λ̃ε,ε,λ̃ε)exists, in which componentsy(t,λ̃ε,ε)andλ̃(ε)satisfy the equationLεy=h(t)λand boundary value conditionsLiy+σiλ=ai,i=1,n+1̅. The issues of limit transfer of the perturbed problem solution to the unperturbed problem solution as a small parameter approaches zero and the existence of the initial jump phenomenon are studied. This research is conducted in two stages. In the first stage, the Cauchy function and boundary functions are introduced. Then, on the basis of the introduced Cauchy function and boundary functions, the solution of the restoration problemLεy=htλ,Liy+σiλ=ai,i=1,n+1̅is obtained from the position of the singularly perturbed problem with the initial jump. Through this process, the formula of the initial jump and the asymptotic estimates of the solution of the considered boundary value problem are identified.

ESTIMATION OF A BOUNDARY VALUE PROBLEM SOLUTION WITH INITIAL JUMP FOR LINEAR DIFFERENTIAL EQUATION

2020 ◽  
Vol 69 (1) ◽  
pp. 168-173
Author(s):  
B. Sharip ◽  
◽  
А.Т. Yessimova ◽  
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Linear Differential Equation ◽  
Special Functions ◽  
Boundary Value ◽  
Analytical Formula ◽  
Singularly Perturbed ◽  
Problem Solution ◽  
Unperturbed Problem ◽  
Linear Differential

The paper considers a boundary value problem for a singularly perturbed linear differential equation with constant third-order coefficients. In this problem, a small parameter is indicated before the highest derivatives that are part of the differential equation and the boundary condition at t = 0.The fundamental system of solutions of a homogeneous singularly perturbed differential equation is constructed on the basis of asymptotic representations obtained for the roots of the corresponding characteristic equation. This system was used to construct the Cauchy function, special functions of boundary value problems, and also the Green function. With the help of these functions, an analytical formula is obtained for solving a singularly perturbed boundary value problem and it turns out that this solution has an initial zero-order jump at t = 0. It is proved that the solution to the considered singularly perturbed boundary value problem tends to the corresponding unperturbed problem obtained from it under .

scholarly journals Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 213 ◽  
Author(s):  
Assiya Zhumanazarova ◽  
Young Im Cho
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Singularly Perturbed Problems ◽  
Cauchy Function ◽  
Higher Derivatives ◽  
Singularly Perturbed Differential Equation ◽  
Perturbed Differential Equation

In this study, the asymptotic behavior of the solutions to a boundary value problem for a third-order linear integro-differential equation with a small parameter at the two higher derivatives has been examined, under the condition that the roots of the additional characteristic equation are negative. Via the scheme of methods and algorithms pertaining to the qualitative study of singularly perturbed problems with initial jumps, a fundamental system of solutions, the Cauchy function, and the boundary functions of a homogeneous singularly perturbed differential equation are constructed. Analytical formulae for the solutions and asymptotic estimates of the singularly perturbed problem are obtained. Furthermore, a modified degenerate boundary value problem has been constructed, and it was stated that the solution of the original singularly perturbed boundary value problem tends to this modified problem’s solution.

scholarly journals Existence results for first derivative dependent ϕ-Laplacian boundary value problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Imran Talib ◽  
Thabet Abdeljawad
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Continuous Functions ◽  
Existence Of Solution ◽  
Existence Results ◽  
Main Concern ◽  
First Derivative ◽  
Boundary Functions ◽  
Carathéodory Function

Abstract Our main concern in this article is to investigate the existence of solution for the boundary-value problem $$\begin{aligned}& (\phi \bigl(x'(t)\bigr)'=g_{1} \bigl(t,x(t),x'(t)\bigr),\quad \forall t\in [0,1], \\& \Upsilon _{1}\bigl(x(0),x(1),x'(0)\bigr)=0, \\& \Upsilon _{2}\bigl(x(0),x(1),x'(1)\bigr)=0, \end{aligned}$$ ( ϕ ( x ′ ( t ) ) ′ = g 1 ( t , x ( t ) , x ′ ( t ) ) , ∀ t ∈ [ 0 , 1 ] , ϒ 1 ( x ( 0 ) , x ( 1 ) , x ′ ( 0 ) ) = 0 , ϒ 2 ( x ( 0 ) , x ( 1 ) , x ′ ( 1 ) ) = 0 , where $g_{1}:[0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ g 1 : [ 0 , 1 ] × R 2 → R is an $L^{1}$ L 1 -Carathéodory function, $\Upsilon _{i}:\mathbb{R}^{3}\rightarrow \mathbb{R} $ ϒ i : R 3 → R are continuous functions, $i=1,2$ i = 1 , 2 , and $\phi :(-a,a)\rightarrow \mathbb{R}$ ϕ : ( − a , a ) → R is an increasing homeomorphism such that $\phi (0)=0$ ϕ ( 0 ) = 0 , for $0< a< \infty $ 0 < a < ∞ . We obtain the solvability results by imposing some new conditions on the boundary functions. The new conditions allow us to ensure the existence of at least one solution in the sector defined by well ordered functions. These ordered functions do not require one to check the definitions of lower and upper solutions. Moreover, the monotonicity assumptions on the arguments of boundary functions are not required in our case. An application is considered to ensure the applicability of our results.

scholarly journals Regularized Asymptotics of the Solution of the Singularly Perturbed First Boundary Value Problem on the Semiaxis for a Parabolic Equation with a Rational “Simple” Turning Point

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 405
Author(s):  
Alexander Yeliseev ◽  
Tatiana Ratnikova ◽  
Daria Shaposhnikova
Keyword(s):  
Parabolic Equation ◽  
Maximum Principle ◽  
Boundary Value Problem ◽  
Regularization Method ◽  
Turning Point ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Asymptotic Convergence ◽  
The Maximum Principle ◽  
Regularized Asymptotics

The aim of this study is to develop a regularization method for boundary value problems for a parabolic equation. A singularly perturbed boundary value problem on the semiaxis is considered in the case of a “simple” rational turning point. To prove the asymptotic convergence of the series, the maximum principle is used.

Sign in / Sign up

Export Citation Format

Share Document