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 A Comparison between the fourth order linear differential equation with its boundary value problem

2020 ◽  
Vol 99 (3) ◽  
pp. 18-25
Author(s):  
Karwan H.F. Jwamer ◽  
◽  
Rando R.Q. Rasul ◽  
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Linear Differential Equation ◽  
Fourth Order ◽  
Upper Bound ◽  
Boundary Value ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Linear Differential

In this paper, we study a fourth order linear differential equation. We found an upper bound for the solutions of this differential equation and also, we prove that all the solutions are in L4(0, ∞). By comparing these results we obtain that all the eigenfunction of the boundary value problem generated by this differential equation are bounded and in L4(0, ∞).

A Method for the Numerical Solution of a Boundary Value Problem for a Linear Differential Equation with Interval Parameters

2019 ◽  
Vol 16 (07) ◽  
pp. 1850115 ◽  
Author(s):  
Nizami A. Gasilov ◽  
Müjdat Kaya
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Linear Differential Equation ◽  
Numerical Method ◽  
Boundary Value ◽  
Real Life ◽  
Boundary Values ◽  
Right Hand ◽  
Linear Differential ◽  
The Right

In many real life applications, the behavior of the system is modeled by a boundary value problem (BVP) for a linear differential equation. If the uncertainties in the boundary values, the right-hand side function and the coefficient functions are to be taken into account, then in many cases an interval boundary value problem (IBVP) arises. In this study, for such an IBVP, we propose a different approach than the ones in common use. In the investigated IBVP, the boundary values are intervals. In addition, we model the right-hand side and coefficient functions as bunches of real functions. Then, we seek the solution of the problem as a bunch of functions. We interpret the IBVP as a set of classical BVPs. Such a classical BVP is constructed by taking a real number from each boundary interval, and a real function from each bunch. We define the bunch consisting of the solutions of all the classical BVPs to be the solution of the IBVP. In this context, we develop a numerical method to obtain the solution. We reduce the complexity of the method from [Formula: see text] to [Formula: see text] through our analysis. We demonstrate the effectiveness of the proposed approach and the numerical method by test examples.

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.

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