## Journal of Applied Mathematics

- J. Appl. Math.
- Volume 2014 (2014), Article ID 956402, 11 pages.

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

#### Abstract

The asymptotic behavior of the solution of the singularly perturbed boundary value problem ${L}_{\epsilon}y=h\left(t\right)\lambda ,{L}_{i}y+\sigma i\lambda ={a}_{i},i=\stackrel{\u0305}{1,n+1}$ is examined. The derivations prove that a unique pair $(\stackrel{\u0303}{y}\left(t,\stackrel{\u0303}{\lambda}\left(\epsilon \right),\epsilon \right),\stackrel{\u0303}{\lambda}\left(\epsilon \right))$ exists, in which components $y(t,\stackrel{\u0303}{\lambda}\left(\epsilon \right),\epsilon )$ and $\stackrel{\u0303}{\lambda}(\epsilon )$ satisfy the equation ${L}_{\epsilon}y=h(t)\lambda $ and boundary value conditions ${L}_{i}y+\sigma i\lambda ={a}_{i},i=\stackrel{\u0305}{1,n+1}$. 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 problem ${L}_{\epsilon}y=h\left(t\right)\lambda ,{L}_{i}y+\sigma i\lambda ={a}_{i},i=\stackrel{\u0305}{1,n+1}$ is 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.

#### Article information

**Source**

J. Appl. Math., Volume 2014 (2014), Article ID 956402, 11 pages.

**Dates**

First available in Project Euclid: 2 March 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.jam/1425305705

**Digital Object Identifier**

doi:10.1155/2014/956402

**Mathematical Reviews number (MathSciNet)**

MR3206901

#### Citation

Nurgabyl, Duisebek. Asymptotic Estimates of the Solution of a Restoration Problem with an Initial Jump. J. Appl. Math. 2014 (2014), Article ID 956402, 11 pages. doi:10.1155/2014/956402. https://projecteuclid.org/euclid.jam/1425305705