# Analytical Solution for a Non-Self-Adjoint and Non-Local-Boundary Value Problem Including a Partial Differential Equation with a Complex Constant Coefficient

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## Abstract

In this paper, we investigate an initial-boundary value problem for a partial differential equation with a complex constant coefficient and non-local-boundary conditions whose related spectral problem is a non-self-adjoint-boundary value problem. For this problem, we first derive the related spectral problem. Then the formal and analytical solution were formulated by using eigenfunctions of the related spectral problem. Finally, the existence, uniqueness and convergence of the series solution will be proved.

## Keywords

Partial differential equations Non-local-boundary value problem Self-adjoint problem Non-self-adjoint operators Fundamental solutions## Mathematics Subject Classification (2010)

35A01 35A08 65N25## Notes

### Acknowledgements

The authors are thankful to the honorable referee for his helpful and useful suggestions to improve the manuscript.

## References

- 1.Almut, B., Chugunova, M.: On computing the instability index of a non-self-adjoint differential operator associated with coating and rimming, flows. SIAM J. Math. Anal.
**43**, 367–388 (2011)zbMATHMathSciNetCrossRefGoogle Scholar - 2.Ashyralyev, A., Yildirim, O.: A note on the second order of accuracy stable difference schemes for the 187 nonlocal boundary value hyperbolic problem. Abstr. Appl. Anal.
**2012**, 846582 (2012), 29 pp.Google Scholar - 3.Avalishvili, G., Avalishvili, M., Gordeziani, D.: On a nonlocal problem with integral boundary conditions for a multidimensional elliptic equation. Appl. Math. Lett.
**24**, 566–571 (2011)zbMATHMathSciNetCrossRefGoogle Scholar - 4.Berikelashvili, G., Khomeriki, N.: On the convergence of difference schemes for one nonlocal boundary-value problem. Lith. Math. J.
**52**, 353–362 (2012)zbMATHMathSciNetCrossRefGoogle Scholar - 5.Bialecki, B., Fairweather, G., Lopez-Marcos, J.C.: The Crank–Nicolson Hermite cubic orthogonal spline collocation method for the heat equation with nonlocal boundary conditions. Adv. Appl. Math. Mech.
**5**, 442–460 (2013)zbMATHMathSciNetGoogle Scholar - 6.Carlson, R.: Adjoint and self-adjoint differential operators on graphs. Electron. J. Differ. Equ.
**1198**, 06(1998). 10pp.Google Scholar - 7.Čiupaila, R., Sapagovas, M., Štikonienė, O.: Numerical solution of nonlinear elliptic equation with nonlocal condition. Nonlinear Anal. Model. Control
**18**, 412–426 (2013)zbMATHMathSciNetGoogle Scholar - 8.Delkhosh, M., Delkhosh, M.: Analytic solutions of some self-adjoint equations by using variable change method and its applications. J. Appl. Math.
**2012**, 180806 (2012). 7 pp.Google Scholar - 9.Everitt, W.N., Markus, L., Plum, M.: An unusual self-adjoint linear partial differential operator. Trans. Am. Math. Soc.
**357**, 1303–1324 (2005)zbMATHMathSciNetCrossRefGoogle Scholar - 10.Gordeziani, D.G., Avalishvili, G.A.: Time-nonlocal problems for Schrödinger type equations: II. Results for specific problems. Differ. Equ.
**41**, 852–859 (2005)zbMATHMathSciNetCrossRefGoogle Scholar - 11.Martín-Vaquero, J., Queiruga-Dios, A., Encinas, A.H.: Numerical algorithms for diffusion–reaction problems with non-classical conditions. Appl. Math. Comput.
**218**, 5487–5492 (2012)zbMATHMathSciNetCrossRefGoogle Scholar - 12.Martín-Vaquero, J., Wade, B.A.: On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions. Appl. Math. Model.
**36**, 3411–3418 (2012)zbMATHMathSciNetCrossRefGoogle Scholar - 13.Myint-U, T., Debnath, L., 4th ed: Linear Partial Differential Equations for Scientists and Engineers, Birkhauser, Boston (2007)Google Scholar
- 14.Sapagovas, M., Jakubelienė, K.: Alternating direction method for two-dimensional parabolic equation with nonlocal integral condition. Nonlinear Anal. Model. Control
**17**, 91–98 (2012)zbMATHMathSciNetGoogle Scholar - 15.Vladimirov, V.S.: Generalized of Mathematical Physics, Mir Publishers, Moscow (1979)Google Scholar
- 16.Wang, A., Sun, J., Zettl, A.: The classification of self-adjoint boundary conditions: Separated, coupled, and mixed. J. Funct. Anal.
**255**, 1554-1573 (2008)zbMATHMathSciNetCrossRefGoogle Scholar

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© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014