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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
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.
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)
Berikelashvili, G., Khomeriki, N.: On the convergence of difference schemes for one nonlocal boundary-value problem. Lith. Math. J. 52, 353–362 (2012)
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)
Carlson, R.: Adjoint and self-adjoint differential operators on graphs. Electron. J. Differ. Equ. 1198, 06(1998). 10pp.
Čiupaila, R., Sapagovas, M., Štikonienė, O.: Numerical solution of nonlinear elliptic equation with nonlocal condition. Nonlinear Anal. Model. Control 18, 412–426 (2013)
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.
Everitt, W.N., Markus, L., Plum, M.: An unusual self-adjoint linear partial differential operator. Trans. Am. Math. Soc. 357, 1303–1324 (2005)
Gordeziani, D.G., Avalishvili, G.A.: Time-nonlocal problems for Schrödinger type equations: II. Results for specific problems. Differ. Equ. 41, 852–859 (2005)
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)
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)
Myint-U, T., Debnath, L., 4th ed: Linear Partial Differential Equations for Scientists and Engineers, Birkhauser, Boston (2007)
Sapagovas, M., Jakubelienė, K.: Alternating direction method for two-dimensional parabolic equation with nonlocal integral condition. Nonlinear Anal. Model. Control 17, 91–98 (2012)
Vladimirov, V.S.: Generalized of Mathematical Physics, Mir Publishers, Moscow (1979)
Wang, A., Sun, J., Zettl, A.: The classification of self-adjoint boundary conditions: Separated, coupled, and mixed. J. Funct. Anal. 255, 1554-1573 (2008)
Acknowledgements
The authors are thankful to the honorable referee for his helpful and useful suggestions to improve the manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jahanshahi, M., Darabadi, M. Analytical Solution for a Non-Self-Adjoint and Non-Local-Boundary Value Problem Including a Partial Differential Equation with a Complex Constant Coefficient. Vietnam J. Math. 43, 677–686 (2015). https://doi.org/10.1007/s10013-014-0113-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10013-014-0113-z
Keywords
- Partial differential equations
- Non-local-boundary value problem
- Self-adjoint problem
- Non-self-adjoint operators
- Fundamental solutions