Green’s Functional Concept for a Nonlocal Problem

Abstract

In this work, by Green’s functional concept, in order to obtain Green’ssolution we concentrate on a new constructive technique by which alinear completely nonhomogeneous nonlocal problem for a second-orderloaded differential equation with generally variable coefficients satisfying some general properties such as p-integrability and boundedness istransformed into one and only one integral equation. A system of threeintegro-algebraic equations called the special adjoint system is obtainedfor this problem. A solution of this special adjoint system is Green’sfunctional which enables us to determine Green’s function and Green’ssolution for the problem. Two illustrative applications are provided.

Keywords

• Green’s function; loaded differential equation; nonlocal condition; adjointproblem.

Green’s Functional Concept for a Nonlocal Problem

Abstract

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Keywords

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References

1. Akhiev, S. S. Representations of the solutions of some linear operator equations, Soviet Math. Dokl., 21(2), 555–558, 1980.
2. Akhiev, S. S. Fundamental solutions of functional differential equations and their representations, Soviet Math. Dokl., 29(2), 180–184, 1984.
3. Akhiev, S. S. and Oru¸co˘ glu, K. Fundamental Solutions of Some Linear Operator Equations and Applications, Acta Applicandae Mathematicae, 71, 1-30, 2002.
4. Akhiev, S. S. Green and Generalized Green’s Functionals of Linear Local and Nonlocal Problems for Ordinary Integro-differential Equations, Acta Applicandae Mathematicae, 95, 73–93, 2007.
5. Alikhanov, A. A., Berezgov, A. M. and Shkhanukov-Lafishev, M. X. Boundary Value Problems for Certain Classes of Loaded Differential Equations and Solving Them by Finite Difference Methods, Computational Mathematics and Mathematical Physics, 48(9), 1581– 1590, 2008.
6. Brown, A. L. and Page, A. Elements of Functional Analysis, New York, 1970.
7. Denche, M. and Kourta, A. Boundary Value Problem for Second-Order Differential Operators with Mixed Nonlocal Boundary Conditions, Journal of Inequalities in Pure and Applied Mathematics, 5(2), 1–16 2004.
8. Fatemi, M. R. and Aliyev, N. A. General Linear Boundary Value Problem for the SecondOrder Integro-Differential Loaded Equation with Boundary Conditions Containing Both Nonlocal and Global Terms, Abstract and Applied Analysis, 2010, Article ID 547526, 1-12, 20

9. H¨ ormander, L. Linear Partial Differential Operators, Springer-Verlag, New York, 1976. Kantorovich, L. V. and Akilov, G. P. Functional Analysis (2nd ed, translated by Howard L. Silcock), Pergamon Press, New York, 1982.
10. Krein, S. G. Linear Equations in Banach Space, Nauka, Moscow, 1971 (in Russian).
11. Naimark, M. A. Linear Differential operators, Nauka, Moscow, 1969 (in Russian).
12. Shilov, G. E. Mathematical Analysis: Second special course, Nauka, Moscow, 1965; English transl., Generalized Functions and Partial Differential Equations, Gordon and Breach, New York, 1968.
13. Stakgold, I. Green’s Functions and Boundary Value Problems, Wiley-Interscience Publications, New York, 1998.
14. Tikhonov, A. N., Vasil’eva, A. B. and Sveshnikov, A. G. Differential Equations, Nauka, Moscow, 1980 (in Russian).