THE EIGENVALUE PROBLEM FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATION WITH TWO-POINT NONLOCAL CONDITIONS

Ahmed Elsaid; Shaimaa M. Helal; Ahmed M. A. El-Sayed

doi:10.11948/2015013

2015 Volume 5 Issue 1

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Ahmed Elsaid, Shaimaa M. Helal, Ahmed M. A. El-Sayed. THE EIGENVALUE PROBLEM FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATION WITH TWO-POINT NONLOCAL CONDITIONS[J]. Journal of Applied Analysis & Computation, 2015, 5(1): 146-158. doi: 10.11948/2015013

Citation:

Ahmed Elsaid, Shaimaa M. Helal, Ahmed M. A. El-Sayed. THE EIGENVALUE PROBLEM FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATION WITH TWO-POINT NONLOCAL CONDITIONS[J]. Journal of Applied Analysis & Computation, 2015, 5(1): 146-158. doi: 10.11948/2015013

THE EIGENVALUE PROBLEM FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATION WITH TWO-POINT NONLOCAL CONDITIONS

1 Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Mansoura, Egypt;
2 Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt

Abstract

Abstract

We study the spectral problem for the system of difference equations of a two-dimensional elliptic partial differential equation with nonlocal conditions. A new form of two-point nonlocal conditions that involve interior points is proposed. The matrix of the difference system is nonsymmetric thus different types of eigenvalues occur. The conditions for the existence of the eigenvalues and their corresponding eigenvectors are presented for the one dimensional problem. Then, these relations are generalized to the twodimensional problem by the separation of variables technique.
- Elliptic partial differential equation /
- Two-point conditions /
- Eigenvalue problem /
- Finite difference method
MSC: 65N06;65F15;35J57

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References

[1]	B. I. Bandyrskii and V. L. Makarov, Sufficient Conditions for the Eigenvalues of the Operator -d/dx²+ q(x) with the Ionkin-Samarskii Conditions to be Real Valued,Comput. Math. Math. Phys, 40(12) (2000), 1715-1728. Google Scholar
[2]	B. I. Bandyrskii, V. L. Makarov and O. L. Ukhanev, Sufficient Conditions for the Convergence of Nonclassical Asymptotic Expansions of the Sturm-Liouville Problem with Periodic Conditions, Differ. Uravn, 35(3) (1999), 369-381. Google Scholar
[3]	J. G. Batten, Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations, Math. Comput., 17(1963), 405-413. Google Scholar
[4]	J. R. Cannon, The solution of the heat equation subject to the specification of energy,Quart. Appl. Math., 21(1963), 55-60. Google Scholar
[5]	R. Ciegis, Parallel numerical algorithms for 3D parabolic problem with nonlocal boundary condition, Informatica, 17(2006), 309-324. Google Scholar
[6]	R. Ciegis, A. Stikonas, O. Stikoniene and O. Subo, A monotonic finitedifference scheme for a parabolic problem with nonlocal conditions,Differ. Equ., 38(2002), 1027-1037. Google Scholar
[7]	R. Ciegis, A. Stikonas, O. Stikoniene and O. Suboc, Stationary problems with nonlocal boundary conditions, Math. Model. Anal., 6(2) (2001), 178-191. Google Scholar
[8]	R. Ciupaila, Z. Jeseviciute and M. Sapagovas, On the Eigenvalue Problem for One-Dimensional Differential Operator with Nonlocal Integral Condition,Nonlinear Analysis:Modelling and Control, 9(2)(2004), 109-116. Google Scholar
[9]	D.G. Gordeziani and G.A. Avalishvili, Investigation of the nonlocal initial boundary value problems for some hyperbolic equations,Hiroshima Math. J.,31(2001), 345-366. Google Scholar
[10]	A. V. Gulin, Asympotic stability of nonlocal difference scheme,Numer. Appl. Math., 2(105) (2011), 34-43. Google Scholar
[11]	A.V. Gulin, V.A.Morozova, On the stability of a nonlocal finite-difference boundary value problem, Differ. Equations, 39(7) (2003), 962-967. Google Scholar
[12]	A.V. Gulin, N. I. Ionkin, V.A. Morozova, Stability of a nonlocal twodimensional finite-difference problem,Differ. Equations, 37(7)(2001) 970-978. Google Scholar
[13]	A.K. Gushin, V.P. Mikhailov, On solvability of nonlocal problems for a second ordered elliptic equation, Sb. Mat., (185) (1994), 121-160. Google Scholar
[14]	V.A. Ilin, On a connection between the form of the boundary conditions and the basis property of eiquiconvergence with a trigonometric series of expansions in root functions of a nonselfadjoint differential operator,Differ. Equations, 30(9) (1994), 1402-1413. Google Scholar
[15]	V.A. Ilin and E. I. Moiseev, Nonlocal boundary-value problem of the 1st kind for a Sturm-Liouville operator in its differential and finite-difference aspects, Differ. Equ., 23(7), (1987) 803-811. Google Scholar
[16]	N. I. Ionkin, Solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition, Differ. Equations, 13(2) (1977), 294-304. Google Scholar
[17]	N. I. Ionkin and V.A. Morozova, Two dimentional heat equation with nonlocal boundary conditions, Differ. Uravn, 36(7) (2000), 884-888. Google Scholar
[18]	Z. Jesevioiute and M. Sapagovas, On the stability of finite-difference schemes for parabolic equations subject to integral conditions with applications to thermoelasticity, Comput. Methods Appl. Math.,8(4) (2008), 360-373. Google Scholar
[19]	G. Kalna and S. McKee, The termostat problem with a nonlocal nonlinear boundary condition, IMA J. Applied Math, 69(2004), 437-462. Google Scholar
[20]	V. L. Makarov, I. I. Lazurchak and B. I. Bandyrsky, Nonclassical Asymptotic Formulas and Approximation with Arbitrary Order of Accuracy of the of Eigenvalues in Sturm-Liouville Problem with Bizadse-Samarsky Conditions, Cybernet. System Anal, 6(2003), 862-879. Google Scholar
[21]	S. Peciulyte, O. Stikoniene and A. Stikonas, Sturm-Liouville problem for stationary differential operator with nonlocal integral boundary condition, Mathem. Modelling and Analysis, 10(4)(2005), 377-392. Google Scholar
[22]	S. Sajavicius, On the eigenvalue problem of the finite difference operators with coupled boundary conditions, Siauliai Math. Semin., 5(13) (2010), 87-100. Google Scholar
[23]	M. Sapagovas, The eigenvalues of some problems with a nonlocal condition, Differ. Equ., 38(7)(2002), 1020-1026. Google Scholar
[24]	M. Sapagovas, On stability of finite-difference schemes for one-dimensional parabolic equations subject to integral conditions, Comput. Appl. Math., 92(2005) 77-90. Google Scholar
[25]	M. Sapagovas, On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems, Lith. Math. J., 48(3)(2008), 339-356. Google Scholar
[26]	M. Sapagovas, Difference method of increased order of accuracy for the Poisson equation with nonlocal conditions, Differ. Equ.,44(7) (2008), 1018-1028. Google Scholar
[27]	M. Sapagovas, G. Kairyte, O. Stikoniene and A. Stikonas, Alternating direction method for a two dimensional parabolic equation with a nonlocal boundary condition, Math.Model. Anal., 12(1) (2007), 131-142. Google Scholar
[28]	M. Sapagovas and O. Stikoniene, Alternating-direction method for a mildly nonlinear elliptic equation with nonlocal integral conditions, Nonlinear Analysis. Model.Cont., 16(2011), 220-230. Google Scholar
[29]	M. P. Sapagovas and A.D. Stikonas, On the structure of the spectrum of a differential operator with a nonlocal condition, Differ. Equations, 41(7) (2005), 1010-1018. Google Scholar
[30]	A.A. Shkalikov, Bases formed by eigenfunctions of ordinary differential operators with integral boundary conditions, Vestnik Moskovsk. Universit, Ser. 1. Matem. Mechan., 6(1982), 12-21. Google Scholar
[31]	Y. Wang, Solutions to nonlinear elliptic equations with a nonlocal boundary condition, Differ. Equ., 2002(2002(5)), 1-16. Google Scholar
[32]	O.S. Zikirov, On boundary-value problem for hyperbolic-type equation of the third order, Lith. Math. J., 47(4)(2007), 484-495. Google Scholar