# zbMATH — the first resource for mathematics

A numerical method for a nonlocal elliptic boundary value problem. (English) Zbl 1149.65099
The problem under consideration is $-\alpha\left(\int_0^1{u(t)\,dt}\right)u''=f(x),\; 0<x<1,\, u(0)=a, \, u(1)=b,$ where $$\alpha(\cdot)$$ is a positive function. The existence and uniqueness of the solution, weak as well as classical, of the problem are proved at different assumptions with respect to the problem data. The numerical problem via a finite difference scheme is investigated. Three numerical examples demonstrate the proposed method.

##### MSC:
 65R20 Numerical methods for integral equations 45J05 Integro-ordinary differential equations 45G10 Other nonlinear integral equations
Full Text:
##### References:
 [1] Robert Stanczy, Nonlocal elliptic equations , Nonlinear Analysis 47 (2001), 3579-3584. · Zbl 1042.35548 · doi:10.1016/S0362-546X(01)00478-3 [2] Corrêa, Francisco Julio S. A., Silvano D.B. Menezes and J. Ferreira, On a class of problems involving a nonlocal operator , Applied Mathematics and Computation, Vol. 147 , Issue 2 (2004) 475–489. · Zbl 1086.35038 · doi:10.1016/S0096-3003(02)00740-3 [3] Corrêa, F.J.S.A., and Daniel C. de Morais Filho, On a class of nonlocal elliptic problems via Galerkin method , Journal of Mathematical Analysis and Applications 310 (2005) 177–187. · Zbl 1136.35378 · doi:10.1016/j.jmaa.2005.01.052 [4] Corrêa, F.J.S.A., and Menezes, S.D.B., Positive solutions for a class of nonlocal elliptic problems , Contributions to nonlinear analysis, 195–206, Progr. Nonlinear Differential Equations Appl., 66 , Birkhaüser, Basel, (2006). · Zbl 1131.35330 · doi:10.1007/3-7643-7401-2_13 [5] Corrêa, F.J.S.A., On positive solutions of nonlocal and nonvariational elliptic problems , Nonlinear Analysis, Vol. 65 , Issue 4 (2006) 864–891. [6] Douglas, Jim, Jr., On the numerical integration of $$\frac\pa^2 u\pa x^2 + \frac\pa^2 u\pa y^2 = \frac\pa u\pa t$$ by implicit methods , J. Soc. Indust. Appl. Math., 3 (1955), 42-65. · Zbl 0067.35802 · doi:10.1137/0103004 [7] Douglas, Jim, Jr., A survey of numerical methods for parabolic differential equations , (1961), Advances in Computers Vol 2 . pp 1–54 Academic Press, N.Y. · Zbl 0133.38503 · doi:10.1016/S0065-2458(08)60140-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.