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A non‐linear and non‐local boundary condition for a diffusion equation in petroleum engineering
Author(s) -
Giroire J.,
HaDuong T.,
Moumas V.
Publication year - 2005
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.622
Subject(s) - mathematics , boundary value problem , uniqueness , neumann boundary condition , mathematical analysis , mixed boundary condition , dirichlet boundary condition , poincaré–steklov operator , elliptic boundary value problem , dirichlet distribution , laplace's equation , boundary (topology) , free boundary problem , contraction mapping , robin boundary condition , fixed point theorem
This article deals with a boundary value problem for Laplace equation with a non‐linear and non‐local boundary condition. This problem comes from petroleum engineering and is used to obtain an estimation of well productivity. The non‐linear and non‐local boundary condition is written on the well boundary. On the outer reservoir boundaries, we have both Dirichlet and Neumann conditions. In this paper, we prove the existence and uniqueness of a solution to this problem. The existence is proved by Schauder theorem and the uniqueness is obtained under more restricted conditions, when the involved operator is a contraction. Copyright © 2005 John Wiley & Sons, Ltd.