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An integral formulation applied to the diffusion and Boussinesq equations
Author(s) -
Taigbenu Akpofure,
Liggett James A.
Publication year - 1986
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620230606
Subject(s) - mathematics , boussinesq approximation (buoyancy) , integral equation , mathematical analysis , kernel (algebra) , partial differential equation , finite element method , flow (mathematics) , representation (politics) , integro differential equation , geometry , first order partial differential equation , heat transfer , physics , natural convection , combinatorics , politics , political science , rayleigh number , law , thermodynamics
A new integral method is proposed here to solve the diffusion equation (confined flow) and the Boussinesq equation (unconfined flow) in a two‐dimensional porous medium. The method, based on Green's theorem, derives its integral representation from the portion of the original differential equation with the highest space derivatives so that the resulting kernel of the integral representation is not time dependent. Compared to an earlier integral formulation, namely the direct Green function, based on the same theorem, the kernel is simpler so that the present theory provides a more efficient numerical model without compromising accuracy. An iterative scheme is employed along with the theory to achieve solutions to the non‐linear Boussinesq equation. Concepts used in the finite difference and finite element methods enable simplification of the temporal derivative. The method is tested with success on a number of numerical examples from groundwater flow.