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Solution of a Diffusion Problem in a Non-Homogeneous Flow and Diffusion Field by the Integral Representation Method (IRM)
Applied and Computational Mathematics
Volume 3, Issue 1, February 2014, Pages: 15-26
Received: Jan. 10, 2014; Published: Feb. 20, 2014
Authors
Hiroshi Isshiki, Institute of Ocean Energy, Saga University, Saga, Japan
Shuichi Nagata, Institute of Ocean Energy, Saga University, Saga, Japan
Yasutaka Imai, Institute of Ocean Energy, Saga University, Saga, Japan
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Abstract
Integral representations are derived from a differential-type boundary value problem using a fundamental solution. A set of integral representations is equivalent to a set of differential equations. If the boundary conditions are substituted into the integral representations, the integral equations are obtained, and the unknown variables are determined by solving the integral equations. In other words, an integral-type boundary value problem is derived from the integral representations. An effective and flexible finite element algorithm is easily obtained from the integral-type boundary value problem. In the present paper, integral representations are obtained for the diffusion of a material or heat in the sea, where the convective velocity and diffusion constant change in space and time. A new numerical solution of an advection-diffusion equation is proposed based integral representations using the fundamental solution of the primary space-differential operator, and the numerical results are shown. An innovative generalization of the integral representation method: generalized integral representation method is also proposed. The numerical examples are given to verify the theory.
Keywords
Advection-Diffusion Problem, Variable Diffusion Constant, Integral Representation Method, Primary Space-Differential Operator, Generalized Fundamental Solution, Generalized Integral Representation Method Component
Hiroshi Isshiki, Shuichi Nagata, Yasutaka Imai, Solution of a Diffusion Problem in a Non-Homogeneous Flow and Diffusion Field by the Integral Representation Method (IRM), Applied and Computational Mathematics. Vol. 3, No. 1, 2014, pp. 15-26. doi: 10.11648/j.acm.20140301.13
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