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
Views 2906 Downloads 127
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
Follow on us
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.
Advection-Diffusion Problem, Variable Diffusion Constant, Integral Representation Method, Primary Space-Differential Operator, Generalized Fundamental Solution, Generalized Integral Representation Method Component
To cite this article
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.
C.A. Brebbia, J.C.F. Telles, L.C. Wrobel, 4.3 Coupled boundary element - Finite difference methods, Boundary Element Techniques, Theory and Applications in Engineering, Springer-Verlag (1984).
B. Nayroles, G. Touzot, P. Villon, Generalizing the finite element method: Diffuse approximation and diffuse elements, Comput. Mechanics 10 (1992) 307-318.
T. Belytschko, Y.Y. Lu, L. Gu, Element-free Galarkin methods, International Journal for Numerical Methods in Engineering 37 (1994) 229-256.
G. Yagawa, Y. Yamda, Free mesh method: A new meshless finite element method, Computational Mechanics 18 (1996) 383-386.
L.B. Lucy, A numerical approach to the testing of the fission hypotheis, The Astronomical Jounal 82 (12) (1977) 1013-1024.
G.R. Liu, M.B. Liu, Smoothed Particle Hydrodyndmics–a meshfree particle method. World Scientific; ISBN 981-238-456-1 (2003).
H. Isshiki, Discrete differential operators on irregular nodes (DDIN), International Journal for Numerical Methods in Engineering 88 (12) (2011) 1323-1343.
H. Isshiki, Random Collocation Method (RCM), IMECE2010-39054, Vancouver, Canada (2010).
J.S. Uhlman, An integral equation formulation of the equations of motion of an incompressible fluid, NUWC-NPT Technical Report 10,086 15 July (1992).