From Integral Representation Method (IRM) to Generalized Integral Representation Method (GIRM)
Applied and Computational Mathematics
Volume 4, Issue 3-1, June 2015, Pages: 1-14
Received: Dec. 26, 2014;
Accepted: Dec. 30, 2014;
Published: Feb. 12, 2015
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Hiroshi Isshiki, IMA, Institute of Mathematical Analysis, Osaka, Japan
Integral Representation Method (IRM) is one of convenient methods to solve Initial and Boundary Value Problems (IBVP). It can be applied to irregular mesh, and the solution is stable and accurate. However, it was originally developed for linear equations with known fundamental solutions. In order to apply to general nonlinear equations, we must generalize the method. In the present paper, a generalization of IRM (GIRM) is discussed and applied to specific problems and the numerical solutions obtained. The numerical results are stable and accurate. The generalized method is called Generalized Integral Representation Method (GIRM). Brief explanations on the relationships with other numerical methods are also given.
From Integral Representation Method (IRM) to Generalized Integral Representation Method (GIRM), Applied and Computational Mathematics. Special Issue: Integral Representation Method and its Generalization.
Vol. 4, No. 3-1,
2015, pp. 1-14.
Wu J.C., Thompson J.F., “Numerical solutions of time-dependent incompressible Navier-Stokes equations using an integro-differential formulations”, Computers & Fluids, (1973), 1, pp. 197-215.
S. J. Uhlman, “An integral equation formulation of the equations of motion of an incompressible fluid”, NUWC-NPT Technical Report 10,086, 15 July, (1992).
H. Isshik, S. Nagata, Y. Imai, “Solution of Viscous Flow around a Circular Cylinder by a New Integral Representation Method (NIRM)”, AJET, 2, 2, (2014), pp. 60-82. file:///C:/Users/l/Downloads/983-5001-1-PB%20(1).pdf
H. Isshik, S. Nagata, Y. Imai, “Solution of a diffusion problem in a non-homogeneous flow and diffusion field by the integral representation method (IRM)”, Applied and Computational Mathematics, 3(1), (2014), pp. 15-26. http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140301.13.pdf
H. Isshiki, Theory and application of the generalized integral representation method (GIRM) in advection diffusion problem, Applied and Computational Mathematics, 3(4), (2014), pp. 137-149. http://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140304.15.pdf
H. Isshiki, T, Takiya and H. Niizato, Application of Generalized Integral representation (GIRM) Method to Fluid Dynamic Motion of Gas or Particles in Cosmic Space Driven by Gravitational Force, Applied and Computational Mathematics, Special Issue: Integral Representation Method and Its Generalization, (2015), under publicaion. http://www.sciencepublishinggroup.com/journal/archive.aspx?journalid=147&issueid=-1
H. Isshiki, A method for Reduction of Spurious or Numerical Oscillations in Integration of Unsteady Boundary Value Problem, AJET, 2, 3, (2014), pp. 190-202. file:///C:/Users/l/Downloads/1360-5725-2-PB%20(2).pdf
H. Isshiki, “Improvement of Stability and Accuracy of Time-Evolution Equation by Implicit Integration”, Asian Journal of Engineering and Technology (AJET), Vol. 2, No. 2 (2014), pp. 1339–160. file:///C:/Users/l/Downloads/1205-5161-1-PB.pdf