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From Integral Representation Method (IRM) to Generalized Integral Representation Method (GIRM)
Issue:
Volume 4, Issue 3-1, June 2015
Pages:
1-14
Received:
26 December 2014
Accepted:
30 December 2014
Published:
12 February 2015
Abstract: 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.
Abstract: 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 gen...
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Application of Generalized Integral Representation (GIRM) Method to Fluid Dynamic Motion of Gas or Particles in Cosmic Space Driven by Gravitational Force
Hiroshi Isshiki,
Toshio Takiya,
Hideyuki Niizato
Issue:
Volume 4, Issue 3-1, June 2015
Pages:
15-39
Received:
22 December 2014
Accepted:
25 December 2014
Published:
12 February 2015
Abstract: Some aspect of the motion of gas or vast-number-of-particles distributed in cosmic space under action of the gravitational force may be treated as a fluid dynamic motion without pressure. Generalized Integral representation Method (GIRM) is applied to fluid dynamic motion of gas or particles to obtain the accurate numerical solutions. In the present theory, the relativistic effects are neglected. The numerical results by GIRM are compared with the solutions by Finite Difference Method (FDM). Spreading and merging of gas or particles and effects of initial velocity distribution are studied numerically. GIRM solutions give reasonable and accurate solutions.
Abstract: Some aspect of the motion of gas or vast-number-of-particles distributed in cosmic space under action of the gravitational force may be treated as a fluid dynamic motion without pressure. Generalized Integral representation Method (GIRM) is applied to fluid dynamic motion of gas or particles to obtain the accurate numerical solutions. In the presen...
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Effects of Generalized Fundamental Solution (GFS) on Generalized Integral Representation Method (GIRM)
Issue:
Volume 4, Issue 3-1, June 2015
Pages:
40-51
Received:
5 February 2015
Accepted:
6 February 2015
Published:
13 March 2015
Abstract: 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. IRM is developed to Generalized Integral Representation Method (GIRM) to treat any kinds of problems including nonlinear problems. In GIRM, Generalized Fundamental Solution (GFS) is used instead of Fundamental Solution (FS) in IRM. Since GFS is not limited to one, the effects of individual GFSs must be clarified. The continuity of GFS is related to the characteristics of individual GFSs.
Abstract: 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. IRM is developed to Generalized Integral Representation Method (GIRM) to treat any kinds of problems including nonlinear problems. In GIRM, Generalized F...
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Application of the Generalized Integral Representation Method (GIRM) to Tidal Wave Propagation
Issue:
Volume 4, Issue 3-1, June 2015
Pages:
52-58
Received:
25 February 2015
Accepted:
25 February 2015
Published:
26 March 2015
Abstract: 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. IRM is developed to Generalized Integral Representation Method (GIRM) to treat any kinds of problems including nonlinear problems. In GIRM, Generalized Fundamental Solution (GFS) is used instead of Fundamental Solution (FS) in IRM. We can use a variety of GFSs in GIRM. The effects of typical GFSs are investigated. In the present paper, an application of GIRM to tidal wave propagation is discussed, and the time evolution involves the second order time derivatives. An explicit time evolution is used successfully in the present paper.
Abstract: 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. IRM is developed to Generalized Integral Representation Method (GIRM) to treat any kinds of problems including nonlinear problems. In GIRM, Generalized F...
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Implementation of One and Two-Step Generalized Integral Representation Methods (GIRMs)
Hideyuki Niizato,
Gantulga Tsedendorj,
Hiroshi Isshiki
Issue:
Volume 4, Issue 3-1, June 2015
Pages:
59-77
Received:
19 March 2015
Accepted:
23 March 2015
Published:
8 April 2015
Abstract: In this study, we summarize and implement one- and two-step Generalized Integral Representation Methods (GIRMs). Although GIRM requires matrix inversion, the solution is stable and the accuracy is high. Moreover, it can be applied to an irregular mesh. In order to validate the theory, we apply one- and two-step GIRMs to the one-dimensional Initial and Boundary Value Problem for advective diffusion. The numerical experiments are conducted and the approximate solutions coincide with the exact ones in both cases. The corresponding computer codes implemented in most popular computational languages are also given.
Abstract: In this study, we summarize and implement one- and two-step Generalized Integral Representation Methods (GIRMs). Although GIRM requires matrix inversion, the solution is stable and the accuracy is high. Moreover, it can be applied to an irregular mesh. In order to validate the theory, we apply one- and two-step GIRMs to the one-dimensional Initial ...
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Application of Generalized Integral Method (GIRM) to Numerical Evaluations of Soliton-to-Soliton and Soliton-to-Bottom Interactions
Gantulga Tsedendorj,
Hiroshi Isshiki,
Rinchinbazar Ravsal
Issue:
Volume 4, Issue 3-1, June 2015
Pages:
78-86
Received:
17 April 2015
Accepted:
17 April 2015
Published:
12 May 2015
Abstract: Numerical evaluations of soliton-soliton and soliton-to-bottom interaction have many applications in various fields. On the other hand, Generalized Integral Representation Method (GIRM) is known as a convenient numerical method for solving Initial and Boundary Value Problem of differential equations such as advective diffusion. In this work, we apply one-step GIRM to numerical evaluations of propagation of a single soliton, soliton-to-soliton interaction and soliton-to-bottom interaction. Firstly, in case of a single soliton, the bottom is considered to be constant in order to understand the behavior of the soliton propagation as it travels in the middle of the sea. Next, in case of soliton-to-bottom, we study behavior of a single soliton propagation when the bottom has different geometries. Finally, we evaluate interaction of two different i.e., big and small solitons. To carry out with the studies, we derive and implement GIRM to numerically solve the Korteweg-de Vries (KdV) equation. In order to verify the theory, numerical experiments are conducted and accurate approximate solutions are obtained in each case of the soliton interactions.
Abstract: Numerical evaluations of soliton-soliton and soliton-to-bottom interaction have many applications in various fields. On the other hand, Generalized Integral Representation Method (GIRM) is known as a convenient numerical method for solving Initial and Boundary Value Problem of differential equations such as advective diffusion. In this work, we app...
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