Special Issue

### Integral Representation Method and its Generalization

• Submission Deadline: 20 April 2015
• Status: Submission Closed
• Lead Guest Editor: Hiroshi Isshiki
About This Special Issue
There are several numerical methods for solving initial boundary value problems such as Finite Difference Method (FDM), Finite Element Method (FEM) usually based on variational or Galerkin approximation, Boundary Element Method (BEM) based on Integral Representation Method (IRM), Smooth Particle Hydrodynamics (SPH), Moving Particle Method (MPM), Vortex Blob Method, Collocation Method and so on. These methods originate from the difference how to discretize the mathematically continuous equations. Each method has its own long and short points. For example, FEM is very flexible to the complex geometry, but mesh-division may invite a serious problem. Since BEM is developed for linear problems, it’s very efficient means of numerical calculation, but it can’t cope with nonlinearity properly. IRM discussed in the special issue is intended to overcome the limit of BEM. In IRM, the fundamental solution is a key to the method, and we usually use the fundamental solution of the linearlized problem. However, this may also invite some difficulties. In case of FEM, the idea using the variational principle is replaced by Galerkin’s method to cope with problems not having the variational principle. Similarly, IRM should also replaced by Generalized Integral representation Method (GIRM) where the fundamental solution is chosen more flexibly.

Generally speaking, physical phenomena are described as boundary value problems in differential equations. We refer to this type of problem as a differential-type boundary value problem. If we use a fundamental solution of the differential equations, we can derive integral representations from the differential-type boundary value problem. If we substitute the boundary conditions into the integral representations, we obtain the integral equations. We can determine the unknown variables by solving the integral equations. The integral representations are equivalent to the differential-type boundary value problem. Hence, we refer to the boundary value problem expressed by the integral representations as the integral-type boundary value problem.

In the FEM, we use simple interpolation functions in the elements. This may reduce the degrees of freedom of the interpolation functions, and we overcome this difficulty by increasing the number of elements. As such, we face a serious problem in the mesh division. In the IRM, since the continuity of unknown variables between the elements is not required explicitly, an easier division into elements and a higher precision interpolation are realized, and a mesh-free approach would be possible.
Lead Guest Editor
• Hiroshi Isshiki

Institute of Mathematicl Anaysis, Sayama, Japan

Guest Editors
• Mahmood Jasim

Head of Mathematics & Physical Sciences, College of Arts & Sciences, University of Nizwa, Oman

• Toshio Takiya

Hitachi Zosen Corporation, Osaka, Japan

• Fabio Da Rocha

Department of Civil Engineering, Federal University of Sergipe, Brazil

Published Articles
• 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
DOI: 10.11648/j.acm.s.2015040301.16
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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... Show More
• 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
DOI: 10.11648/j.acm.s.2015040301.15
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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 ... Show More
• Hiroshi Isshiki

Issue: Volume 4, Issue 3-1, June 2015
Pages: 52-58
Received: 25 February 2015
Accepted: 25 February 2015
Published: 26 March 2015
DOI: 10.11648/j.acm.s.2015040301.14
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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... Show More
• Hiroshi Isshiki

Issue: Volume 4, Issue 3-1, June 2015
Pages: 40-51
Received: 5 February 2015
Accepted: 6 February 2015
Published: 13 March 2015
DOI: 10.11648/j.acm.s.2015040301.13
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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... Show More
• 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
DOI: 10.11648/j.acm.s.2015040301.12
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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... Show More
• Hiroshi Isshiki

Issue: Volume 4, Issue 3-1, June 2015
Pages: 1-14
Received: 26 December 2014
Accepted: 30 December 2014
Published: 12 February 2015
DOI: 10.11648/j.acm.s.2015040301.11
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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... Show More