Applied and Computational Mathematics

Special Issue

Integral Representation Method and its Generalization

  • Submission Deadline: Apr. 20, 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
  • 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: Apr. 17, 2015
    Accepted: Apr. 17, 2015
    Published: May 12, 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
  • 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: Mar. 19, 2015
    Accepted: Mar. 23, 2015
    Published: Apr. 08, 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
  • Application of the Generalized Integral Representation Method (GIRM) to Tidal Wave Propagation

    Hiroshi Isshiki

    Issue: Volume 4, Issue 3-1, June 2015
    Pages: 52-58
    Received: Feb. 25, 2015
    Accepted: Feb. 25, 2015
    Published: Mar. 26, 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
  • Effects of Generalized Fundamental Solution (GFS) on Generalized Integral Representation Method (GIRM)

    Hiroshi Isshiki

    Issue: Volume 4, Issue 3-1, June 2015
    Pages: 40-51
    Received: Feb. 05, 2015
    Accepted: Feb. 06, 2015
    Published: Mar. 13, 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
  • 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: Dec. 22, 2014
    Accepted: Dec. 25, 2014
    Published: Feb. 12, 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
  • From Integral Representation Method (IRM) to Generalized Integral Representation Method (GIRM)

    Hiroshi Isshiki

    Issue: Volume 4, Issue 3-1, June 2015
    Pages: 1-14
    Received: Dec. 26, 2014
    Accepted: Dec. 30, 2014
    Published: Feb. 12, 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