DOI: 10.1002/num.10033. Polynomial particular solutions for certain partial differential operators

Polynomial particular solutions for certain partial differential operators

10.1002/num.10033

Crossref journal-article

Wiley

Numerical Methods for Partial Differential Equations (311)

Abstract

AbstractIn this article, we consider a variant of the Dual Reciprocity Method (DRM) for solving boundary value problems based on approximating source terms by polynomials other than the traditional basis functions. The use of pseudo‐spectral approximations and symbolic methods enables us to obtain highly accurate results without solving the often ill‐conditioned equations that occur when radial basis function approximations are used. When the given partial differential equation is either Poisson's equation or an inhomogeneous Helmholtz‐type equation, we are able to obtain either closed form particular solutions or efficient recursive algorithms. Using the particular solutions, we convert the inhomogeneous equations to homogeneous. The resulting homogeneous equations are then amenable to solution by boundary‐type methods such as the Boundary Element Method (BEM) or the Method of Fundamental Solutions (MFS). Using the MFS, we provide numerical solutions to a variety of boundary value problems in R2 and R3. Using this approach, we can achieve high accuracy with a modest number of interpolation and collocation points. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 112–133, 2003

Bibliography

Golberg, M. A., Muleshkov, A. S., Chen, C. S., & Cheng, A. H. âD. (2002). Polynomial particular solutions for certain partial differential operators. Numerical Methods for Partial Differential Equations, 19(1), 112â133. Portico.

Authors 4

M. A. Golberg (first)
A. S. Muleshkov (additional)
C. S. Chen (additional)
A. H.‐D. Cheng (additional)

References 21 Referenced 64

10.1007/978-3-662-11273-1_22
{'key': 'e_1_2_1_3_2', 'volume-title': 'The dual reciprocity boundary element method', 'author': 'Partridge P. W.', 'year': '1992'} / The dual reciprocity boundary element method by Partridge P. W. (1992)
10.1007/s004660050420
{'key': 'e_1_2_1_5_2', 'volume-title': 'Discrete projection methods for integral equations', 'author': 'Golberg M. A.', 'year': '1997'} / Discrete projection methods for integral equations by Golberg M. A. (1997)
10.1093/imanum/5.3.319
10.1016/0955-7997(94)90024-8
{'key': 'e_1_2_1_8_2', 'first-page': '27', 'volume-title': 'Advances in computational engineering and sciences', 'author': 'Muleshkov A. S.', 'year': '2000'} / Advances in computational engineering and sciences by Muleshkov A. S. (2000)
10.1007/s004660050344
{'key': 'e_1_2_1_10_2', 'first-page': '103', 'volume-title': 'Boundary integral methods: numerical and mathematical aspects', 'author': 'Golberg M. A.', 'year': '1999'} / Boundary integral methods: numerical and mathematical aspects by Golberg M. A. (1999)
10.1216/jiea/1181075987
{'key': 'e_1_2_1_12_2', 'series-title': 'Boundary element technology', 'first-page': '377', 'author': 'Chen C. S.', 'year': '1999'} / Boundary element technology by Chen C. S. (1999)
10.1007/978-3-642-84108-8
{'key': 'e_1_2_1_14_2', 'volume-title': 'Spectral methods in fluid dynamics', 'author': 'Boyd J. P.', 'year': '2001'} / Spectral methods in fluid dynamics by Boyd J. P. (2001)
10.1016/S1570-8659(97)80003-8 / Handbook of numerical analysis by Bernardi C. (1997)
10.1090/S0025-5718-1982-0637287-3
{'key': 'e_1_2_1_17_2', 'series-title': 'Boundary element methods', 'first-page': '313', 'author': 'Chen C. S.', 'year': '1999'} / Boundary element methods by Chen C. S. (1999)
10.1023/A:1018981221740
10.1137/0722040
10.1016/S0895-7177(99)00233-2
10.1023/A:1012873712701
10.1016/S0955-7997(96)00033-1

Dates

Type	When
Created	22 years, 8 months ago (Dec. 11, 2002, 8:59 a.m.)
Deposited	1 year, 9 months ago (Nov. 17, 2023, 7:36 p.m.)
Indexed	4 months ago (April 14, 2025, 10:59 p.m.)
Issued	22 years, 8 months ago (Dec. 10, 2002)
Published	22 years, 8 months ago (Dec. 10, 2002)
Published Online	22 years, 8 months ago (Dec. 10, 2002)
Published Print	22 years, 7 months ago (Jan. 1, 2003)

Funders 0

None

BibTeX

@article{Golberg_2002, title={Polynomial particular solutions for certain partial differential operators}, volume={19}, ISSN={1098-2426}, url={http://dx.doi.org/10.1002/num.10033}, DOI={10.1002/num.10033}, number={1}, journal={Numerical Methods for Partial Differential Equations}, publisher={Wiley}, author={Golberg, M. A. and Muleshkov, A. S. and Chen, C. S. and Cheng, A. H.‐D.}, year={2002}, month=dec, pages={112–133} }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        4,
        15
      ]
    ],
    "date-time": "2025-04-15T02:59:59Z",
    "timestamp": 1744685999228
  },
  "reference-count": 21,
  "publisher": "Wiley",
  "issue": "1",
  "license": [
    {
      "start": {
        "date-parts": [
          [
            2002,
            12,
            10
          ]
        ],
        "date-time": "2002-12-10T00:00:00Z",
        "timestamp": 1039478400000
      },
      "content-version": "vor",
      "delay-in-days": 0,
      "URL": "http://onlinelibrary.wiley.com/termsAndConditions#vor"
    }
  ],
  "content-domain": {
    "domain": [],
    "crossmark-restriction": false
  },
  "published-print": {
    "date-parts": [
      [
        2003,
        1
      ]
    ]
  },
  "abstract": "<jats:title>Abstract</jats:title><jats:p>In this article, we consider a variant of the Dual Reciprocity Method (DRM) for solving boundary value problems based on approximating source terms by polynomials other than the traditional basis functions. The use of pseudo\u2010spectral approximations and symbolic methods enables us to obtain highly accurate results without solving the often ill\u2010conditioned equations that occur when radial basis function approximations are used. When the given partial differential equation is either Poisson's equation or an inhomogeneous Helmholtz\u2010type equation, we are able to obtain either closed form particular solutions or efficient recursive algorithms. Using the particular solutions, we convert the inhomogeneous equations to homogeneous. The resulting homogeneous equations are then amenable to solution by boundary\u2010type methods such as the Boundary Element Method (BEM) or the Method of Fundamental Solutions (MFS). Using the MFS, we provide numerical solutions to a variety of boundary value problems in <jats:bold>R<jats:sup>2</jats:sup></jats:bold> and <jats:bold>R<jats:sup>3</jats:sup></jats:bold>. Using this approach, we can achieve high accuracy with a modest number of interpolation and collocation points. \u00a9 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 112\u2013133, 2003</jats:p>",
  "DOI": "10.1002/num.10033",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2002,
        12,
        11
      ]
    ],
    "date-time": "2002-12-11T13:59:17Z",
    "timestamp": 1039615157000
  },
  "page": "112-133",
  "source": "Crossref",
  "is-referenced-by-count": 64,
  "title": "Polynomial particular solutions for certain partial differential operators",
  "prefix": "10.1002",
  "volume": "19",
  "author": [
    {
      "given": "M. A.",
      "family": "Golberg",
      "sequence": "first",
      "affiliation": []
    },
    {
      "given": "A. S.",
      "family": "Muleshkov",
      "sequence": "additional",
      "affiliation": []
    },
    {
      "given": "C. S.",
      "family": "Chen",
      "sequence": "additional",
      "affiliation": []
    },
    {
      "given": "A. H.\u2010D.",
      "family": "Cheng",
      "sequence": "additional",
      "affiliation": []
    }
  ],
  "member": "311",
  "published-online": {
    "date-parts": [
      [
        2002,
        12,
        10
      ]
    ]
  },
  "reference": [
    {
      "key": "e_1_2_1_2_2",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/978-3-662-11273-1_22"
    },
    {
      "key": "e_1_2_1_3_2",
      "volume-title": "The dual reciprocity boundary element method",
      "author": "Partridge P. W.",
      "year": "1992"
    },
    {
      "key": "e_1_2_1_4_2",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/s004660050420"
    },
    {
      "key": "e_1_2_1_5_2",
      "volume-title": "Discrete projection methods for integral equations",
      "author": "Golberg M. A.",
      "year": "1997"
    },
    {
      "key": "e_1_2_1_6_2",
      "doi-asserted-by": "publisher",
      "DOI": "10.1093/imanum/5.3.319"
    },
    {
      "key": "e_1_2_1_7_2",
      "doi-asserted-by": "publisher",
      "DOI": "10.1016/0955-7997(94)90024-8"
    },
    {
      "key": "e_1_2_1_8_2",
      "first-page": "27",
      "volume-title": "Advances in computational engineering and sciences",
      "author": "Muleshkov A. S.",
      "year": "2000"
    },
    {
      "key": "e_1_2_1_9_2",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/s004660050344"
    },
    {
      "key": "e_1_2_1_10_2",
      "first-page": "103",
      "volume-title": "Boundary integral methods: numerical and mathematical aspects",
      "author": "Golberg M. A.",
      "year": "1999"
    },
    {
      "key": "e_1_2_1_11_2",
      "doi-asserted-by": "publisher",
      "DOI": "10.1216/jiea/1181075987"
    },
    {
      "key": "e_1_2_1_12_2",
      "series-title": "Boundary element technology",
      "first-page": "377",
      "author": "Chen C. S.",
      "year": "1999"
    },
    {
      "key": "e_1_2_1_13_2",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/978-3-642-84108-8"
    },
    {
      "key": "e_1_2_1_14_2",
      "volume-title": "Spectral methods in fluid dynamics",
      "author": "Boyd J. P.",
      "year": "2001"
    },
    {
      "key": "e_1_2_1_15_2",
      "series-title": "Handbook of numerical analysis",
      "doi-asserted-by": "crossref",
      "first-page": "209",
      "DOI": "10.1016/S1570-8659(97)80003-8",
      "author": "Bernardi C.",
      "year": "1997"
    },
    {
      "key": "e_1_2_1_16_2",
      "doi-asserted-by": "publisher",
      "DOI": "10.1090/S0025-5718-1982-0637287-3"
    },
    {
      "key": "e_1_2_1_17_2",
      "series-title": "Boundary element methods",
      "first-page": "313",
      "author": "Chen C. S.",
      "year": "1999"
    },
    {
      "key": "e_1_2_1_18_2",
      "doi-asserted-by": "publisher",
      "DOI": "10.1023/A:1018981221740"
    },
    {
      "key": "e_1_2_1_19_2",
      "doi-asserted-by": "publisher",
      "DOI": "10.1137/0722040"
    },
    {
      "key": "e_1_2_1_20_2",
      "doi-asserted-by": "publisher",
      "DOI": "10.1016/S0895-7177(99)00233-2"
    },
    {
      "key": "e_1_2_1_21_2",
      "doi-asserted-by": "publisher",
      "DOI": "10.1023/A:1012873712701"
    },
    {
      "key": "e_1_2_1_22_2",
      "doi-asserted-by": "publisher",
      "DOI": "10.1016/S0955-7997(96)00033-1"
    }
  ],
  "container-title": "Numerical Methods for Partial Differential Equations",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "https://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1002%2Fnum.10033",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "text-mining"
    },
    {
      "URL": "https://onlinelibrary.wiley.com/doi/pdf/10.1002/num.10033",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2023,
        11,
        18
      ]
    ],
    "date-time": "2023-11-18T00:36:34Z",
    "timestamp": 1700267794000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://onlinelibrary.wiley.com/doi/10.1002/num.10033"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        2002,
        12,
        10
      ]
    ]
  },
  "references-count": 21,
  "journal-issue": {
    "issue": "1",
    "published-print": {
      "date-parts": [
        [
          2003,
          1
        ]
      ]
    }
  },
  "alternative-id": [
    "10.1002/num.10033"
  ],
  "URL": "http://dx.doi.org/10.1002/num.10033",
  "relation": {},
  "ISSN": [
    "0749-159X",
    "1098-2426"
  ],
  "subject": [],
  "container-title-short": "Numerical Methods Partial",
  "published": {
    "date-parts": [
      [
        2002,
        12,
        10
      ]
    ]
  }
}