View in EDS

Legendre spectral collocation method for one- and two-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations.

Saved in:
Bibliographic Details
Title: Legendre spectral collocation method for one- and two-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations.
Authors: Ezz-Eldien, S. S.1 (AUTHOR) s_sezeldien@yahoo.com, Alalyani, A.2 (AUTHOR)
Source: International Journal of Modern Physics C: Computational Physics & Physical Computation. Feb2026, Vol. 37 Issue 2, p1-11. 11p.
Subjects: Nonlinear equations, Pantograph, Legendre's polynomials, Volterra equations, Integro-differential equations, Numerical analysis, Approximation algorithms
Abstract: The integration of functional, integral and delay components of pantograph Volterra–Fredholm integro-differential equations provides a powerful framework for modeling complex systems with interdependent dynamics, particularly where nonlinearity influences and proportional feedback are essential. Numerical approximations to multi-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations present significant challenges due to the integration of nonlinearity, proportional delays and mixed integral terms, necessitating adaptive methods to achieve highly accurate approximations. In this work, we extend the Legendre spectral approximation to the one- and two-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations. In this method, the Legendre differentiation matrix and the pantograph operational matrix are used to manage the proportional delay terms inherent in these equations. To demonstrate the superiority of the proposed scheme, we present comparisons with other established spectral methods, highlighting the advantages of the Legendre spectral approach. [ABSTRACT FROM AUTHOR]
Copyright of International Journal of Modern Physics C: Computational Physics & Physical Computation is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
FullText Text:
  Availability: 0
Header DbId: egs
DbLabel: Engineering Source
An: 189477022
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: Legendre spectral collocation method for one- and two-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations.
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Ezz-Eldien%2C+S%2E+S%2E%22">Ezz-Eldien, S. S.</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> s_sezeldien@yahoo.com</i><br /><searchLink fieldCode="AR" term="%22Alalyani%2C+A%2E%22">Alalyani, A.</searchLink><relatesTo>2</relatesTo> (AUTHOR)
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="JN" term="%22International+Journal+of+Modern+Physics+C%3A+Computational+Physics+%26+Physical+Computation%22">International Journal of Modern Physics C: Computational Physics & Physical Computation</searchLink>. Feb2026, Vol. 37 Issue 2, p1-11. 11p.
– Name: Subject
  Label: Subjects
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Nonlinear+equations%22">Nonlinear equations</searchLink><br /><searchLink fieldCode="DE" term="%22Pantograph%22">Pantograph</searchLink><br /><searchLink fieldCode="DE" term="%22Legendre's+polynomials%22">Legendre's polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Volterra+equations%22">Volterra equations</searchLink><br /><searchLink fieldCode="DE" term="%22Integro-differential+equations%22">Integro-differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Approximation+algorithms%22">Approximation algorithms</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: The integration of functional, integral and delay components of pantograph Volterra–Fredholm integro-differential equations provides a powerful framework for modeling complex systems with interdependent dynamics, particularly where nonlinearity influences and proportional feedback are essential. Numerical approximations to multi-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations present significant challenges due to the integration of nonlinearity, proportional delays and mixed integral terms, necessitating adaptive methods to achieve highly accurate approximations. In this work, we extend the Legendre spectral approximation to the one- and two-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations. In this method, the Legendre differentiation matrix and the pantograph operational matrix are used to manage the proportional delay terms inherent in these equations. To demonstrate the superiority of the proposed scheme, we present comparisons with other established spectral methods, highlighting the advantages of the Legendre spectral approach. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of International Journal of Modern Physics C: Computational Physics & Physical Computation is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=189477022
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1142/S0129183125500615
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 11
        StartPage: 1
    Subjects:
      – SubjectFull: Nonlinear equations
        Type: general
      – SubjectFull: Pantograph
        Type: general
      – SubjectFull: Legendre's polynomials
        Type: general
      – SubjectFull: Volterra equations
        Type: general
      – SubjectFull: Integro-differential equations
        Type: general
      – SubjectFull: Numerical analysis
        Type: general
      – SubjectFull: Approximation algorithms
        Type: general
    Titles:
      – TitleFull: Legendre spectral collocation method for one- and two-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations.
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Ezz-Eldien, S. S.
      – PersonEntity:
          Name:
            NameFull: Alalyani, A.
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 02
              Text: Feb2026
              Type: published
              Y: 2026
          Identifiers:
            – Type: issn-print
              Value: 01291831
          Numbering:
            – Type: volume
              Value: 37
            – Type: issue
              Value: 2
          Titles:
            – TitleFull: International Journal of Modern Physics C: Computational Physics & Physical Computation
              Type: main
ResultId 1