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A Computational Scheme for 1D Time-Dependent Singularly Perturbed Parabolic Differential-Difference Equations.

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Title: A Computational Scheme for 1D Time-Dependent Singularly Perturbed Parabolic Differential-Difference Equations.
Authors: Prasad, E. Siva1 (AUTHOR) emineni@yahoo.co.in, Phaneendra, K.2 (AUTHOR) kollojuphaneendra@yahoo.co.in
Source: Computational Mathematics & Mathematical Physics. Feb2025, Vol. 65 Issue 2, p236-251. 16p.
Subjects: Differential-difference equations, Taylor's series, Euler method, Discretization methods, Splines
Abstract: In this research, we proposed a fitted numerical scheme for time-dependent singularly perturbed parabolic equations with small retarded terms in the reaction terms. When the delay and advanced terms are of small order of perturbation, Taylor's series expansion is used to approximate delay terms. The resulting equations is solved by using the classical backward Euler method for the discretization of time variable and the adaptive cubic spline method in spatial discretize the variables on a uniform mesh. The suggested numerical technique is shown to be a parameter uniformly convergent of first order in time and second order in spatial direction. Numerical tests are given to validate the effectiveness of the adaptive cubic spline method, as well as to validate theoretical investigations and compare the numerical results with the other methods available in the literature. It is observed that the suggested method produces more accurate approximation findings and has a higher rate of convergence than the other methods currently accessible in the literature. [ABSTRACT FROM AUTHOR]
Copyright of Computational Mathematics & Mathematical Physics is the property of Springer Nature 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.)
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  Data: A Computational Scheme for 1D Time-Dependent Singularly Perturbed Parabolic Differential-Difference Equations.
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  Data: <searchLink fieldCode="JN" term="%22Computational+Mathematics+%26+Mathematical+Physics%22">Computational Mathematics & Mathematical Physics</searchLink>. Feb2025, Vol. 65 Issue 2, p236-251. 16p.
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  Label: Abstract
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  Data: In this research, we proposed a fitted numerical scheme for time-dependent singularly perturbed parabolic equations with small retarded terms in the reaction terms. When the delay and advanced terms are of small order of perturbation, Taylor's series expansion is used to approximate delay terms. The resulting equations is solved by using the classical backward Euler method for the discretization of time variable and the adaptive cubic spline method in spatial discretize the variables on a uniform mesh. The suggested numerical technique is shown to be a parameter uniformly convergent of first order in time and second order in spatial direction. Numerical tests are given to validate the effectiveness of the adaptive cubic spline method, as well as to validate theoretical investigations and compare the numerical results with the other methods available in the literature. It is observed that the suggested method produces more accurate approximation findings and has a higher rate of convergence than the other methods currently accessible in the literature. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Computational Mathematics & Mathematical Physics is the property of Springer Nature 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.)
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        Value: 10.1134/S096554252470194X
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        Text: English
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      – SubjectFull: Differential-difference equations
        Type: general
      – SubjectFull: Taylor's series
        Type: general
      – SubjectFull: Euler method
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      – SubjectFull: Discretization methods
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      – SubjectFull: Splines
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      – TitleFull: A Computational Scheme for 1D Time-Dependent Singularly Perturbed Parabolic Differential-Difference Equations.
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              Text: Feb2025
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