Numerical Simulations of Third-Order Singularly Perturbed Boundary Value Problems

Azhar Iqbal; Tayyaba Akram; Waseem Ahmad Khan; Ajmal Ali

doi:10.1007/978-981-97-2089-7_8

Numerical Simulations of Third-Order Singularly Perturbed Boundary Value Problems

Azhar Iqbal^*
, Tayyaba Akram
, Waseem Ahmad Khan
, Ajmal Ali

^*Corresponding author for this work

Department of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

This paper introduces a quartic trigonometric B-spline (QTBS) for the purpose of producing numerical solutions for a third-order singularly perturbed boundary value problem (SpBvp) characterized by the presence of a minuscule parameter multiplying the highest-order derivative. The QTBS basis function, denoted as T_i(x), is applied at various nodal points. This approach is employed subsequent to the modification of the problem at the singularity point through the utilization of the L’Hopital rule. Resulting system of equations is solved by Mathematica to get the solution. These present numerical results are compared with the numerical results that exist in the literature like the method presented by Mishra (Am J Numer Anal 3(1):18–24, 2015) [1]. This demonstrates that the approximate solutions produced by the numerical algorithm developed using the QTBS are better than the regular quartic B-spline (QBS).

Original language	English
Title of host publication	Soft Computing
Subtitle of host publication	Theories and Applications - Proceedings of SoCTA 2023
Editors	Rajesh Kumar, Ajit Kumar Verma, Om Prakash Verma, Tanu Wadehra
Publisher	Springer Science and Business Media Deutschland GmbH
Pages	79-91
Number of pages	13
ISBN (Print)	9789819720880
DOIs	https://doi.org/10.1007/978-981-97-2089-7_8
State	Published - 2024
Event	7th International Conference on Soft Computing: Theories and Applications, SoCTA 2023 - Una, India Duration: 21 Dec 2023 → 23 Dec 2023

Publication series

Name	Lecture Notes in Networks and Systems
Volume	971 LNNS
ISSN (Print)	2367-3370
ISSN (Electronic)	2367-3389

Conference

Conference	7th International Conference on Soft Computing: Theories and Applications, SoCTA 2023
Country/Territory	India
City	Una
Period	21/12/23 → 23/12/23

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024.

Keywords

Basis function
Quartic trigonometric B-spline
Singularly perturbation

ASJC Scopus subject areas

Control and Systems Engineering
Signal Processing
Computer Networks and Communications

Access to Document

10.1007/978-981-97-2089-7_8

Cite this

Iqbal, A., Akram, T., Khan, W. A., & Ali, A. (2024). Numerical Simulations of Third-Order Singularly Perturbed Boundary Value Problems. In R. Kumar, A. K. Verma, O. P. Verma, & T. Wadehra (Eds.), Soft Computing: Theories and Applications - Proceedings of SoCTA 2023 (pp. 79-91). (Lecture Notes in Networks and Systems; Vol. 971 LNNS). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/978-981-97-2089-7_8

Iqbal, Azhar ; Akram, Tayyaba ; Khan, Waseem Ahmad et al. / Numerical Simulations of Third-Order Singularly Perturbed Boundary Value Problems. Soft Computing: Theories and Applications - Proceedings of SoCTA 2023. editor / Rajesh Kumar ; Ajit Kumar Verma ; Om Prakash Verma ; Tanu Wadehra. Springer Science and Business Media Deutschland GmbH, 2024. pp. 79-91 (Lecture Notes in Networks and Systems).

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abstract = "This paper introduces a quartic trigonometric B-spline (QTBS) for the purpose of producing numerical solutions for a third-order singularly perturbed boundary value problem (SpBvp) characterized by the presence of a minuscule parameter multiplying the highest-order derivative. The QTBS basis function, denoted as Ti(x), is applied at various nodal points. This approach is employed subsequent to the modification of the problem at the singularity point through the utilization of the L{\textquoteright}Hopital rule. Resulting system of equations is solved by Mathematica to get the solution. These present numerical results are compared with the numerical results that exist in the literature like the method presented by Mishra (Am J Numer Anal 3(1):18–24, 2015) [1]. This demonstrates that the approximate solutions produced by the numerical algorithm developed using the QTBS are better than the regular quartic B-spline (QBS).",

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Iqbal, A, Akram, T, Khan, WA & Ali, A 2024, Numerical Simulations of Third-Order Singularly Perturbed Boundary Value Problems. in R Kumar, AK Verma, OP Verma & T Wadehra (eds), Soft Computing: Theories and Applications - Proceedings of SoCTA 2023. Lecture Notes in Networks and Systems, vol. 971 LNNS, Springer Science and Business Media Deutschland GmbH, pp. 79-91, 7th International Conference on Soft Computing: Theories and Applications, SoCTA 2023, Una, India, 21/12/23. https://doi.org/10.1007/978-981-97-2089-7_8

Numerical Simulations of Third-Order Singularly Perturbed Boundary Value Problems. / Iqbal, Azhar; Akram, Tayyaba; Khan, Waseem Ahmad et al.
Soft Computing: Theories and Applications - Proceedings of SoCTA 2023. ed. / Rajesh Kumar; Ajit Kumar Verma; Om Prakash Verma; Tanu Wadehra. Springer Science and Business Media Deutschland GmbH, 2024. p. 79-91 (Lecture Notes in Networks and Systems; Vol. 971 LNNS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - This paper introduces a quartic trigonometric B-spline (QTBS) for the purpose of producing numerical solutions for a third-order singularly perturbed boundary value problem (SpBvp) characterized by the presence of a minuscule parameter multiplying the highest-order derivative. The QTBS basis function, denoted as Ti(x), is applied at various nodal points. This approach is employed subsequent to the modification of the problem at the singularity point through the utilization of the L’Hopital rule. Resulting system of equations is solved by Mathematica to get the solution. These present numerical results are compared with the numerical results that exist in the literature like the method presented by Mishra (Am J Numer Anal 3(1):18–24, 2015) [1]. This demonstrates that the approximate solutions produced by the numerical algorithm developed using the QTBS are better than the regular quartic B-spline (QBS).

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Iqbal A, Akram T, Khan WA, Ali A. Numerical Simulations of Third-Order Singularly Perturbed Boundary Value Problems. In Kumar R, Verma AK, Verma OP, Wadehra T, editors, Soft Computing: Theories and Applications - Proceedings of SoCTA 2023. Springer Science and Business Media Deutschland GmbH. 2024. p. 79-91. (Lecture Notes in Networks and Systems). doi: 10.1007/978-981-97-2089-7_8