Numerical integration is one of the important branch of mathematics. Singular integrals arises in different applications in applied and engineering mathematics. The evaluation of singular integrals is one of the most challenging jobs. Earlier different techniques were developed for evaluating such integrals, but these were not straightforward. Recently various order straightforward formulae have been developed for evaluating such integrals but; all these integral formulae depend on Romberg technique for more accurate results. Based on these integral formulae, different order (up to fifth) implicit methods have been developed for solving singular initial value problems. These implicit methods give better results than those obtained by implicit Runge-Kutta methods but; the derivation of such higher order formulae are not so easy. In this article, a new third order straightforward integral formula has been proposed for evaluating singular integrals. This new formula is able to evaluate more efficiently than others existing formulae, moreover it has the independent ability to calculate very near accurate result to the exact value of the numerical integrals. Based on this new integral formula a new third order implicit method has been proposed for solving singular initial value problems. The new method provides significantly better results than other existing methods.
| Published in | American Journal of Applied Mathematics (Volume 8, Issue 5) |
| DOI | 10.11648/j.ajam.20200805.14 |
| Page(s) | 265-270 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Singular Integrals, Romberg Scheme, Singular Initial Value Problems, Implicit Runge-Kutta Methods
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APA Style
Md. Habibur Rahaman, M. Kamrul Hasan, Md. Ayub Ali, Md. Shamsul Alam. (2020). Approximate Numerical Solution of Singular Integrals and Singular Initial Value Problems. American Journal of Applied Mathematics, 8(5), 265-270. https://doi.org/10.11648/j.ajam.20200805.14
ACS Style
Md. Habibur Rahaman; M. Kamrul Hasan; Md. Ayub Ali; Md. Shamsul Alam. Approximate Numerical Solution of Singular Integrals and Singular Initial Value Problems. Am. J. Appl. Math. 2020, 8(5), 265-270. doi: 10.11648/j.ajam.20200805.14
AMA Style
Md. Habibur Rahaman, M. Kamrul Hasan, Md. Ayub Ali, Md. Shamsul Alam. Approximate Numerical Solution of Singular Integrals and Singular Initial Value Problems. Am J Appl Math. 2020;8(5):265-270. doi: 10.11648/j.ajam.20200805.14
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Download@article{10.11648/j.ajam.20200805.14,
author = {Md. Habibur Rahaman and M. Kamrul Hasan and Md. Ayub Ali and Md. Shamsul Alam},
title = {Approximate Numerical Solution of Singular Integrals and Singular Initial Value Problems},
journal = {American Journal of Applied Mathematics},
volume = {8},
number = {5},
pages = {265-270},
doi = {10.11648/j.ajam.20200805.14},
url = {https://doi.org/10.11648/j.ajam.20200805.14},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200805.14},
abstract = {Numerical integration is one of the important branch of mathematics. Singular integrals arises in different applications in applied and engineering mathematics. The evaluation of singular integrals is one of the most challenging jobs. Earlier different techniques were developed for evaluating such integrals, but these were not straightforward. Recently various order straightforward formulae have been developed for evaluating such integrals but; all these integral formulae depend on Romberg technique for more accurate results. Based on these integral formulae, different order (up to fifth) implicit methods have been developed for solving singular initial value problems. These implicit methods give better results than those obtained by implicit Runge-Kutta methods but; the derivation of such higher order formulae are not so easy. In this article, a new third order straightforward integral formula has been proposed for evaluating singular integrals. This new formula is able to evaluate more efficiently than others existing formulae, moreover it has the independent ability to calculate very near accurate result to the exact value of the numerical integrals. Based on this new integral formula a new third order implicit method has been proposed for solving singular initial value problems. The new method provides significantly better results than other existing methods.},
year = {2020}
}
TY - JOUR T1 - Approximate Numerical Solution of Singular Integrals and Singular Initial Value Problems AU - Md. Habibur Rahaman AU - M. Kamrul Hasan AU - Md. Ayub Ali AU - Md. Shamsul Alam Y1 - 2020/09/21 PY - 2020 N1 - https://doi.org/10.11648/j.ajam.20200805.14 DO - 10.11648/j.ajam.20200805.14 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 265 EP - 270 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20200805.14 AB - Numerical integration is one of the important branch of mathematics. Singular integrals arises in different applications in applied and engineering mathematics. The evaluation of singular integrals is one of the most challenging jobs. Earlier different techniques were developed for evaluating such integrals, but these were not straightforward. Recently various order straightforward formulae have been developed for evaluating such integrals but; all these integral formulae depend on Romberg technique for more accurate results. Based on these integral formulae, different order (up to fifth) implicit methods have been developed for solving singular initial value problems. These implicit methods give better results than those obtained by implicit Runge-Kutta methods but; the derivation of such higher order formulae are not so easy. In this article, a new third order straightforward integral formula has been proposed for evaluating singular integrals. This new formula is able to evaluate more efficiently than others existing formulae, moreover it has the independent ability to calculate very near accurate result to the exact value of the numerical integrals. Based on this new integral formula a new third order implicit method has been proposed for solving singular initial value problems. The new method provides significantly better results than other existing methods. VL - 8 IS - 5 ER -
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