NEW EFFECTIVE TRANSFORMATIONAL COMPUTATIONAL METHODS

Jun Zhang; Ruzong Fan; Fangyang Shen; Junyi Tu

doi:10.11948/20230222

2024 Volume 14 Issue 1

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Jun Zhang, Ruzong Fan, Fangyang Shen, Junyi Tu. NEW EFFECTIVE TRANSFORMATIONAL COMPUTATIONAL METHODS[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 317-333. doi: 10.11948/20230222

Citation:

Jun Zhang, Ruzong Fan, Fangyang Shen, Junyi Tu. NEW EFFECTIVE TRANSFORMATIONAL COMPUTATIONAL METHODS[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 317-333. doi: 10.11948/20230222

NEW EFFECTIVE TRANSFORMATIONAL COMPUTATIONAL METHODS

1.
Department of Computer Science and Engineering Technology, University of Maryland Eastern Shore, Princess Anne, MD 21853, USA
2.
Department of Biostatistics, Bioinformatics, and Biomathematics, Georgetown University, Washington, DC 20057, USA
3.
Department of Computer Systems Technology, New York City College of Technology, CUNY, Brooklyn, NY 11201, USA
4.
Department of Computer Science, Salisbury University, Salisbury, MD 21801, USA

Author Bio: Email: rf740@georgetown.edu(R. Fan); Email: fshen@citytech.cuny.edu(F. Shen); Email: jxtu@salisbury.edu(J. Tu)
Corresponding author: Email: jzhang@umes.edu(J. Zhang)

Abstract

Abstract

Mathematics serves as a fundamental intelligent theoretic basis for computation, and mathematical analysis is very useful to develop computational methods to solve various problems in science and engineering. Integral transforms such as Laplace Transform have been playing an important role in computational methods. In this paper, we will introduce Sumudu Transform in a new computational approach, in which effective computational methods will be developed and implemented. Such computational methods are straightforward to understand, but powerful to incorporate into computational science to solve different problems automatically. We will provide computational analysis and essentiality by surveying and summarizing some related recent works, with additional automatic proof details by applying system built-in functions. Applications include the computation of coefficients of Taylor's expansions, calculation of generating functions, mathematical identity proofs, solving differential equations and integral equations. For demonstration purposes, some of the methods were implemented in Maple with demonstrational results matching the expected values.
- Sumudu transform /
- automatic calculation /
- computation of coefficient /
- generating function /
- identity proof /
- differential equation /
- integral equation /
- problem solving
MSC: 33F10, 34-04, 40-04, 44-04, 44A10, 44A30, 45-04

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References

[1]	S. K. Al-Omari, The q-Sumudu transform and its certain properties in a generalized q-calculus theory, Advances in Difference Equations, 2021, 2021, Article number: 10. doi: 10.1186/s13662-020-03147-1 CrossRef Google Scholar
[2]	S. Alfaqeih, G. Bakıcıerler and E. Misirli, Conformable double Sumudu transform with applications, Journal of Applied and Computational Mechanics, 2021, 7(2), 578–586. Google Scholar
[3]	A. Atangana, A new defy for iteration methods, Journal of Applied Analysis and Computation, 2015, 5(3), 273–283. doi: 10.11948/2015025 CrossRef Google Scholar
[4]	F. B. M. Belgacem, Introducing and analysing deeper Sumudu properties, Nonlinear Studies, 2006, 13(1), 23–41. Google Scholar
[5]	F. B. M. Belgacem and A. A. Karaballi, Sumudu transform fundamental properties investigations and applications, Journal of Applied Mathematics and Stochastic Analysis, 2006, 2006, 1–23. Google Scholar
[6]	C. Epstein and J. Schotland, The Bad Truth about Laplace's Transform, SIAM Review, 2008, 50(3), 504–552. doi: 10.1137/060657273 CrossRef Google Scholar
[7]	A. Golmankhaneh and C. Tunç, Sumudu transform in fractal calculus, Applied Mathematics and Computation, 2019, 350, 386–401. doi: 10.1016/j.amc.2019.01.025 CrossRef Google Scholar
[8]	M. Kapoor, Sumudu transform for time fractional physical models an analytical aspect, Journal of Applied Analysis and Computation, 2023, 13(3), 1255–1273. Google Scholar
[9]	A. Kilicman and H. Hassan Eltayeb, Some Remarks on the Sumudu and Laplace Transforms and Applications to Differential Equations, ISRN Applied Mathematics, 2012. Google Scholar
[10]	K. Kuhlman, Review of inverse Laplace transform algorithms for Laplace-space numerical approaches, Numerical Algorithms, 2013, 63, 339–355. doi: 10.1007/s11075-012-9625-3 CrossRef Google Scholar
[11]	M. S. Mechee and A. J. Naeemah, A study of triple Sumudu transform for solving partial differential equations with some applications, Multidisciplinary European Academic Journal, 2020, 2(2). Google Scholar
[12]	M. Petkovsek, H. S. Wilfe and D. Zeilberger, A = B, A. K. Peters, Ltd, 1996. Google Scholar
[13]	R. A. Ravenscroft, Rational Generating Function Applications in Maple V: Mathematics and its Application, R. Lopez (ed.), Birkhaser, 1994. Google Scholar
[14]	R. A. Ravenscroft and E. A. Lamagna, Symbolic summation with generating functions, Proceedings of the International Symposium on Symbolic and Algebraic Computation, Association for Computing Machinery, 1989, 228–233. Google Scholar
[15]	M. S. Rawashdeh and S. Maitama, Solving nonlinear ordinary differential equations using the NDM, Journal of Applied Analysis and Computation, 2015, 5(1), 77–88. Google Scholar
[16]	D. V. Widder, An Introduction to Transform Theory, Academic Press, 1971. Google Scholar
[17]	J. Zhang, Sumudu Transform based coefficients calculation, Nonlinear Studies, 2008, 15(4), 355–372. Google Scholar
[18]	J. Zhang, R. Fan and F. Shen, New method for the computation of generating functions with applications, The 2021 International Conference on Computational Science and Computational Intelligence, IEEE, 2021, Las Vegas, USA. Google Scholar
[19]	J. Zhang and F. Shen, Sumudu transform for automatic mathematical proof and identity generation, The 14th International Conference on Information Technology: New Generations, Springer, 2017, Las Vegas, USA. Google Scholar
[20]	J. Zhang, F. Shen and Y. Waguespack, Incorporating generating functions to computational science education, The 2016 International Conference on Computational Science and Computational Intelligence, IEEE, 2016, Las Vegas, USA. Google Scholar
[21]	J. Zhang, W. Zhu and F. Shen, New techniques to solve differential equations automatically, The 2017 International Conference on Computational Science and Computational Intelligence, IEEE, 2017, Las Vegas, USA. Google Scholar
[22]	J. Zhang, W. Zhu and F. Shen, New algorithms to solve integral equations automatically, The 2019 International Conference on Computational Science and Computational Intelligence, IEEE, 2019, Las Vegas, USA. Google Scholar
[23]	W. Zhao and S. Maitama, Beyond Sumudu transform and natural transform: Transform properties and applications, Journal of Applied Analysis and Computation, 2020, 10(4), 1223–1241. Google Scholar