An efficient algorithm for the numerical evaluation of pseudo differential operator with error estimation

Amit K. Pandey; Manoj P. Tripathi; Harendra Singh; Pentyala S. Rao; Devendra Kumar; D. Baleanu; Amit K. Pandey; Manoj P. Tripathi; Harendra Singh; Pentyala S. Rao; Devendra Kumar; D. Baleanu

doi:10.3934/math.2022982

AIMS Mathematics

2022, Volume 7, Issue 10: 17829-17842. doi: 10.3934/math.2022982

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An efficient algorithm for the numerical evaluation of pseudo differential operator with error estimation

1.
Department of Mathematics and Computing, Indian Institute of Technology (ISM) Dhanbad-826004, India
2.
Department of Mathematics, Udai Pratap Autonomous College, Varanasi-221002, India
3.
Department of Mathematics, Post-Graduate College, Ghazipur-233001, Uttar Pradesh, India
4.
Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India
5.
Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Eskisehir Yolu 29. Km, Yukarıyurtcu Mahallesi Mimar Sinan Caddesi No: 406790, Etimesgut, Turkey
6.
Institute of Space Sciences, Magurele-Bucharest, Romania

Received: 15 April 2022 Revised: 26 June 2022 Accepted: 05 July 2022 Published: 03 August 2022
MSC : 33F05, 35S99, 65K99

In this paper we introduce an efficient and new numerical algorithm for evaluating a pseudo differential operator. The proposed algorithm is time saving and fruitful. The theoretical as well as numerical error estimation of the algorithm is established, together with its stability analysis. We have provided numerical illustrations and established that the numerical findings echo the analytical findings. The proposed technique has a convergence rate of order three. CPU time of computation is also listed. Trueness of numerical findings are validated using figures.
- pseudo differential operator,
- numerical algorithm,
- Hat functions
Citation: Amit K. Pandey, Manoj P. Tripathi, Harendra Singh, Pentyala S. Rao, Devendra Kumar, D. Baleanu. An efficient algorithm for the numerical evaluation of pseudo differential operator with error estimation[J]. AIMS Mathematics, 2022, 7(10): 17829-17842. doi: 10.3934/math.2022982

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Abstract

In this paper we introduce an efficient and new numerical algorithm for evaluating a pseudo differential operator. The proposed algorithm is time saving and fruitful. The theoretical as well as numerical error estimation of the algorithm is established, together with its stability analysis. We have provided numerical illustrations and established that the numerical findings echo the analytical findings. The proposed technique has a convergence rate of order three. CPU time of computation is also listed. Trueness of numerical findings are validated using figures.

References

[1]	A. H. Zemanian, Distribution theory and transform analysis, McGraw-Hill, McGraw-Hill, New York, 1965.
[2]	L. Schwartz, Theorie des distributions, Hermann, Paris, 1997.
[3]	S. Zaidman, Distribution and pseudo-differential operators, Longman, Essex, England, 1991.
[4]	O. P. Singh, J. N. Pandey, The Fourier-Bessel series representation of the pseudo differential operator ${\left({ - {x.{ - 1}}D} \right).v}$, Proc. Amer. Math. Soc., 115 (1992), 969–976. https://doi.org/10.1090/S0002-9939-1992-1107924-6
[5]	R. S. Pathak, P. K. Pandey, A class of pseudo-differential operators associated with Bessel operators, J. Math. Anal. Appl., 196 (1995), 736–747. https://doi.org/10.1006/jmaa.1995.1437 doi: 10.1006/jmaa.1995.1437
[6]	R. S. Pathak, S. K. Upadhyay, Pseudo-differential operators involving Hankel transforms, J. Math. Anal. Appl., 213 (1997), 133–147. https://doi.org/10.1006/jmaa.1997.5495 doi: 10.1006/jmaa.1997.5495
[7]	M. P. Tripathi, O. P. Singh, A Hankel transform approach to inverse quasi-static steady-state thermal stresses in a thick circular plate, Int. J. Appl. Comput. Math., 2 (2016), 609–624. https://doi.org/10.1007/s40819-015-0081-3 doi: 10.1007/s40819-015-0081-3
[8]	H. Singh, Numerical simulation for fractional delay differential equations, Int. J. Dyn. Control, 9 (2020), 463–474. https://doi.org/10.1007/s40435-020-00671-6 doi: 10.1007/s40435-020-00671-6
[9]	H. Singh, A. M. Wazwaz, Computational method for reaction diffusion-model arising in a spherical catalyst, Int. J. Appl. Comput. Math., 7 (2021), 65. https://doi.org/10.1007/s40819-021-00993-9 doi: 10.1007/s40819-021-00993-9
[10]	H. Singh, A. K. Singh, R. K. Pandey, D. Kumar, J. Singh, An efficient computational approach for fractional Bratu's equation arising in electrospinning process, Math. Methods Appl. Sci., 44 (2021), 10225–10238. https://doi.org/10.1002/mma.7401 doi: 10.1002/mma.7401
[11]	L. S. Dube, J. N. Pandey, On the Hankel transformation of distributions, Tohoku Math. J., 27 (1975), 337–354. https://doi.org/10.2748/tmj/1203529246 doi: 10.2748/tmj/1203529246
[12]	W. Y. Lee, On Schwartz's Hankel transformation of certain spaces of distributions, SIAM J. Math. Anal., 6 (1975), 427–432. https://doi.org/10.1137/0506037 doi: 10.1137/0506037
[13]	A. L. Schwartz, An inversion theorem for Hankel transforms, Proc. Amer. Math. Soc., 22 (1969), 713–717. https://doi.org/10.2307/2037465 doi: 10.1090/S0002-9939-1969-0243294-0
[14]	E. Babolian, M. Mordad, A numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions, Comput. Math. Appl., 62 (2011), 187–198. https://doi.org/10.1016/j.camwa.2011.04.066 doi: 10.1016/j.camwa.2011.04.066
[15]	O. P. Singh, Orthogonal expansions of certain pseudo differential operator, Int. J. Math. Sci., 3 (2004), 129–142.

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