Numerical simulation of multiple roots of van der Waals and CSTR problems with a derivative-free technique

Sunil Kumar; Jai Bhagwan; Lorentz Jäntschi; Sunil Kumar; Jai Bhagwan; Lorentz Jäntschi

doi:10.3934/math.2023731

AIMS Mathematics

2023, Volume 8, Issue 6: 14288-14299. doi: 10.3934/math.2023731

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Numerical simulation of multiple roots of van der Waals and CSTR problems with a derivative-free technique

1.
Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Channai 601103, India
2.
Department of Mathematics, UCRD, Chandigarh University, Mohali 140413, India
3.
Department of Mathematics, Government Post Graduate College, Rohtak 124001, India
4.
Department of Physics and Chemistry, Technical University of Cluj-Napoca, Cluj-Napoca 400641, Romania

Received: 10 February 2023 Revised: 19 March 2023 Accepted: 30 March 2023 Published: 17 April 2023
MSC : 41A25, 49M15, 65H05

In this paper, a derivative-free one-point iterative technique is proposed, with memory for finding multiple roots of practical problems, such as van der Waals and continuous stirred tank reactor problems, whose multiplicity is unknown in the literature. The new technique has an order of convergence of 1.84 and requires two function evaluations. It can be used as a seed to produce higher-order methods with similar properties, and it increases the efficiency of a similar procedure without memory due to Schröder. After studying its order of convergence, its stability is checked by applying it to the considered problems and comparing with the technique of the same nature for finding multiple roots. The geometrical behavior of the numerical results of the techniques is also studied.
- nonlinear equations,
- multiple roots,
- derivative-free method,
- convergence
Citation: Sunil Kumar, Jai Bhagwan, Lorentz Jäntschi. Numerical simulation of multiple roots of van der Waals and CSTR problems with a derivative-free technique[J]. AIMS Mathematics, 2023, 8(6): 14288-14299. doi: 10.3934/math.2023731

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Abstract

In this paper, a derivative-free one-point iterative technique is proposed, with memory for finding multiple roots of practical problems, such as van der Waals and continuous stirred tank reactor problems, whose multiplicity is unknown in the literature. The new technique has an order of convergence of 1.84 and requires two function evaluations. It can be used as a seed to produce higher-order methods with similar properties, and it increases the efficiency of a similar procedure without memory due to Schröder. After studying its order of convergence, its stability is checked by applying it to the considered problems and comparing with the technique of the same nature for finding multiple roots. The geometrical behavior of the numerical results of the techniques is also studied.

References

[1]	I. K. Argyros, Convergence and applications of Newton-type iterations, New York: Springer, 2008. http://doi.org/10.1007/978-0-387-72743-1
[2]	J. F. Traub, Iterative methods for the solution of equations, Englewood Cliffs: New Jersey, 1964.
[3]	P. Jarratt, D. Nudds, The use of rational functions in the iterative solution of equations on a digital computer, The Computer Journal, 8 (1965), 62–65. http://doi.org/10.1093/comjnl/8.1.62 doi: 10.1093/comjnl/8.1.62
[4]	C. F. Gerald, P. O. Wheatley, Applied numerical analysis, Reading, MA: Addison-Wesley, 1994.
[5]	D. E. Muller, A method of solving algebraic equations using an automatic computer, Math. Comput., 10 (1956), 208–215. http://doi.org/10.1090/S0025-5718-1956-0083822-0 doi: 10.1090/S0025-5718-1956-0083822-0
[6]	J. R. Sharma, A family of methods for solving nonlinear equations using quadratic interpolation, Comput. Math. Appl., 48 (2004), 709–714. http://doi.org/10.1016/j.camwa.2004.05.004 doi: 10.1016/j.camwa.2004.05.004
[7]	J. R. Sharma, S. Kumar, C. Cesarano, An efficient derivative free one-point method with memory for solving nonlinear equations, Mathematics, 7 (2019), 604. http://doi.org/10.3390/math7070604 doi: 10.3390/math7070604
[8]	R. Behl, A. Cordero, J. R. Torregrosa, A new higher-order optimal derivative-free scheme for multiple roots, J. Comput. Appl. Math., 404 (2022), 113773. https://doi.org/10.1016/j.cam.2021.113773 doi: 10.1016/j.cam.2021.113773
[9]	S. Kumar, D. Kumar, J. R. Sharma, C. Cesarano, P. Aggarwal, Y. M. Chu, An optimal fourth order derivative-free numerical algorithm for multiple roots, Symmetry, 12 (2020), 1038. http://doi.org/10.3390/sym12061038 doi: 10.3390/sym12061038
[10]	S. Kumar, D. Kumar, J. R. Sharma, I. K. Argyros, An efficient class of fourth-order derivative-free method for multiple roots, Int. J. Nonlin. Sci. Num., 24 (2021), 265–275. http://doi.org/10.1515/ijnsns-2020-0161 doi: 10.1515/ijnsns-2020-0161
[11]	F. Zafar, A. Cordero, J. R. Torregrosa, A family of optimal fourth-order method for multiple roots of nonlinear equations, Math. Method. Appl. Sci., 43 (2020), 7869–7884. http://doi.org/10.1002/mma.5384 doi: 10.1002/mma.5384
[12]	J. R. Sharma, H. Arora, A family of fifth-order iterative methods for finding multiple roots of nonlinear equations, Numer. Analys. Appl., 14 (2021), 186–199. http://doi.org/10.1134/S1995423921020075 doi: 10.1134/S1995423921020075
[13]	S. Akram, F. Akram, M. Junjua, M. Arshad, T. Afzal, A family of optimal eighth order iterative function for multiple roots and its dynamics, J. Math., 2021 (2021), 5597186. http://doi.org/10.1155/2021/5597186 doi: 10.1155/2021/5597186
[14]	E. Schröder, Über unendlich viele algorithmen zur auflösung der gleichungen, Math. Ann., 2 (1870), 317–365. http://doi.org/10.1007/BF01444024 doi: 10.1007/BF01444024
[15]	A. Cordero, B. Neta, J. R. Torregrosa, Memorizing schröder's method as an efficient strategy for estimating roots of unknown multiplicity, Mathematics, 9 (2021), 2570. http://doi.org/10.3390/math9202570 doi: 10.3390/math9202570
[16]	J. M. Ortega, W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, 2008. http://doi.org/10.1016/C2013-0-11263-9
[17]	S. Hayashi, A. Constantinides, N. Mostoufi, Numerical methods for chemical engineers with MATLAB application, 1999. Available from: https://www.cambridge.org/us/academic/subjects/engineering/chemical-engineering/numerical-methods-chemical-engineering-applications-matlab?format = HB.
[18]	J. M. Douglas, Process dynamics and control, Vol. 2, control system synthesis, Englewood Cliffs: Prentice Hall, 1972.
[19]	A. Cordero, J. R. Torregrosa, Variants of Newton method using fifth order quadrature formulas, Appl. Math. Comput., 190 (2007), 686–698. http://doi.org/10.1016/j.amc.2007.01.062 doi: 10.1016/j.amc.2007.01.062
[20]	L. Jäntschi, Modelling of acids and bases revisited. Stud. U. Babes-Bol. Che., 67 (2022), 73–92. http://doi.org/10.24193/subbchem.2022.4.05 doi: 10.24193/subbchem.2022.4.05

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