A piecewise homotopy Padé technique to approximate an arbitrary function

Mourad S. Semary; Aisha F. Fareed; Hany N. Hassan; Mourad S. Semary; Aisha F. Fareed; Hany N. Hassan

doi:10.3934/math.2023578

AIMS Mathematics

2023, Volume 8, Issue 5: 11425-11439. doi: 10.3934/math.2023578

Previous Article Next Article

Research article

A piecewise homotopy Padé technique to approximate an arbitrary function

Department of Basic Engineering Sciences, Benha Faculty of Engineering, Benha University, Egypt

Received: 06 December 2022 Revised: 03 February 2023 Accepted: 14 February 2023 Published: 13 March 2023
MSC : 65C30, 65L12

The Padé approximation and its enhancements provide a more accurate approximation of functions than the Taylor series truncation. A new technique for approximating functions into rational functions is proposed in this paper. This technique is based on the homotopy Padé technique and introduces new parameters known as merging parameters. These parameters are added to the Tayler series before the Padé process is computed. To control error, the merging parameters and dividing the interval into subintervals are used. Two illustrative examples are used to demonstrate the validity and reliability of the proposed novel approximation. The robustness and efficiency of the proposed approximation were demonstrated by computing the absolute error and comparing the results to those of the standard Padé technique and the generalized restrictive Padé technique. Also, Hard-core scattering problem and Debye-Hukel function are tested by the proposed technique. The piecewise homotopy Padé method is an excellent path to approximate any function. The proposed new approximation's efficacy and accuracy have been validated using Mathematica 12.
- rational approximation,
- homotopy Padé technique,
- Padé approximation,
- the generalized restrictive Padé technique
Citation: Mourad S. Semary, Aisha F. Fareed, Hany N. Hassan. A piecewise homotopy Padé technique to approximate an arbitrary function[J]. AIMS Mathematics, 2023, 8(5): 11425-11439. doi: 10.3934/math.2023578

Related Papers:

Abstract

The Padé approximation and its enhancements provide a more accurate approximation of functions than the Taylor series truncation. A new technique for approximating functions into rational functions is proposed in this paper. This technique is based on the homotopy Padé technique and introduces new parameters known as merging parameters. These parameters are added to the Tayler series before the Padé process is computed. To control error, the merging parameters and dividing the interval into subintervals are used. Two illustrative examples are used to demonstrate the validity and reliability of the proposed novel approximation. The robustness and efficiency of the proposed approximation were demonstrated by computing the absolute error and comparing the results to those of the standard Padé technique and the generalized restrictive Padé technique. Also, Hard-core scattering problem and Debye-Hukel function are tested by the proposed technique. The piecewise homotopy Padé method is an excellent path to approximate any function. The proposed new approximation's efficacy and accuracy have been validated using Mathematica 12.

References

[1]	G. A. Baker, Essentials of Padé approximants, New York: Academic Press, 1975.
[2]	G. A. Baker, P. Graves-Morris, Padé approximants, New York: Addison-Wesley, 1982.
[3]	D. Boito, P. Masjuan, F. Oliani, Higher-order QCD corrections to hadronic τ decays from Padé approximants, J. High Energy Phys., (2018), 1–41. https://doi.org/10.1007/JHEP08(2018)075 doi: 10.1007/JHEP08(2018)075
[4]	C. Ingo, T. R. Barrick, A. G. Webb, I. Ronen, Accurate Padé global approximations for the Mittag-Leffler function, its inverse, and its partial derivatives to efficiently compute convergent power series, Int. J. Appl. Comput. Math., 3 (2017), 347–362. https://doi.org/10.1007/s40819-016-0158-7 doi: 10.1007/s40819-016-0158-7
[5]	F. Martin-Vergara, F. Rus, F. R. Villatoro, Padé schemes with Richardson extrapolation for the sine-Gordon equation, Commun. Nonlinear Sci. Numer. Simul., 85 (2020), 105243. https://doi.org/10.1016/j.cnsns.2020.105243 doi: 10.1016/j.cnsns.2020.105243
[6]	T. A. Abassy, M. A. El-Tawil, H. E. Zoheiry, Solving nonlinear partial differential equations using the modified variational iteration Padé technique, J. Comput. Appl. Math., 207 (2007), 73–91. https://doi.org/10.1016/j.cam.2006.07.024. doi: 10.1016/j.cam.2006.07.024
[7]	K. S. Nisar, J. Ali, M. K. Mahmood, D. Ahmad, S. Ali, Hybrid evolutionary padé approximation approach for numerical treatment of nonlinear partial differential equations, Alex. Eng. J., 60 (2021), 4411–4421. https://doi.org/10.1016/j.aej.2021.03.030. doi: 10.1016/j.aej.2021.03.030
[8]	S. Ahsan, R. Nawaz, M. Akbar, K. S. Nisar, E. E. Mahmoud, M. M. Alqarni, Numerical solution of 2D-fuzzy Fredholm integral equations using optimal homotopy asymptotic method, Alex. Eng. J., 60 (2021), 2483–2490. https://doi.org/10.1016/j.aej.2020.12.049. doi: 10.1016/j.aej.2020.12.049
[9]	N. Pareek, A. Gupta, D. L. Suthar, G. Agarwal, K. S. Nisar, Homotopy analysis approach to study the dynamics of fractional deterministic Lotka-Volterra model, Arab. Basic Appl. Sci., 29 (2022), 121–128. https://doi.org/10.1080/25765299.2022.2071027 doi: 10.1080/25765299.2022.2071027
[10]	H. N. A. Ismail, A. Y. H. Elmekkawy, Restrictive padé approximation for solving ﬁrst-order hyperbolic in two space dimensions, In: Proceeding of the 9^th ASAT Conference, 9 (2001), 51–59. https://doi.org/10.21608/asat.2001.24759
[11]	J. Gilewicza, M. Pindorb, J. J. Telega, S. Tokarzewski, N-point Padé approximants and two-sided estimates of errors on the real axis for Stieltjes functions, J. Comput. Appl. Math., 178 (2005), 247–253. https://doi.org/10.1016/j.cam.2003.12.051 doi: 10.1016/j.cam.2003.12.051
[12]	S. J. Liao, K. F. Cheung, Homotopy analysis of nonlinear progressive waves in deep water, J. Eng. Math., 45 (2003), 105–116. https://doi.org/10.1023/A:1022189509293 doi: 10.1023/A:1022189509293
[13]	Y. Chakir, J. Abouir, B. Benouahmane, Multivariate homogeneous two-point Padé approximants and continued fractions, Comput. Appl. Math., 39 (2020), 1–16, https://doi.org/10.1007/s40314-019-0929-y doi: 10.1007/s40314-019-0929-y
[14]	A. A. Gonchar, E. A. Rakhmanov, V. N. Sorokin, Hermite-Padé approximations for systems of Markov-type functions, Sb. Math., 188 (1997), 33–58. https://doi.org/10.1070/SM1997v188n05ABEH000225 doi: 10.1070/SM1997v188n05ABEH000225
[15]	G. L. Lagomasino, S. M. Peralta, On the convergence of type 1 Hermite-Padé approximants, Adv. Math., 273 (2015), 124–148. https://doi.org/10.1016/j.aim.2014.12.025 doi: 10.1016/j.aim.2014.12.025
[16]	C. B. Zeng, Y. Q. Chen, Global Padé approximations of the generalized Mittag-Leffler function and its inverse, Fract. Calc. Appl. Anal., 18 (2015), 1492–1506. https://doi.org/10.1515/fca-2015-0086 doi: 10.1515/fca-2015-0086
[17]	I. O. Sarumi, K. M. Furati, A. Q. M. Khaliq, Highly accurate global Padé approximations of generalized Mittag-Leffler function and its inverse, J. Sci. Comput., 82 (2020), 46. https://doi.org/10.1007/s10915-020-01150-y doi: 10.1007/s10915-020-01150-y
[18]	L. Y. Ming, C. Yong, Adomian decomposition method and Padé approximation for nonlinear differential-difference equations, Commun. Theor. Phys., 51 (2009), 581–587. https://doi.org/10.1088/0253-6102/51/4/02 doi: 10.1088/0253-6102/51/4/02
[19]	R. J. Betancourt, A. Marco, G. Perez, E. E. Barocio, L. J. Arroyo, Analysis of inter-area oscillations in power systems using Adomian-Padé approximation method, 2010 9th IEEE/IAS International Conference on Industry Applications-INDUSCON 2010, Sao Paulo, Brazil, 1–6. https://doi.org/10.1109/INDUSCON.2010.5740043
[20]	V. Turut, N. Güzel, On solving partial differential equations of fractional order by using the variational iteration method and multivariate Padé approximations, Eur. J. Pure Appl. Math., 6 (2013), 147–171.
[21]	M. G. Ibrahim, Numerical simulation for non-constant parameters effects on blood flow of Carreau-Yasuda nanofluid flooded in gyrotactic microorganisms: DTM-Pade application, Arch. Appl. Mech., 92 (2022), 1643–1654. https://doi.org/10.1007/s00419-022-02158-6 doi: 10.1007/s00419-022-02158-6
[22]	M. S. Semary, M. T. M. Elbarawy, A. F. Fareed, Discrete Temimi-Ansari method for solving a class of stochastic nonlinear differential equations, AIMS Math., 7 (2022), 5093–5105. https://doi.org/10.3934/math.2022283 doi: 10.3934/math.2022283
[23]	M. S. Semary, H. N. Hassan, A. G. Radwan, Controlled Picard method for solving nonlinear fractional reaction-diffusion models in porous catalysts, Chem. Eng. Commun., 204 (2017), 635–647. https://doi.org/10.1080/00986445.2017.1300151 doi: 10.1080/00986445.2017.1300151
[24]	A. F. Fareed, M. A. Elsisy, M. S. Semary, M. T. M. M. Elbarawy, Controlled Picard's transform technique for solving a type of time fractional Navier-Stokes equation resulting from incompressible fluid flow, Int. J. Appl. Comput. Math., 8 (2022), 184. https://doi.org/10.1007/s40819-022-01361-x doi: 10.1007/s40819-022-01361-x
[25]	A. F. Fareed, M. S. Semary, H. N. Hassan, An approximate solution of fractional order Riccati equations based on controlled Picard's method with Atangana-Baleanu fractional derivative, Alex. Eng. J., 61 (2022), 3673–3678. https://doi.org/10.1016/j.aej.2021.09.009 doi: 10.1016/j.aej.2021.09.009
[26]	A. F. Fareed, M. T. M. Elbarawy, M. S. Semary, Fractional discrete Temimi-Ansari method with singular and nonsingular operators: applications to electrical circuits, Adv. Cont. Discr. Mod., 2023 (2023), 5. https://doi.org/10.1186/s13662-022-03742-4 doi: 10.1186/s13662-022-03742-4
[27]	M. S. Semary, H. N. Hassan, The homotopy analysis method for q-difference equations, Ain Shams Eng. J., 9 (2018), 415–421. https://doi.org/10.1016/j.asej.2016.02.005 doi: 10.1016/j.asej.2016.02.005
[28]	M. S. Semary, H. N. Hassan, A. G. Radwan, Modified methods for solving two classes of distributed order linear fractional differential equations, Appl. Math. Comput., 323 (2018), 106–119. https://doi.org/10.1016/j.amc.2017.11.047 doi: 10.1016/j.amc.2017.11.047
[29]	S. Abbasbandy, E. Shivanian, K. Vajravelu, S. Kumar, A new approximate analytical technique for dual solutions of nonlinear differential equations arising in mixed convection heat transfer in a porous medium, Int. J. Numer. Methods Heat Fluid Flow, 27 (2017), 486–503. https://doi.org/10.1108/HFF-11-2015-0479 doi: 10.1108/HFF-11-2015-0479
[30]	Z. K. Bojdi, S. Ahmadi-Asl, A. Aminataei, A new extended Padé approximation and its application, Adv. Numer. Anal., 2013 (2013), 1–8. https://doi.org/10.1155/2013/263467 doi: 10.1155/2013/263467
[31]	M. S. Semary, H. N. Hassan, An effective approach for solving MHD viscous flow due to a shrinking sheet, Appl. Math. Inf. Sci., 10 (2016), 1425–1432. https://doi.org/10.18576/amis/100421 doi: 10.18576/amis/100421
[32]	H. N. A. Ismail, On the convergence of the restrictive Padé approximation to the exact solutions of IBVP of parabolic and hyperbolic types, Appl. Math. Comput., 162 (2005), 1055–1064. https://doi.org/10.1016/j.amc.2004.01.023 doi: 10.1016/j.amc.2004.01.023
[33]	H. N. A. Ismail, Unique solvability of restrictive Padé and restrictive Taylor's approximations, App. Math. Comput., 152 (2004), 89–97. https://doi.org/10.1016/S0096-3003(03)00546-0 doi: 10.1016/S0096-3003(03)00546-0
[34]	R. Jedynak, J. Gilewicz, Computation of the 𝑐-table related to the Padé approximation, J. Appl. Math., 2013 (2013), 1–10. http://dx.doi.org/10.1155/2013/185648 doi: 10.1155/2013/185648
[35]	C. Brezinski, M. Redivo-Zaglia, Padé-type rational and barycentric interpolation, Numer. Math., 125 (2013), 89–113. https://doi.org/10.1007/s00211-013-0535-7 doi: 10.1007/s00211-013-0535-7
[36]	H. N. Hassan, M. S. Semary, An analytic solution to a parameterized problems arising in heat transfer equations by optimal homotopy analysis method, Walailak J. Sci. Technol., 11 (2014), 659–677. https://doi.org/10.14456/WJST.2014.87 doi: 10.14456/WJST.2014.87
[37]	B. Wu, C. Li, Explicit determinant formulas of generalized restrictive Padé approximation, J. Inform. Comput. Sci., 9 (2012), 2959–2967.
[38]	G. A. Baker, J. L. Gammel, The Padé approximant, J. Math Anal. Appl., 2 (1961), 21–30. https://doi.org/10.1016/0022-247X(61)90042-7 doi: 10.1016/0022-247X(61)90042-7
[39]	L. D. Landau, E. M. Lifshitz, Statistical physics, Moscow: Nauka, 1976.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)