Perturbation-iteration method applied to partial differential equations

Pakdemirli, Mehmet

doi:10.47974/JIM-1744

Open Access Research Article

Perturbation-iteration method applied to partial differential equations

^*Mehmet PakdemirliCorresponding authorpakdemirli@gmail.comDepartment of Mechanical EngineeringManisa Celal Bayar UniversityYunusemre, Manisa, 45140, Turkey0000-0003-1221-9387View full profile →

* Corresponding author · click or hover a name for details

pp. 425–439Vol. 28Issue 2March 2025DOI: 10.47974/JIM-1744XML

Received:: 09 Nov 2022
Published Online:: 15 Mar 2025
Article type:: Research Article
Language:: EN
Article no.:: JIM-1744
Pages:: 425–439

Abstract

Partial differential equations are treated with the Perturbation Iteration Method (PIM) Sample equations possessing exact analytical solutions were treated for comparison with the iteration solutions. Two different iteration algorithms are applied to the differential equations. Linear as well as nonlinear problems are treated. Exact solutions are contrasted with the approximate solutions for each sample problem. It is shown that the Perturbation Iteration Method can be successfully applied to partial differential equations in construction of admissible approximate analytical solutions.

Keywords

Perturbation techniques Iteration algorithms Partial differential equations Approximate solutions

Subject Classifications

35B2035C0535C2035F1535F2035F3035G1535G3041A3041A58

References

[1] Y. Aksoy and M. Pakdemirli, “New Perturbation-Iteration Solutions for Bratu-type Equations”, Computers & Mathematics with Applications Vol. 59, pp. 2802-2808 (2010).
[2] Y. Aksoy, M. Pakdemirli, S. Abbasbandy, H. Boyacı, “New Perturbation-Iteration Solutions for Nonlinear Heat Transfer Equations”, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 22, pp. 814-828 (2012).
[3] K. M. M. Al-Abrahemee, “Modification of high performance training algorithm for solving singular perturbation partial differential equations with cubic convergence”, Journal of Interdisciplinary Mathematics, vol 24, no. 7, pp 2035–2047 (2021).
[4] A. J. Al Saif and A. J. Harfash, “A comparison between the reduced differential transform method and perturbation-iteration algorithm for solving two dimensional unsteady incompressible Navier-Stokes equations”, Journal of Applied Mathematics and Physics, vol. 6, pp. 2518-2543 (2018).
[5] A. J. Al-Saif and A. J. Harfash, “Perturbation Iteration Algorithm for solving heat and mass transfer in the unsteady squeezing flow between parallel plates,” Journal of Applied Computational Mechanics, vol. 5, no. 4, pp 804-815 (2019).
[6] M. M. Bahşi and M. Çevik, “Numerical solution of Pantograph type delay differential equations using Perturbation Iteration Algorithms”, Journal of Applied Mathematics, vol. 2015, p. 139821, (2015).
[7] N. Bildik, “Optimal Perturbation-Iteration method for solving Telegraph equations”, International Journal of Applied Physics and Mathematics, vol. 7, no. 3, pp. 165-172 (2017).
[8] N. Bildik, “General convergence analysis for the perturbation-iteration technique”, Turkish Journal of Mathematics and Computer Science, vol. 6, pp. 1-9 (2017).
[9] N. Bildik and S. Deniz, “A new efficient method for solving delay differential equations and a comparison with other methods”, The European Physical Journal Plus, vol. 132, p. 51 (2017).
[10] N. Bildik and S. Deniz, “New analytic approximate solutions to the generalized regularized long wave equations”, Bulletin of Korean Mathematical Society, vol. 55, no. 3, pp. 749-762 (2018).
[11] N. Bildik and S. Deniz, “Solving the Burgers and long wave equations using the new perturbation iteration technique”, Numerical Methods for Partial Differential Equations, vol. 34, pp. 1489-1501 (2018).
[12] N. Bildik and S. Deniz, ”Comparative study between optimal homotopy asymptotic method and perturbation iteration technique for different types of Nonlinear equations”, Iranian Journal of Science and Technology, Transactions A-Science, vol. 42, pp. 647-654 (2018).
[13] S. Deniz and N. Bildik, “A new analytical technique for solving Lane-Emden type equations arising in astrophysics”, Bulletin of Belgium Mathematical Society-Simon Stevin, vol. 24. pp. 305-320 (2017).
[14] S. Deniz and N. Bildik, “Optimal Perturbation-Iteration method for Bratu type problems”, Journal of King Saud University-Science, vol. 30, pp. 91-99 (2018).
[15] I. T. Dolapci, M. Şenol and M. Pakdemirli, “New Perturbation Iteration Solutions for Fredholm and Volterra Integral Equations”, Journal of Applied Mathematics, vol. 2013, p. 682537 (2013), http://dx.doi.org/10.1155/2013/682537.
[16] M. Gahamanyi, J. M. Ntganda and M. S. D. Hapgar, “Perturbation Iteration Method for solving mathematical model of glucose and insulin in diabetic human during physical activity”, Open Journal of Applied Sciences, vol. 6, pp. 826-838 (2016).
[17] A. J. Harfash and A. J. Al-Saif, “MHD flow of fourth grade fluid solve by PIA algorithm”, Journal of Advanced Research in Fluid Mechanics and Thermal Sciences, vol. 59, no. 2, pp. 220-231 (2019).
[18] M. Khalid, M. Sultana, F. Zaidi and J. Khan, “A solution of a water quality model in a uniform stream channel using new iterative method”, International Journal of Computer Applications, vol. 115, no. 6, pp. 1-4 (2015).
[19] M. Khalid, M. Sultana, F. Zaidi and U. Arshaul, “An effective perturbation-iteration algorithm for solving Riccati differential equations”, International Journal of Computer Applications, vol. 111, no. 10, pp. 1-5 (2015).
[20] M. Khalid, M. Sultana, F. Zaidi and F. S. Khan, “Solving polluted lake system by using perturbation-iteration method”,. International Journal of Computer Applications, vol. 114, no. 4, pp. 1-7 (2015).
[21] M. Khalid, M. Sultana, F. Zaidi and A. Uroosa, “Solving linear and non-linear Klein-Gordon equations by new perturbation iteration transform method”, TWMS Journal of Applied Engineering Mathematics, vol. 6, no. 1, pp. 115-125 (2016).
[22] M. Pakdemirli, “Review of the new Perturbation-Iteration method”, Mathematical and Computational Applications, vol. 18, pp. 139-151 (2013).
[23] M. Pakdemirli, “Application of the Perturbation-Iteration method to boundary layer problems”, Springer Plus, vol. 5, p. 208 (2016).
[24] M. Pakdemirli, “Perturbation–iteration method for strongly nonlinear vibrations”. Journal of Vibration and Control, vol. 23, vol. 6, pp. 959-969 (2017).
[25] M. Pakdemirli and Y. Aksoy, “Theorems on implementation of the perturbation iteration method”, Applied Mathematical Sciences, vol. 18, no. 7, pp 343-357 (2024).
[26] M. Pakdemirli, Y. Aksoy and H. Boyacı, “A New Perturbation-Iteration Approach for First Order Differential Equations”. Mathematical and Computational Applications, vol. 16, pp. 890-899 (2011).
[27] M. Pakdemirli and H. Boyacı, “Generation of root finding algorithms via perturbation theory and some formulas”, Applied Mathematics and Computation, vol. 184, pp 783-788 (2007).
[28] M. Pakdemirli, H. Boyacı and H. A. Yurtsever, “Perturbative derivation and comparisons of root-finding algorithms with fourth order derivatives”, Mathematical and Computational Applications, vol. 12. pp. 117-124 (2007).
[29] M. Pakdemirli, H. Boyacı and H. A. Yurtsever, “A root finding algorithm with fifth order derivatives”. Mathematical and Computational Applications, vol. 13, pp. 123-128 (2008).
[30] R. P. Singh and Y. N. Reddy, “Perturbation Iteration method for solving differential difference equations having boundary layer”, Communications in Mathematics and Applications, vol. 11, no. 4, pp. 617-633 (2020).
[31] H. M. Srivastava, S. Deniz and K. M. Saad, “An efficient semi-analytical method for solving the generalized regularized long wave equations with a new fractional derivative operator”, Journal of King Saud University-Science, vol. 33, p. 101345 (2021).
[32] M. Şenol, “On the numerical solutions of a wave equation”. International Journal of Advanced Engineering Research and Science, vol. 5, no. 12, vol. 213-216 (2018).
[33] M. Şenol, M. Alquran and H. D. Kasmaci, “On the comparison of perturbation-iteration algorithms and residual power series method to solve fractional Zakharov-Kuznetsov equation”, Results in Physics, vol. 9, pp. 321-327 (2018).
[34] M. Şenol, S. Atpinar, Z. Zararsız, S. Salahshour and A. Ahmadian, “Approximate solution of time fractional fuzzy partial differential equations”. Computational and Applied Mathematics, vol. 38., no. 18 (2019).
[35] M. Şenol and I. T. Dolapçı, “On the Perturbation Iteration Algorithm for fractional differential equations”. Journal of King Saud University-Science, vol. 28, pp. 69-74 (2016).
[36] M. Şenol, I. T. Dolapci, Y. Aksoy and M. Pakdemirli, “Perturbation-Iteration Method for First-Order Differential Equations and Systems”, Abstract and Applied Analysis, vol. 2013, p. 704137 (2013), http://dx.doi.org/10.1155/2013/704137.
[37] M. Şenol and H. D. Kasmaci, “On the numerical solution of nonlinear fractional integro-differential equations”. New Trends in Mathematical Sciences, vol. 5, no. 3, pp. 118-127 (2017).
[38] M. Şenol and H. D. Kasmaci, “Perturbation-Iteration Algorithm for systems of fractional differential equations and convergence analysis”, Progress in Fractional Differentiation and Applications, vol. 3, vol. 4, pp. 271-279 (2017).
[39] O. Tasbozan, M. Şenol, A. Kurt and O. Özkan, “New solutions of fractional Drinfeld-Sokolov-Wilson system in shallow water waves”, Ocean Engineering, vol. 161, pp. 62-68 (2018).
[40] V. Yıldız, M. Pakdemirli and Y. Aksoy, “Parallel plate flow of a third grade fluid and a Newtonian fluid with variable viscosity”, Zeitschrift fur Naturforschung A, vol. 71, no. 7, pp. 595-606 (2016).

Views: 160Downloads: 77Citations: 2