Numerical solution of second order Painlevé differential equation
Volume 21, Issue 2, pp 150--157
http://dx.doi.org/10.22436/jmcs.021.02.06
Publication Date: April 11, 2020
Submission Date: October 18, 2019
Revision Date: February 03, 2020
Accteptance Date: March 03, 2020
-
1254
Downloads
-
2593
Views
Authors
Hijaz Ahmad
- Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan.
Tufail A. Khan
- Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan.
Shao-Wen Yao
- School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China.
Abstract
In this paper, the second order Painlevé differential equation is solved by variational iteration algorithm-I with an auxiliary parameter (VI-I with AP), how to optimally find the auxiliary parameter and Pade approximates for the numerical solution are explained. The effectiveness and suitability of the proposed method are shown by solving two types of second order Painlevé differential equation and the proposed method is compared with other methods to illustrate the accuracy and efficiency of the method.
Share and Cite
ISRP Style
Hijaz Ahmad, Tufail A. Khan, Shao-Wen Yao, Numerical solution of second order Painlevé differential equation, Journal of Mathematics and Computer Science, 21 (2020), no. 2, 150--157
AMA Style
Ahmad Hijaz, Khan Tufail A., Yao Shao-Wen, Numerical solution of second order Painlevé differential equation. J Math Comput SCI-JM. (2020); 21(2):150--157
Chicago/Turabian Style
Ahmad, Hijaz, Khan, Tufail A., Yao, Shao-Wen. "Numerical solution of second order Painlevé differential equation." Journal of Mathematics and Computer Science, 21, no. 2 (2020): 150--157
Keywords
- Painlevé equation
- second order Painlevé differential equation
- VIA-I with AP
- RK4
MSC
References
-
[1]
H. Ahmad, Variational iteration algorithm--II with an auxiliary parameter and its optimal determination, Nonlinear Sci. Lett. A, 9 (2018), 62--72
-
[2]
H. Ahmad, Variational iteration method with an auxiliary parameter for solving differential equations of the fifth order, Nonlinear Sci. Lett. A, 9 (2018), 27--35
-
[3]
H. Ahmad, Variational Iteration Algorithm--I with an Auxiliary Parameter for Solving Fokker--Planck Equation, Earthline J. Math. Sci., 2 (2019), 29--37
-
[4]
H. Ahmad, T. A. Khan, Variational iteration algorithm-I with an auxiliary parameter for wave-like vibration equations, J. Low Frequency Noise Vibr. Active Control, 38 (2019), 1113--1124
-
[5]
H. Ahmad, T. A. Khan, Variational iteration algorithm I with an auxiliary parameter for the solution of differential equations of motion for simple and damped mass–spring systems, Noise & Vibration Worldwide, 51 (2020), 12--20
-
[6]
H. Ahmad, T. A. Khan, C. Cesarano, Numerical Solutions of Coupled Burgers' Equations, Axioms, 8 (2019), 16 pages
-
[7]
H. Ahmad, A. R. Seadawy, T. A. Khan, Numerical solution of Korteweg-de Vries-Burgers equation by the modified variational Iteration algorithm--II arising in shallow water waves, Phys. Scr., 95 (2020), 12 pages
-
[8]
N. Anjum, J.-H. He, Laplace transform: Making the variational iteration method easier, Appl. Math. Lett., 92 (2019), 134--138
-
[9]
S. S. Behzadi, Convergence of Iterative Methods for Solving Painlevé Equation, Appl. Math. Sci. (Ruse), 4 (2010), 1489--1507
-
[10]
A. V. Borisov, N. A. Kudryashov, Paul Painlevé and his contribution to science, Regul. Chaotic Dyn., 19 (2014), 1--19
-
[11]
J. P. Boyd, Padé approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain, Comput. Phys., 11 (1997), 299--303
-
[12]
P. A. Clarkson, Painlevé equations--nonlinear special functions, J. Comput. Appl. Math., 153 (2003), 127--140
-
[13]
E. Hashemizadeh, F. Mahmoudi, Numerical solution of Painlevé equation by Chebyshev polynomials, J. Interpolat. Approx. Sci. Comput., 2016 (2016), 26--31
-
[14]
J.-H. He, A fractal variational theory for one-dimensional compressible flow in a microgravity space, Fractals, 27 (2019), 11 pages
-
[15]
J.-H. He, A modified Li--He's variational principle for plasma, Int. J. Numer. Methods Heat Fluid Flow, 2019 (2019), 11 pages
-
[16]
J.-H. He, Lagrange Crisis and Generalized Variational Principle for 3D unsteady flow, Int. J. Numer. Methods Heat Fluid Flow, 2019 (2019), 15 pages
-
[17]
J.-H. He, The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators, J. Low Frequency Noise Vibr. Active Control, 2019 (2019), 12 pages
-
[18]
J.-H. He, Generalized Variational Principles for Buckling Analysis of Circular Cylinders, Acta Mech., 231 (2020), 899--906
-
[19]
J.-H. He, H. Latifizadeh, A general numerical algorithm for nonlinear differential equations by the variational iteration method, Int. J. Numer. Methods Heat Fluid Flow, 2020 (2020), 11 pages
-
[20]
C.-H. He, Y. Shen, F.-Y. Ji, J.-H. He, Taylor series solution for fractal Bratu-type equation arising in electrospinning process, Fractals, 28 (2020), 13 pages
-
[21]
J.-H. He, C. Sun, A variational principle for a thin film equation, J. Math. Chem., 57 (2019), 2075--2081
-
[22]
E. Hesameddini, A. Peyrovi, The Use of Variational Iteration Method and Homotopy Perturbation Method for Painlevé Equation I, Appl. Math. Sci., 3 (2009), 1861--1871
-
[23]
M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, Pergamon Press, Oxford (1978)
-
[24]
M. Nadeem, H. Ahmad, Variational Iteration Method for Analytical Solution of the Lane-Emden Type Equation with Singular Initial and Boundary Conditions, Earthline J. Math. Sci., 2 (2019), 127--142
-
[25]
M. Nadeem, F. Li, H. Ahmad, He's variational iteration method for solving non--homogeneous Cauchy Euler differential equations, Nonlinear Sci. Lett. A Math. Phys. Mech., 9 (2018), 231--237
-
[26]
M. Nadeem, F. Li, H. Ahmad, Modified Laplace variational iteration method for solving fourth--order parabolic partial differential equation with variable coefficients, Comput. Math. Appl., 2019 (2019), 10 pages
-
[27]
D.-N. Yu, J.-H. He, A. G. Garcıa, Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators, J. Low Frequency Noise Vibr. Active Control, 2018 (2018), 11 pages