Document Type : Research Articles
Authors
- Azar Shabani ^{1}
- Alireza Fatehi ^{} ^{} ^{2}
- Fahimeh Soltanian ^{} ^{} ^{3}
- Reza Jamilnia ^{4}
^{1} Department of Mathematics, Payame Noor University, Tehren, Iran
^{2} APAC Research Group, Faculty of Electrical Eng., K.N. Toosi University of Technology, Tehran, Iran
^{3} Department of Mathematics, Payame Noor University, Tehran, Iran
^{4} Department of Mechanical Engineering, University of Guilan, Rasht, Iran.
Abstract
In this paper, two semi-analytical techniques are introduced to compute the solutions of differential-algebraic equations (DAEs), called the Least Squares Repetitive Homotopy Perturbation Method (LSRHPM) and the Least Squares Span Repetitive Homotopy Perturbation Method (LSSRHPM). The truncated series solution by the homotopy perturbation method only is suitable for small-time intervals. Therefore, to extend it for long time intervals, we consider the Repetitive Homotopy Perturbation Method (RHPM). To improve the accuracy of the solutions obtained by RHPM and to reduce the residual errors, least squares methods and span set are combined with RHPM. The proposed methods are applied to solve nonlinear differential-algebraic equations and optimal control problems. The results of the proposed methods are compared using some illustrative examples. The obtained results demonstrate the effectiveness and high accuracy of the new modifications. The effect of the parameters on the accuracy and performance of the methods are studied through some illustrative examples
Keywords
- differential-algebraic equations
- semi-analytical homotopy perturbation method
- least squares method
- span set
- optimal control
Main Subjects
solutions of systems of differential-algebraic equations by
Laplace homotopy analysis method”, Mathematics and
Computer Science Series, Vol. 39(2), pp. 191-199, 2012.
[2] M. Heydari, G. Barid Loghmani, S.M. Hosseini, and S.M.
Karbassi, “Direct method to solve differential-algebraic
equations by using the operational matrices of chebyshev
cardinal functions”, Journal of Mathematical Extension,
Vol. 7, No. 2, pp. 25-47, 2013.
[3] L. Hong, and S. Yongzhong, “On a regularization of index
2 differential-algebraic equations with properly stated
leading term”, Acta Mathematica Scientia, Vol. 31B(2), PP.
383-398, 2011.
[4] S. Yongzhong, “Solvability of higher index time-varying linear differential-algebraic equations”, Acta Mathematica
Scientia, Vol. 21B(1), pp. 77-92, 2001.
[5] M. Saravi, E. Babolian, R. England, and R. Bromilow,
“System of linear ordinary differential and differential-
algebraic equations and pseudo-spectral method”, Comput.
Math. Appl, Vol. 59, pp. 1524-1531, 2010.
[6] M. Ghovatmand, M. Hosseini, and M. Jafari, “Reducing
index, and pseudo spectral method for high index
differential-algebraic equations”, Journal of Advanced
Research in Scientific Computing, Vol. 3, Issue.2, pp. 42-
55, 2011.
[7] M. M. Hosseini, “Reducing index method for differential-
algebraic equations with constraint singularities”, Applied
Mathematics and Computation, vol. 153, pp. 205-214, 2004.
[8] U.M. Ascher,and L.R. Petzold, “Projected implicit Runge–
Kutta methods for differential-algebraic equations”, SIAM
J. Numerical Analysis, Vol. 28, pp. 1097-1120, , 1991.
[9] K.F. Brenan, S.L Campbell, L.R. Petzold, “Numerical
Solution of Initial-Value Problems in Differential Algebraic
Equations”, Elsevier, NewYork, 1989.
[10] L. Jay, “Specialized Runge–Kutta methods for index 2
differential-algebraic equations”, Math. Comput, Vol. 75,
pp. 641-654, 2005.
[11] S.K, Vanani, and A. Aminataei, “Numerical solution of
differential algebraic equations using a multiquadric
approximation scheme”, Mathematical and Computer
Modelling, Vol. 53, No. 5-6, pp. 659 -666, 2011.
[12] H. Vazquez-leal, “Exact solutions for differential–algebraic
equations”, Miskolc mathematical notes, Vol. 15, No. 1, pp.
227-238, 2014.
[13] Ch. Hachtel, A. Bartela, M. G Ìˆunthera, and A. Sandu,
“Multirate Implicit Euler Schemes for a Class of
Differential-Algebraic Equations of Index-1”, Journal of
Computational and Applied Mathematics, Vol. 19, pp. 1-19,
2019.
[14] M. Al-Jawary, and S. Hatif, “A semi-analytical iterative
method for solving differential algebraic equations”, Ain
Shams Engineering Journal, Vol. 9, pp. 2581-2586, 2018.
[15] M. Bulatov, and L. Solovarova, “Collocation-variation
difference schemes with several collocation points for
differential-algebraic equation”, Applied Numerical
Mathematics, article in press, 2019.
[16] S. Hussain, A. Shah, S. Ayub, and A. Ullah, “An
approximate analytical solution of the Allen-Cahn equation
usinghomotopy perturbation method and homotopy
analysis method”, Heliyon, Vol. 5, e03060, 2019.
[17] G. Lingyun, “A result of systems of nonlinear complex
algebraic differential equations”, Acta Mathematica
Scientia, Vol. 30B(5), pp. 1507–1513, 2010.
[18] F. Soltanian, M. Dehghan, and S.M Karbassi S M, “Solution
of the differential algebraic equations via homotopy
perturbation method and their engineering applications”,
International Journal of Computer Mathematics, Vol. 87,
No. 9, pp.1950-1974, 2010 [19] M.M Hosseini, “Adomian decomposition method for
solution of differential algebraic equations”, J. Comput.
Appl. Math, Vol. 197, pp. 495-501, 2006.
[20] M.M Hosseini, “Adomian decomposition method for
solution of nonlinear differential algebraic equations”,
Appl. Math. Comput, Vol.181, pp. 1737-1744, 2006.
[21] H. Ghaneai, and M.M Hosseini, “Solving differential-
algebraic equations through variational iteration method
with an auxiliary parameter”, Applied Mathematical
Modelling, Vol. 40, pp. 3991-4001, 2016.
[22] M. Ghovatmand, M.M Hosseini, and M. Nilli, “application
of Chebyshev approximation in the process of variational
iteration method for solving differential algebraic
equatons”, Mathematical and Computational Applications,
Vol.16, No. 4, pp. 969-978, 2011.
[23] F. Soltanian F, S.M Karbassi, and M.M Hosseini,
“Application of He’s variational iteration method for
solution of differential-algebraic equations”, Chaos
Solitons and Fractals, Vol. 41, pp. 436-445, 2009.
[24] F. Awawdeh, H.M Jaradat, and O, Alsayyed, “Solving
system of DAEs by homotopy analysis method”, Vol. 42,
1422-1427, 2009.
[25] A.S Shabani , A. Fatehi, F. Soltanian, and R. Jamilni,
“Design of Continuous Nonlinear Predictive Controllers by
Solving Differential-Algebraic Equations with Boundary
Conditions by Homotopy Perturbation Method”, Control
Journal, Vol. 12, Is. 4, pp. 1-14, 2019, in persian.
[26] S. Bothayna, S.A Kashkari, El-Tantawy, H. Alvaro,
Salas, and L.S. El-Sherif, “Homotopy perturbation method
for studying dissipative nonplanar solitons in an
electronegative complex plasma”, Chaos, Solitons and
Fractals, Vol. 130, pp. 1-10, 2020.
[27] M. Turkyilmazoglu, “Convergence of the Homotopy
Perturbation Method” ,Int. J. Nonlinear Sci. Numer. Simul,
Vol. 12, pp. 9-14, 2011.
[28] Z. Ayati, and J. Biazar, “On the convergence of Homotopy
perturbation method”, Journal of the Egyptian
Mathematical Society, Vol. 23, pp. 424-428, 2015.
[29] J.H He, “Homotopy perturbation technique”, Comput.
Methods Appl. Mech. Engrg, Vol.178, pp. 257-262, 1999.
[30] J.H He, “An elementary introduction to the homotopy
perturbation method”, Comput. Math. Appl, Vol. 57, pp.
410-412, 2009.
[31] C. Bota, and B. Caruntu, “Approximate analytical solutions
of nonlinear differential equations using the Least Squares
Homotopy Perturbation Method”, Journal of Mathematical
Analysis and Applications, Vol. 448, Is.1, pp. 401-408,
2017.
[32] H. Ching-Lai, and M.Abu syed, “Multi objective Decision
Making Methods and Applications”, springer-verloy, 1978.
[33] M. McAsey, L. Mou , and W. Han W, “Convergence of the
forward-backward sweep method in optimal control”,
Computational Optimization and Applications, Vol. 53,
Iss.1, PP. 207-226, 2012.