Variational Iteration Method for Partial Differential Equations with Piecewise Constant Arguments
Keywords:
Variational iteration method; Partial differential equations; Piecewise constant arguments; Approximate solutions
Abstract
In this paper, the variational iteration method is applied to solve the partial differential equations with piecewise constant arguments. This technique provides a sequence of functions which converges to the exact solutions of the problem and is based on the use of Lagrange
multipliers for identification of optimal value of a parameter in a functional. Employing this technique, we obtain the approximate solutions of the above mentioned equation in every interval [n, n + 1) (n = 0, 1, · · ·). Illustrative examples are given to show the efficiency of the
method.
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References
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[3] S.H. Chang. Convergence of variational iteration method applied to two-point diffusion problems. Appl. Math. Model., 40: 6805-6810, 2016.
[4] M.M. Khader. Numerical and theoretical treatment for solving linear and nonlinear delay differential equations using variational iteration method. Arab. J. Math. Sci., 19: 243-256, 2013.
[5] H. Ahmad, T.A. Khan, and P.S. Stanimirovic. Modified variational iteration algorithm-II: convergence and applications to diffusion models. Complexity, 2020, DOI: 10.1155/2020/8841718.
[6] M. Turkyilmazoglu. An optimal variational iteration method. Appl. Math. Lett., 24: 762-765, 2011.
[7] M. Dehghan, R. Salehi. Solution of a nonlinear time-delay model in biology via semi-analytical approaches. Comput. Phys. Commun., 181: 1255-1265, 2010.
[8] I. Ali, N.A. Malik. Hilfer fractional advection-diffusion equations with power-law initial condition; a numerical study using variational iteration method. Comput. Math. Appl., 68: 1161-1179, 2014.
[9] A. Akkouche, A. Maidi, and M. Aidene. Optimal control of partial differential equations based on the variational iteration method. Comput. Math. Appl., 68: 622-631, 2014.
[10] M.G. Sakar, H. Ergoren. Alternative variational iteration method for solving the time-fractional Fornberg-Whitham equation. Appl. Math. Model., 39: 3972-3979, 2015.
[11] H. Ghaneai, M.M. Hosseini. Solving differential-algebraic equations through variational iteration method with an auxiliary parameter. Appl. Math. Model., 40: 3991-4001, 2016.
[12] O. Martin. A modified variational iteration method for the analysis of viscoelastic beams. Appl. Math. Model., 40: 7988-7995, 2016.
[13] H. Ahmad, A.R. Seadawy, and T.A. Khan. Study on numerical solution of dispersive water wave phenomena by using a reliable modification of variational iteration algorithm. Math. Comput. Simul., 177: 13-23, 2020.
[14] S. Busenberg, K. Cooke. Vertically Transmitted Diseases: Models and Dynamics. Springer, Berlin, 1993.
[15] T. Kupper, R. Yuang. On quasi-periodic solutions of differential equations with piecewise constant argument. J. Math. Anal. Appl., 267: 173-193, 2002.
[16] M.Z. Liu, M.H. Song, and Z.W. Yang. Stability of Runge-Kutta methods in the numerical solution of equation u′(t) = au(t) + a0u([t]). J. Comput. Appl. Math., 166: 361-370, 2004.
[17] Q. Wang, Q.Y. Zhu, and M.Z. Liu. Stability and oscillations of numerical solutions for differential equations with piecewise continuous arguments of alternately advanced and retarded type. J. Comput. Appl. Math., 235: 1542-1552, 2011.
[18] M.H. Song, X. Liu. The improved linear multistep methods for differential equations with piecewise continuous arguments. Appl. Math. Comput., 217: 4002-4009, 2010.
[19] Y. Xie, C.J. Zhang. A class of stochastic one-parameter methods for nonlinear SFDEs with piecewise continuous arguments. Appl. Numer. Math., 135: 1-14, 2019.
[20] C.J. Zhang, C. LI, and J.Y. Jiang. Extended block boundary value methods for neutral equations with piecewise constant argument. Appl. Numer. Math., 150: 182-193, 2020.
[21] J. Wiener. Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993.
[22] J.H. He. Variational iteration method for delay differential equations. Commun. Nonlinear Sci. Numer. Simulat., 2: 235-236, 1997.
[23] J.H. He. Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comput. Methods Appl. Mech. Engng., 167: 69-73, 1998.
[24] J.H. He. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Engng., 167, 57-68, 1998.
Published
2020-12-19
How to Cite
Wang, Q. (2020). Variational Iteration Method for Partial Differential Equations with Piecewise Constant Arguments. Journal of Progressive Research in Mathematics, 17(1), 100-108. Retrieved from http://www.scitecresearch.com/journals/index.php/jprm/article/view/1981
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