The solution of nonlinear parabolic equation using variational iteration method
DOI:
https://doi.org/10.20956/jmsk.v16i3.8468Keywords:
Variational iteration method, nonlinear parabolic differential equation, semi analytical methodAbstract
Variational iteration method is a semi analytic solution used to solve the parabolic differential equation both of homogen or nonhomogen. In the process of determining an approximation solution, this method did not use a linearization and a small pertubation. In this paper, the variational iteration method is implemented in the parabolic differential equation in the form of ut = uxx + f(u) + g(x, t) with appropriate intial condition. Furthermore, some examples of special parabolic differential equations are given to test the reliability and convergence of the method. Based on the result of study shows that the variational iteration method is able to solve the parabolic differential equation with a good accuration.
References
Polyanin, A. D., & Zaitsev, V. F.2012.. Handbook of Nonlinear Partial Differential Equations, CRC Press, Boca Raton.
Adomian, G. 1988. A Review of the Decomposition Method in Applied Mathematics. Journal of Mathematical Analysis and Applications, Vol. 135, hal. 501 – 544.
Soufyane A,. & Boulamlf, M. 2005. Solution of Linear and Nonlinear Parabolic Equations by the Decomposition Method. Applied Mathematics and Computation, Vol. 162, hal. 687-693.
Biazar, J., & Ayati, Z. 2006. An Approximation to the Solution of Parabolic Equation by Adomian Decomposition Method and Comparing the Result with Crank-Nicolson Method. International Mathematical Forum, Vol. 1, No. 39, hal. 1925 – 1933.
Wartono & Muda, Y. 2011. Aproksimasi Metode Dekomposisi Adomian pada Persamaan Hiperbolik Nonlinier. Jurnal Sains, Teknologi dan Industri, Vol. 9, No. 1, hal. 97 – 103.
Wartono & Muhaijir, M. N. 2013. Penyelesaian Persamaan Riccati dengan Menggunakan Metode Dekomposisi Adomian Laplace. Jurnal Sains, Teknologi dan Industri, Vol. 10, No. 2, hal. 97 – 101.
Liao, S. 1998. Homotopy Analysis Method: A New Analytic Method for Nonlinear Problems. Applied Mathematics and Mechanics, Vol. 19, No. 10, hal. 957 – 962.
Liao, S. 2004. On the Homotopy Analysis Method for Nonlinear Problems. Applied Mathematics and Computation, Vol. 147, hal. 499 – 513.
Jafari, H., Chun, C., Seifi, S., & Saeidy, M. 2009. Analytical Solution for Nonlinear Gas Dynamic Equation by Homotopy Analysis Method. Applications and Applied Mathematics, Vol. 4, No. 1, hal. 149 – 154.
Chen, C. L., & Liu, Y. C. 1998. Solution of Two-Point Boundary-Velue Problems Using the Differential Transformation Method. Journal of Optimization Theory and Applications, Vol. 99, No. 1, hal. 23 – 35.
Mirzaee, F. 2011. Differential Tranform Method for Solving Linear and Nonlinear of Ordinary Differential Equations. Applied Mathematical Sciences, Vol. 5, No. 70, hal. 3465 – 3472.
He, J. 1999. Homotopy Pertubation Technique. Computer Methods in Applied Mechanics ans Engineering, Vol. 178, hal. 257 – 262.
Mantifar, M., & Ghanbari, M. 2009. Homotopy Pertubation Method fot the Fisher’s Equation and Its Generalized. International Journal of Nonlinear Science, Vol. 8, No. 4, hal. 448 – 455.
He, J. 1997. A New Approach to Nonlinear Partial Differential Equations. Communications in Nonlinear Science and Numerical Simulation, Vol. 2, No. 4, hal. 230 – 235.
_____. 1999. Variational Iteration Method – A Kind of Non-Linear Analytical Technique : Some Examples. International Journal of Non-Linear Mechanics, Vol. 34, hal. 699 – 708.
Batiha, B., Noorani, M. S. M., & Hashim, I. 2008. The Solution of the One Species Lotka-Volterra Equation Using Variational Iteration Method. Malaysian Journal of Mathematical Sciences, Vol 2, No. 2, hal. 55 – 60.
Ghotbi, A. R., Barari, A., Omidvar, M., & Ganji, D. D. 2009. Aplication of Variatioanl Iteration Method to Parabolic Problem. Applied Mathematical Sciences, Vol. 3, No. 19, hal. 927-934.
Biazar, J. 2010. Variational Iteration Method for Linear and Nonlinear System of Integro-Differential Equations. Interantional Mathematical Forum, Vol. 5, No. 64, hal. 3185 – 3191.
Wazwaz, A. M., & Khuri, S. A. 2015. The Variational Iteration Method for Solving the Volterra Integro-Differential Forms of the Lane-Emnden and the Emnden-Fowler Problems with Intial and Boundary Value Conditions. Open Engineering, Vol. 5, hal. 31 – 41.
Narayanamoorthy, S., & Mathankumar, S. 2018. Variational Iterative Method: An Approriate Numerical Scheme for Solving System of Linear Volterra Fuzzy Integro-Differential Equations. Advances in Difference Equations, Vol. 394, hal. 1 – 15.
Tatari, M., & Dehghan, M. 2007. On the Convergence of He’s Variational Iteration Method. Journal of Computational and Applied Mathematics, Vol 207, hal. 121 – 128.
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