# The Integral Iterative Method for Approximate Solution of Newell-Whitehead-Segel Equation

### Abstract

In this paper, the Newell-Whitehead-Segel (NWS) equation is solved using the integral iterative method (IIM) to determine the accuracy and effectiveness of the method. Comparison of results obtained by IIM with the exact solution and other existing results obtained by other methods such as new iterative method (NIM), Adomian decomposition method (ADM) and Laplace Adomian decomposition method (LADM) revealed the accuracy and effectiveness of the method. The approximation results obtained by IIM is comparable with the others. IIM is reliable and easier in solving the nonlinear problems since this method is simple, straightforward and does not require calculating multiple integral and demand less computational work.

### References

[1] A. C. Newell and J. A. Whitehead, ”Finite bandwitdth, finite amplitude convection,” J. Fluid Mech., vol. 38, no. 2, pp. 279-303, 1969.

[2] L. A. Segel, ”Distant side-walls cause slow amplitude modulation of cellular convection,” J. Fluid. Mech., vol 38, no. 1, pp. 203-224, 1969.

[3] R. Ezzati, and K. Shakibi. "Using adomian’s decomposition and multiquadric quasi-interpolation methods for solving Newell–Whitehead equation." Procedia. Comput. Sci., vol. 3, pp. 1043-1048, 2011.

[4] S. A. Manaa, "An approximate aolution to the Newell-Whitehead equation by Adomian decomposition method." Raf. J. of Comp. & Math’s., vol. 8, no. 1, pp. 171-18, 2011.

[5] P. Pue-on, "Laplace Adomian decomposition method for solving Newell-Whitehead-Segel equation." Appl. Math. Sci., vol. 7, no. 132, pp. 6593-6600, 2013.

[6] S. S. Nourazar, S. Salman, M. Soori, and A. Nazari-Golshan. "On the exact solution of Newell-Whitehead-Segel equation using the homotopy perturbation method." Aust. J. Basic & Appl. Sci., vol. 5, no. 8, pp. 1400-1411, 2011.

[7] A. Prakash, and K. Manoj, "He’s variational iteration method for the solution of nonlinear Newell–Whitehead–Segel equation." J. Appl. Anal. Comput., vol. 6, no.3, pp. 738-748, 2016.

[8] J. Patade, and S. Bhalekar. "Approximate analytical solutions of Newell-Whitehead-Segel equation using a new iterative method." Int. J. Simul. Model., vol. 11, no. 2, pp. 94-103, 2015.

[9] A. Saravanan, and N. Magesh. "A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell–Whitehead–Segel equation," J. Egypt. Math. Soc., vol. 21, no. 3, pp. 259-265, 2013.

[10] H. K. Jassim, "Homotopy perturbation algorithm using Laplace transform for Newell-Whitehead-Segel equation." Int. J. Adv. Appl. Math. Mech., vol. 2, no. 4, pp. 8-12, 2015.

[11] B. Latif, M. S. Selamat, A. N. Rosli, A. I. Yusoff, and N. M. Hasan, “The semi analytics iterative method for solving Newell-Whitehead-Segel Equation,” Math Stat., vol. 8, no. 2, pp. 87-94, 2020.

[12] M. Almousa, I. Al-Zuhairi, Y. Al-Qudah, and H. Qoqazeh, “Mahgoub Adomian decomposition method for solving Newell-Whitehead-Segel equation. Int. J. Math. Stat. Invent., vol. 8, no. 1, pp. 22-24, 2020.

[13] N. S. Elgazery, "A Periodic Solution of the Newell-Whitehead-Segel (NWS) Wave Equation via Fractional Calculus." J. Appl. Comput. Mech, vol. 6, pp. 1293-1300, 2020.

[14] A. A. Hemeda, “A friendly iterative technique for solving nonlinear integro-differential and systems of nonlinear integro-differential equations,” Int. J. Comput. Methods, vol. 25, no. 1, pp. 1850016, 2018

[15] A. A. Hemeda, and E. E. Eladdad, ”New iterative methods for solving Fokker-Planck equation,” Math. Probl. Eng., 2018

[16] A. A. Hemeda, “Iterative methods for solving fractional gas dynamics and coupled Burgers’ equation,” Int. J. Math. Model. Numer. Opt., vol. 4, no. 4, pp. 282-298, 2015.

[17] A. A. Nahla and A. A. Hemeda, “Picard iteration and Padé approximations for stiff fractional point kinetics equations,” Appl. Math. Comput., vol. 293, pp. 72-80, 2017.

[2] L. A. Segel, ”Distant side-walls cause slow amplitude modulation of cellular convection,” J. Fluid. Mech., vol 38, no. 1, pp. 203-224, 1969.

[3] R. Ezzati, and K. Shakibi. "Using adomian’s decomposition and multiquadric quasi-interpolation methods for solving Newell–Whitehead equation." Procedia. Comput. Sci., vol. 3, pp. 1043-1048, 2011.

[4] S. A. Manaa, "An approximate aolution to the Newell-Whitehead equation by Adomian decomposition method." Raf. J. of Comp. & Math’s., vol. 8, no. 1, pp. 171-18, 2011.

[5] P. Pue-on, "Laplace Adomian decomposition method for solving Newell-Whitehead-Segel equation." Appl. Math. Sci., vol. 7, no. 132, pp. 6593-6600, 2013.

[6] S. S. Nourazar, S. Salman, M. Soori, and A. Nazari-Golshan. "On the exact solution of Newell-Whitehead-Segel equation using the homotopy perturbation method." Aust. J. Basic & Appl. Sci., vol. 5, no. 8, pp. 1400-1411, 2011.

[7] A. Prakash, and K. Manoj, "He’s variational iteration method for the solution of nonlinear Newell–Whitehead–Segel equation." J. Appl. Anal. Comput., vol. 6, no.3, pp. 738-748, 2016.

[8] J. Patade, and S. Bhalekar. "Approximate analytical solutions of Newell-Whitehead-Segel equation using a new iterative method." Int. J. Simul. Model., vol. 11, no. 2, pp. 94-103, 2015.

[9] A. Saravanan, and N. Magesh. "A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell–Whitehead–Segel equation," J. Egypt. Math. Soc., vol. 21, no. 3, pp. 259-265, 2013.

[10] H. K. Jassim, "Homotopy perturbation algorithm using Laplace transform for Newell-Whitehead-Segel equation." Int. J. Adv. Appl. Math. Mech., vol. 2, no. 4, pp. 8-12, 2015.

[11] B. Latif, M. S. Selamat, A. N. Rosli, A. I. Yusoff, and N. M. Hasan, “The semi analytics iterative method for solving Newell-Whitehead-Segel Equation,” Math Stat., vol. 8, no. 2, pp. 87-94, 2020.

[12] M. Almousa, I. Al-Zuhairi, Y. Al-Qudah, and H. Qoqazeh, “Mahgoub Adomian decomposition method for solving Newell-Whitehead-Segel equation. Int. J. Math. Stat. Invent., vol. 8, no. 1, pp. 22-24, 2020.

[13] N. S. Elgazery, "A Periodic Solution of the Newell-Whitehead-Segel (NWS) Wave Equation via Fractional Calculus." J. Appl. Comput. Mech, vol. 6, pp. 1293-1300, 2020.

[14] A. A. Hemeda, “A friendly iterative technique for solving nonlinear integro-differential and systems of nonlinear integro-differential equations,” Int. J. Comput. Methods, vol. 25, no. 1, pp. 1850016, 2018

[15] A. A. Hemeda, and E. E. Eladdad, ”New iterative methods for solving Fokker-Planck equation,” Math. Probl. Eng., 2018

[16] A. A. Hemeda, “Iterative methods for solving fractional gas dynamics and coupled Burgers’ equation,” Int. J. Math. Model. Numer. Opt., vol. 4, no. 4, pp. 282-298, 2015.

[17] A. A. Nahla and A. A. Hemeda, “Picard iteration and Padé approximations for stiff fractional point kinetics equations,” Appl. Math. Comput., vol. 293, pp. 72-80, 2017.

Published

2022-05-25

How to Cite

SELAMAT, Mat Salim bin et al.
The Integral Iterative Method for Approximate Solution of Newell-Whitehead-Segel Equation.

**Mathematical Sciences and Informatics Journal**, [S.l.], v. 3, n. 1, p. 1-10, may 2022. ISSN 2735-0703. Available at: <https://myjms.mohe.gov.my/index.php/mij/article/view/13009>. Date accessed: 27 may 2024. doi: https://doi.org/10.24191/mij.v3i1.13009.
Issue

Section

Articles