A New Computational Method Based on Integral Transform for Solving Linear and Nonlinear Fractional Systems

  • Diyar Hashim Malo University of Duhok, Duhok, Iraq
  • Rogash Younis Masiha Van Yüzüncü Yıl Üniversitesi, Van, Turkey
  • Muhammad Amin Sadiq Murad University of Duhok, Duhok, Iraq
  • Sadeq Taha Abdulazeez University of Duhok, Duhok, Iraq
Keywords: Fractional stiff systems, Differential equations, Elzaki homotopy perturbation methods, Kernel Hilbert space method, Approximate solutions

Abstract

In this article, the Elzaki homotopy perturbation method is applied to solve fractional stiff systems. The Elzaki homotopy perturbation method (EHPM) is a combination of a modified Laplace integral transform called the Elzaki transform and the homotopy perturbation method. The proposed method is applied for some examples of linear and nonlinear fractional stiff systems. The results obtained by the current method were compared with the results obtained by the kernel Hilbert space KHSM method. The obtained result reveals that the Elzaki homotopy perturbation method is an effective and accurate technique to solve the systems of differential equations of fractional order.

Downloads

Download data is not yet available.

References

C. F. Curtiss and J. O. Hirschfelder, “Integration of Stiff Equations.,” Proc. Natl. Acad. Sci. U. S. A., vol. 38, no. 3, pp. 235–243, Mar. 1952, doi: 10.1073/pnas.38.3.235.

J. E. Flaherty and R. E. O’Malley, “The Numerical Solution of Boundary Value Problems for Stiff Differential Equations,” Math. Comput., vol. 31, no. 137, p. 66, 1977, doi: 10.2307/2005781.

D. Zwillinger, “Stiff Equations” Handb. Differ. Equations, vol. 38, no. 1950, pp. 690–694, 1992, doi: 10.1016/b978-0-12-784391-9.50181-0.

R. V Slonevskii and R. R. Stolyarchuk, “Rational-fractional methods for solving stiff systems of differential equations,” J. Math. Sci., vol. 150, no. 5, pp. 2434–2438, 2008, doi: 10.1007/s10958-008-0141-x.

H. Jafari and V. Daftardar-Gejji, “Solving a system of nonlinear fractional differential equations using Adomian decomposition,” J. Comput. Appl. Math., vol. 196, no. 2, pp. 644–651, 2006, doi: 10.1016/j.cam.2005.10.017.

H. Aminikhah and M. Hemmatnezhad, “An effective modification of the homotopy perturbation method for stiff systems of ordinary differential equations,” Appl. Math. Lett., vol. 24, no. 9, pp. 1502–1508, 2011, doi: 10.1016/j.aml.2011.03.032.

A. Rani, M. Saeed, Q. M. Ul-Hassan, M. Ashraf, M. Y. Khan, and K. Ayub, “Solving system of differential equations of fractional order by homotopy analysis method,” J. Sci. Arts, vol. 17, no. 3, pp. 457–468, 2017.

M. T. Darvishi, F. Khani, and A. A. Soliman, “The numerical simulation for stiff systems of ordinary differential equations,” Comput. Math. with Appl., vol. 54, no. 7–8, pp. 1055–1063, 2007.

J. Biazar, M. A. Asadi, and F. Salehi, “Rational homotopy perturbation method for solving stiff systems of ordinary differential equations,” Appl. Math. Model., vol. 39, no. 3–4, pp. 1291–1299, 2015, doi: 10.1016/j.apm.2014.09.003.

O. A. Akinfenwa, B. Akinnukawe, and S. B. Mudasiru, “A family of Continuous Third Derivative Block Methods for solving stiff systems of first order ordinary differential equations,” J. Niger. Math. Soc., vol. 34, no. 2, pp. 160–168, 2015, doi: 10.1016/j.jnnms.2015.06.002.

O. H. Mohammed and H. A. Salim, “Computational methods based laplace decomposition for solving nonlinear system of fractional order differential equations,” Alexandria Eng. J., vol. 57, no. 4, pp. 3549–3557, 2018, doi: 10.1016/j.aej.2017.11.020.

M. H. T. Alshbool and I. Hashim, “Multistage Bernstein polynomials for the solutions of the Fractional Order Stiff Systems,” J. King Saud Univ. - Sci., vol. 28, no. 4, pp. 280–285, 2016, doi: 10.1016/j.jksus.2015.06.001.

A. Freihet, S. Hasan, M. Al-Smadi, M. Gaith, and S. Momani, “Construction of fractional power series solutions to fractional stiff system using residual functions algorithm,” Adv. Differ. Equations, vol. 2019, no. 1, 2019, doi: 10.1186/s13662-019-2042-3.

M. Kumar and A. S. Saxena, “New iterative method for solving higher order KDV equations,” pp. 246–257.

M. Javidi and B. Ahmad, “Numerical solution of fourth-order time-fractional partial differential equations with variable coefficients,” J. Appl. Anal. Comput., vol. 5, no. 1, pp. 52–63, 2015, doi: 10.11948/2015005.

D. H. Shou, “The homotopy perturbation method for nonlinear oscillators,” Comput. Math. with Appl., vol. 58, no. 11–12, pp. 2456–2459, 2009, doi: 10.1016/j.camwa.2009.03.034.

J. Biazar, B. Ghanbari, M. G. Porshokouhi, and M. G. Porshokouhi, “He’s homotopy perturbation method: A strongly promising method for solving non-linear systems of the mixed Volterra–Fredholm integral equations,” Comput. Math. with Appl., vol. 61, no. 4, pp. 1016–1023, 2011, doi: https://doi.org/10.1016/j.camwa.2010.12.051.

J. Biazar and H. Ghazvini, “Homotopy perturbation method for solving hyperbolic partial differential equations,” Comput. Math. with Appl., vol. 56, no. 2, pp. 453–458, 2008, doi: https://doi.org/10.1016/j.camwa.2007.10.032.

J. Biazar, F. Badpeima, and F. Azimi, “Application of the homotopy perturbation method to Zakharov–Kuznetsov equations,” Comput. Math. with Appl., vol. 58, no. 11, pp. 2391–2394, 2009, doi: https://doi.org/10.1016/j.camwa.2009.03.102.

T. M. Elzaki and J. Biazar, “Homotopy perturbation method and Elzaki transform for solving system of nonlinear partial differential equations,” World Appl. Sci. J., vol. 24, no. 7, pp. 944–948, 2013, doi: 10.5829/idosi.wasj.2013.24.07.1041.

A. C. Loyinmi and T. K. Akinfe, “Exact solutions to the family of Fisher’s reaction‐diffusion equation using Elzaki homotopy transformation perturbation method,” Eng. Reports, vol. 2, no. 2, pp. 1–32, 2020, doi: 10.1002/eng2.12084.

J. Ul Rahman, D. Lu, M. Suleman, J. H. He, and M. Ramzan, “He-elzaki method for spatial diffusion of biological population,” Fractals, vol. 27, no. 5, 2019, doi: 10.1142/S0218348X19500695.

N. Anjum, M. Suleman, D. Lu, J. H. He, and M. Ramzan, “Numerical iteration for nonlinear oscillators by Elzaki transform,” J. Low Freq. Noise Vib. Act. Control, 2019, doi: 10.1177/1461348419873470.

D. Lu, M. Suleman, J. H. He, U. Farooq, S. Noeiaghdam, and F. A. Chandio, “Elzaki projected differential transform method for fractional order system of linear and nonlinear fractional partial differential equation,” Fractals, vol. 26, no. 3, 2018, doi: 10.1142/S0218348X1850041X.

R. M. Jena and S. Chakraverty, “Solving time-fractional Navier–Stokes equations using homotopy perturbation Elzaki transform,” SN Appl. Sci., vol. 1, no. 1, pp. 1–13, 2019, doi: 10.1007/s42452-018-0016-9.

A. C. Loyinmi and T. K. Akinfe, “An algorithm for solving the Burgers–Huxley equation using the Elzaki transform,” SN Appl. Sci., vol. 2, no. 1, p. 7, 2020.

T. M. Elzaki, “The New Integral Transform ‘ELzaki Transform’,” vol. 7, no. 1, pp. 57–64, 2011.

E. M. A. Hilal, “Elzaki and Sumudu Transforms for Solving Some,” vol. 8, no. 2, pp. 167–173, 2012.

T. M. Elzaki, “Solution of Nonlinear Differential Equations Using Mixture of Elzaki Transform and Differential Transform Method,” vol. 7, no. 13, pp. 631–638, 2012.

D. Ziane and M. H. Cherif, “Resolution of Nonlinear Partial Differential Equations by Elzaki Transform Decomposition Method Laboratory of mathematics and its applications (LAMAP),” vol. 5, pp. 17–30, 2015.

O. E. Ige, R. A. Oderinu, and T. M. Elzaki, “Adomian polynomial and Elzaki transform method for solving sine-gordon equations,” IAENG Int. J. Appl. Math., vol. 49, no. 3, pp. 1–7, 2019.

Q. Branch, N. Branch, and M. Benchohra, “Applications of homotopy perturbation method and Elzaki transform for solving nonlinear partial differential equations of fractional order,” vol. 2015, no. 6, pp. 91–104, 2016.

N. Shawagfeh, “Decomposition method for fractional partial differential equations,” no. December, 2017, doi: 10.5829/idosi.wasj.2019.18.24.

A. Prakash and V. Verma, “Numerical method for fractional model of newell-whitehead-segel equation,” Front. Phys., vol. 7, no. FEB, pp. 1–10, 2019, doi: 10.3389/fphy.2019.00015.

O. Abu Arqub and M. Al-Smadi, “Numerical algorithm for solving time-fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions,” Numer. Methods Partial Differ. Equ., vol. 34, no. 5, pp. 1577–1597, 2018, doi: 10.1002/num.22209.

CROSSMARK
Published
2021-05-31
DIMENSIONS
How to Cite
MaloD. H., MasihaR. Y., MuradM. A. S., & AbdulazeezS. T. (2021). A New Computational Method Based on Integral Transform for Solving Linear and Nonlinear Fractional Systems. Jurnal Matematika MANTIK, 7(1), 9-19. https://doi.org/10.15642/mantik.2021.7.1.9-19