NUMERICAL APPROXIMATION OF OSCILLATORY INITIAL VALUE PROBLEMS USING THE HOMOTOPY ANALYSIS ALGORITHM

  • Olayemi Obafaiye Federal university lokoja
  • Sunday O. Imoni
  • David I. Lanlege
Keywords: Numerical Analysis, Initial Value Problems, Differential Equations, Homotopy Analysis

Abstract

In this paper, we present numerical approximation for oscillatory initial value problems (IVPs) using the homotopy analysis algorithm. The convergence of the method is discussed and numerical experiments are presented to illustrate the computational effectiveness of the algorithm. The results obtained are in good agreement with the exact solutions and Adomian decomposition method (ADM). These results show that the algorithm introduced here is easy to apply without linearization.

References

Abbasbandy, S. (2006). The Application of Homotopy Analysis Method to Nonlinear Equations Arising in Heat Transfer. Physics Letters A, 360(1):109–113. DOI: https://doi.org/10.1016/j.physleta.2006.07.065

Abbassbandy, S. (2017).The Homotopy Analysis Method for Solving Non linear Problems. Modelling and Analysis of Modern Fluid Problems, 7(2).

Akyildiz, F. and Vajravelu, K. (2008). Magnetohydrodynamic Flow of a Viscoelastic Fluid. Phys. Lett. A., 372:3380 –3384. DOI: https://doi.org/10.1016/j.physleta.2008.01.073

Bataineh, A. S., Noorani, M. S. M., and Hashim, I. (2009). Direct Solution of nth Order ivps by Homotopy Analysis Method. Differential Equations and Nonlinear Mechanics, 2009:1–15. DOI: https://doi.org/10.1155/2009/842094

Chioma, I., Ugonna, E., Michael, U., Andrew, O., Ifeyinwa, M., and Ijeoma, U. (2019). Application of Homotopy Analysis Method for solving an seirs Epidemic Model. Mathematical Modelling and Applications, 4(3):36–48. DOI: https://doi.org/10.11648/j.mma.20190403.11

Fadugba, S. E. and Edeki, S. O. (2022). Homotopy Analysis Method for Fractional Barrier Option pde. Journal of Physics: Conference Series, 2199(1):012008. DOI: https://doi.org/10.1088/1742-6596/2199/1/012008

Fallahzadeh, A. and Shakibi, K. (2015).A Method to Solve Convection-Diffusion Equation Based on Homotopy Analysis Method. Journal of Interpolation and Approximation in Scientific Computing, 2015(1):1–8. DOI: https://doi.org/10.5899/2015/jiasc-00074

Ghanbari, B. (2014). The Convergence Study of the Homotopy Analysis Method for Solving Nonlinear Volterra-Fredholm Integrodifferential Equations. The Scientific World Journa, 2014:1–7. DOI: https://doi.org/10.1155/2014/465951

Ghoreishi, M., Ismail, A., and Alomari, A. (2011). Application of the Homotopy Analysis Method for Solving a Model For Hiv Infection of cd4+ t-cells. Mathematical and Computer Modelling, 54(11-12):3007–3015. DOI: https://doi.org/10.1016/j.mcm.2011.07.029

Imoni S.O, F.O Otunla and Ramamohan (2006) Embedded Implicit Runge-kutta Algorithmic Method for Solving Second-Order Differential Equations – Int.] or Compute mathematics vol 8.3 No. 11 (777 – 784) DOI: https://doi.org/10.1080/00207160601084505

Imoni S.O, Ikhile M.N.O (2014) Zero Dissipated RIKKN Point of Order 5 for Solving Special Second Order IVPS A uni-Dalakin-Volume: fab. Rev-nat Mathematics 53. No. 2 (53-69)

Imoni S.O (2020). Diagonally Implicit Runge Kutta-Nystrom (RKN) Method for Solving Second Order ODEs on Partial Computers, FUDMA Journal of Science, Vol 4, No.3PP.513-522 DOI: https://doi.org/10.33003/fjs-2020-0403-371

Khan, M. (2019). Analytical Solution of Van Der Pols Differential Equation Using Homotopy Perturbation Method. Journal of Applied Mathematics and Physics, 7:1– 12. DOI: https://doi.org/10.4236/jamp.2019.71001

Liang, S. and Jeffrey, D. (2009). Comparison of Homotopy Analysis Method and Homotopy Perturbation Method through an Evaluation Equation. Commun. Nonlinear Sci. Numer. Simulat. 14:4057 –4064. Liao, S. (1992). The Proposed Homotopy Analysis Technique for the Solution of Non-Linear Problems. PhD dissertation. DOI: https://doi.org/10.1016/j.cnsns.2009.02.016

Liao, S. (1997). Homotopy Analysis Method: A New Analytical Technique for Nonlinear Problems. Direct Science, 2(2):95–100. Liao, S. (1999). An Explicit, Totally Analytic Approximation of Blasius Viscous Flow Problems. Int. J. Nonlin. Mech., 34:759 –778. DOI: https://doi.org/10.1016/S0020-7462(98)00056-0

Liao, S. (2010). An Optimal Homotopy-Analysis Approach For Strongly Nonlinear dif- Ferential Equations. Commun. Nonlinear Sci. Numer. Simulat, 15:2003–2016. Liao, S. (2012). The Homotopy Analysis Method in Nonlinear Differential Equations. Higher Education Press and Springer, Beijing and Heidelberg. DOI: https://doi.org/10.1007/978-3-642-25132-0_3

Maitama, S. and Zhao, W. (2019). New Homotopy Analysis Transform Method for Solving Multidimensional Fractional Diffusion Equations. Arab Journal of Basic and Applied Sciences, 27(1):27–44. DOI: https://doi.org/10.1080/25765299.2019.1706234

Marinca, V. and Herisanu, N. (2008). Application of Optimal Homotopy Asymptotic Method for Solving Nonlinear Equations Arising in Heat Transfer. Int. Commun. Heat Mass., 35:710 –715. DOI: https://doi.org/10.1016/j.icheatmasstransfer.2008.02.010

kharrib, H. and Salem, T. (2021). New Algorithm of the Optimal Homotopy Asymptotic Method for Solving Lane-Emden Equations. Journal of Applied Mathematics and Computation, 5(4):237–246. DOI: https://doi.org/10.26855/jamc.2021.12.001

Mohyud-Din, S., Hussain, A., and Yildirim, A. (2010). Homotopy Analysis Method for Parametric Differential Equations. World Applied Sciences Journa, 11(7):851–856.

Motsa, S., Sibanda, P., and Shateyi, S. (2010). A New Spectral Homotopy Analysis Method for Solving a Nonlinear Second order bvp. Commun. Nonlinear Sci. Numer. Simulat, 15:2293–2302. DOI: https://doi.org/10.1016/j.cnsns.2009.09.019

Nandeppanavar, M. (2016). Applications of homotopy Analysis Method in Science and Engineering Research Problems. Journal of Information Engineering and Applications, (4):21–26.

Okposo, N. and Jonathan, A. (2020). Application of the Homotopy Analysis Transform Method to the Solution of a Fractional Attraction Keller Segel Chemo taxis Model. Science World Journa, 15(3):1597–:6343.

Omar, H. A. (2021a). Homotopy Analysis Based Hybrid Genetic Algorithm and Secant Method to solve ivps and Higher Order bvp. Institute of Electrical and Electronics Engineers (IEEE), 9:65101–65115. DOI: https://doi.org/10.1109/ACCESS.2021.3076300

Omar, H. A. (2021b). An Integrated Genetic Algorithm and Homotopy Analysis Method to Solve Nonlinear Equation Systems. Mathematical Problems in Engineering, 2021:1–14.

Omar, H. A. (2021c). An Integrated Genetic Algorithm and Homotopy Analysis Method to Solve Nonlinear Equation Systems. Mathematical Problems in Engineering, 2021:1–14. DOI: https://doi.org/10.1155/2021/5589322

Siddiqui, A. M., Haroon, T., Bhatti, S., and Ansari, A. R. (2010). A Comparison of the Adomian and Homotopy Perturbation Methods in Solving the Problem of Squeezing Flow between Two Circular plates. Mathematical Modelling and Analysis, 15(4):491–504. DOI: https://doi.org/10.3846/1392-6292.2010.15.491-504

Siddiqui, M. and Iqbal, A. (2019). Solution of Non-Linear ito System of Equations by Homotopy Analysis Method (ham). International Journal of Engineering and Technology, 8(1.10):144–150.

Singh, M., Naseem, M., Kumar, A., and Kumar, S. (2016). Homotopy Analysis Transform Algorithm to Solve Time-Fractional Foam Drainage Equation. Nonlinear Engineering, 5(3):161– 166. DOI: https://doi.org/10.1515/nleng-2016-0014

Vishal, K., Kumar, S., and Das, S. (2012). Application of Homotopy Analysis Method for Fractional Swift Hohenberg Equation. Applied Mathematical Modelling, 36:3630– 3637. DOI: https://doi.org/10.1016/j.apm.2011.10.001

Yabushita, K., Yamashita, M., and Tsuboi, K. (2017). An Analytic Solution of Projectile Motion with the Quadratic Resistance Law using the Homotopy Analysis Method. J. Phys. A. Math. Theory, 40:8403 –8416. DOI: https://doi.org/10.1088/1751-8113/40/29/015

Yuan, L.-G. and Alam, Z. (2016). An Optimal Homotopy Analysis Method Based on Particle Swarm Optimization: Application on fractional-order differential equation. Journal of Applied Analysis and Computation

Published
2023-04-30
How to Cite
ObafaiyeO., Imoni S. O., & Lanlege D. I. (2023). NUMERICAL APPROXIMATION OF OSCILLATORY INITIAL VALUE PROBLEMS USING THE HOMOTOPY ANALYSIS ALGORITHM. FUDMA JOURNAL OF SCIENCES, 7(2), 227 - 234. https://doi.org/10.33003/fjs-2023-0702-1743