RESOLVING DELAY DIFFERENTIAL EQUATIONS WITH HOMOTOPY PERTURBATION AND SUMUDU TRANSFORM
DOI:
https://doi.org/10.11113/jurnalteknologi.v85.18937Keywords:
Homotopy Perturbation Method, Delay Differential Equations, Sumudu Transform, Homotopy Perturbation Sumudu Transform MethodAbstract
A novel proposition has been introduced in this study for resolving delay differential equations (DDEs) of nature that is a composite in reference to Homotopy perturbation method (HPM) along with Sumudu transform. A rare transform called the Sumudu transform is used alongside the perturbation theory. Demonstration of this new methodology is shown by solving a few numerical cases. Reducing the complication of computational tasks associated to the conservative means is the objective of this research. Results display the amount of valuation being reduced and is as good as in the previous studies as well in comparison.
References
Alomari, A. K., Noorani, M. S. M., & Nazar, R. 2009. Solution of Delay Differential Equation by Means of Homotopy Analysis Method. Acta Applicandae Mathematicae. 108(2): 395-412. https://doi.org/10.1007/s10440-008-9318-z.
Dehghan, M., & Shakeri, F. 2007. Solution of a Partial Differential Equation Subject to Temperature Overspecification by He's Homotopy Perturbation Method. Physica Scripta. 75(6): 778. https://doi.org/10.1088/0031-8949/75/6/007.
Dehghan, M., & Shakeri, F. 2008. Use of He's Homotopy Perturbation Method for Solving a Partial Differential Equation Arising in Modeling of Flow in Porous Media. Journal of Porous Media. 11(8). https://doi.org/10.1615/JPorMedia.v11.i8.50.
Dehghan, M., & Shakeri, F. 2008. Solution of an Integro-differential Equation Arising in Oscillating Magnetic Fields using He's homotopy Perturbation Method. Progress in Electromagnetics Research. 78: 361-376. https://doi.org/10.2528/PIER07090403.
Ghorbani, A., & Saberi-Nadjafi, J. 2007. He's Homotopy Perturbation Method for Calculating Adomian Polynomials. International Journal of Nonlinear Sciences and Numerical Simulation. 8(2): 229-232. https://doi.org/10.1515/IJNSNS.2007.8.2.229.
He, J. H. 1999. Homotopy Perturbation Technique. Computer Methods in Applied Mechanics and Engineering. 178(3-4): 257-262. https://doi.org/10.1016/S0045-7825(99)00018-3.
He, J. H. 2003. Homotopy Perturbation Method: A New Nonlinear Analytical Technique. Applied Mathematics and Computation. 135(1): 73-79. https://doi.org/10.1016/S0096-3003(01)00312-5.
He, J. H. 2006. Homotopy Perturbation Method for Solving Boundary Value Problems. Physics letters A. 350(1-2): 87-88. https://doi.org/10.1016/j.physleta.2005.10.005.
Lakshmanan, S., Rihan, F. A., Rakkiyappan, R., & Park, J. H. 2014. Stability Analysis of the Differential Genetic Regulatory Networks Model with Time-varying Delays and Markovian Jumping Parameters. Nonlinear analysis: Hybrid systems. 14: 1-15. https://doi.org/10.1016/j.nahs.2014.04.003.
Rakkiyappan, R., Velmurugan, G., Rihan, F. A., & Lakshmanan, S. 2016. Stability Analysis of Memristor‐based Complex‐valued Recurrent Neural Networks with Time Delays. Complexity. 21(4): 14-39. https://doi.org/10.1002/cplx.21618.
Ratib Anakira, N., Alomari, A. K., & Hashim, I. 2013. Optimal Homotopy Asymptotic Method for Solving Delay Differential Equations. Mathematical Problems in Engineering. 2013. https://doi.org/10.1063/1.4858786.
Rihan, F. A., Abdelrahman, D. H., Al-Maskari, F., Ibrahim, F., & Abdeen, M. A. 2014. Delay Differential Model for Tumour-immune Response with Chemoimmunotherapy and Optimal Control. Computational and Mathematical Methods in Medicine. 2014. https://doi.org/10.1155/2014/982978.
Shakeri, F., & Dehghan, M. 2007. Inverse Problem of Diffusion Equation by He's Homotopy Perturbation Method. Physica Scripta. 75(4): 551. https://doi.org/10.1088/0031-8949/75/4/031.
Watugala, G. 1993. Sumudu Transform: A New Integral Transform to Solve Differential Equations and Control Engineering Problems. Integrated Education. 24(1): 35-43. https://doi.org/10.1080/0020739930240105.
Ganji, D. D., & Sadighi, A. 2006. Application of He's Homotopy-perturbation Method to Nonlinear Coupled Systems of Reaction-diffusion Equations. International Journal of Nonlinear Sciences and Numerical Simulation. 7(4): 411-418. https://doi.org/10.1515/IJNSNS.2006.7.4.411.
Singh, J., Gupta, P. K., & Rai, K. N. 2011. Homotopy Perturbation Method to Space-time Fractional Solidification in a Finite Slab. Applied Mathematical Modelling. 35(4): 1937-1945.
https://doi.org/10.1016/j.apm.2010.11.005.
Liu, Y., Li, Z., & Zhang, Y. 2011. Homotopy Perturbation Method to Fractional Biological Population Equation. Fractional Differential Calculus. 1(1): 117-124. https://doi.org/10.7153/fdc-01-07.
Rekha, S., Balaganesan, P., & Renuka, J. 2021. Homotopy Perturbation Method for Mathematical Modelling of Dengue Fever. Journal of Physics: Conference Series. 1724(1): 012056. IOP Publishing. https://doi.org/10.1088/1742-6596/1724/1/012056.
Domairry, G., & Aziz, A. 2009. Approximate Analysis of MHD Squeeze Flow between Two Parallel Disks with Suction or Injection by Homotopy Perturbation Method. Mathematical Problems in Engineering. 2009. https://doi.org/10.1155/2009/603916.
T. A. Biala, O. O. Asim, and Y. O. Afolabi. 2014. A Combination of the Laplace Transform and the Variational Iteration Method for the Analytical Treatment of Delay Differential Equations. International Journal of Differential Equations and Applications. 13(3).
Senu, N., Lee, K. C., Ahmadian, A., & Ibrahim, S. N. I. 2022. Numerical Solution of Delay Differential Equation using Two-Derivative Runge-Kutta Type Method with Newton Interpolation. Alexandria Engineering Journal. 61(8): 5819-5835. https://doi.org/10.1016/j.aej.2021.11.009.
Anakira, N., Jameel, A., Hijazi, M., Alomari, A. K., & Man, N. 2022. A New Approach for Solving Multi-pantograph Type Delay Differential Equations. International Journal of Electrical & Computer Engineering. 12(2): 2088-8708.
https://doi.org/10.11591/ijece.v12i2.pp1859-1868.
Jameel, A. F., Anakira, N. R., Alomari, A. K., Al-Mahameed, M., & Saaban, A. 2019. A New Approximate Solution of the Fuzzy Delay Differential Equations. International Journal of Mathematical Modelling and Numerical Optimisation. 9(3): 221-240. https://doi.org/10.1504/IJMMNO.2019.100476.
Izadi, M., & Srivastava, H. M. 2021. An Efficient Approximation Technique Applied to a Non-linear Lane-emden Pantograph Delay Differential Model. Applied Mathematics and Computation. 401: 126123. https://doi.org/10.1016/j.amc.2021.126123.
Anakira, N. R., Abdelkarim, H., & Abu-Dawas, M. 2020. Homotopy Sumudu Transformation Method for Solving Fractional Delay Differential Equations. Gen. Lett. Math. 9(1): 33-41. https://doi.org/10.31559/GLM2020.9.1.4.
Downloads
Published
Issue
Section
License
Copyright of articles that appear in Jurnal Teknologi belongs exclusively to Penerbit Universiti Teknologi Malaysia (Penerbit UTM Press). This copyright covers the rights to reproduce the article, including reprints, electronic reproductions, or any other reproductions of similar nature.
