HOMOTOPY PERTURBATIONS METHOD: THEORETICAL ASPECTS & APPLICATIONS
Abstract
The application of the homotopy perturbation method (HPM) in two different research's area has been proposed in this paper. First, the HPM has been used for approximate solving of the well-known implicit equation for electrostatic surface potential of MOSFET transistor. The approximate analytical solution obtained in this case has relative simple mathematical form, and simultaneously high degree of accuracy. Next, HPM has been applied in determination of the invariant measures (IMs) of the non-linear dynamical systems with chaotic behavior. The convergence and efficiency of this method have been confirmed and illustrated in some characteristic examples of chaotic mappings.
References
Adamu, G., Bawa, M., Jiya, M., & Chado, U. 2017. A Mathematical Model for the Dynamics of Zika Virus via Homotopy Perturbation Method. Journal of Applied Sciences and Environmental Management, 21(4), pp. 615-623, doi:10.4314/jasem.v21i4.1
Biazar, J., Badpeima, F., & Azimi, F. 2009. Application of the homotopy perturbation method to Zakharov–Kuznetsov equations. Computers & Mathematics with Applications, 58(11-12), pp. 2391-2394. doi:10.1016/j.camwa.2009.03.102
Bota, C., & Căruntu, B. 2017. Approximate analytical solutions of nonlinear differential equations using the Least Squares Homotopy Perturbation Method. Journal of Mathematical Analysis and Applications, 448(1), pp. 401-408. doi:10.1016/j.jmaa.2016.11.031
Chen, T.L., & Gildenblat, G. 2001. Analytical approximation for the MOSFET surface potential. Solid-State Electronics, 45(2), pp. 335-339. doi:10.1016/s0038-1101(00)00283-5
Dong, C., Chen, Z., & Jiang, W. 2013. A modified homotopy perturbation method for solving the nonlinear mixed Volterra–Fredholm integral equation. Journal of Computational and Applied Mathematics, 239, pp. 359-366. doi:10.1016/j.cam.2012.09.003
El-Sayed, A.M.A., Elsaid, A., El-Kalla, I.L., & Hammad, D. 2012. A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains. Applied Mathematics and Computation, 218(17), pp. 8329-8340. doi:10.1016/j.amc.2012.01.057
Gadallah, M.R., & Elzaki, T.M. 2017. An application of improvement of new homotopy perturbation method for solving third order nonlinear singular partial differential equations. Universal Journal of Mathematics, 2 (1), pp. 110-124.
He, J. 1999. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178(3-4), pp. 257-262. doi:10.1016/s0045-7825(99)00018-3
He, J. 2000. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics, 35(1), pp. 37-43. doi:10.1016/s0020-7462(98)00085-7
He, J. 2003. Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and Computation, 135(1), pp. 73-79. doi:10.1016/s0096-3003(01)00312-5
He, J. 2006. Homotopy perturbation method for solving boundary value problems. Physics Letters A, 350(1-2), pp. 87-88. doi:10.1016/j.physleta.2005.10.005
He, J.H. 2008. Resent development of the homotopy perturbation method. Topological Methods in Nonlinear Analysis, 31 (2), pp. 205-209.
Hetmaniok, E., Nowak, I., Słota, D., & Wituła, R. 2013. A study of the convergence of and error estimation for the homotopy perturbation method for the Volterra–Fredholm integral equations. Applied Mathematics Letters, 26(1), pp. 165-169. doi:10.1016/j.aml.2012.08.005
Hetmaniok, E., Słota, D., & Wituła, R. 2012. Convergence and error estimation of homotopy perturbation method for Fredholm and Volterra integral equations. Applied Mathematics and Computation, 218(21), pp. 10717-10725. doi:10.1016/j.amc.2012.04.041
Kevkić, T., Stojanović, V., & Petković, D. 2018. Solving Schrödinger equation for a particle in one-dimensional lattice: An homotopy perturbations approach. Romanian Reports in Physics, in press, accepted manuscript.
Kevkić, T., Stojanović, V., & Randjelović, D. 2017. Application of homotopy perturbation method in solving coupled Schrödinger and Poisson equation in accumulation layer. Romanian Journal of Physics, Article, No. 122. 62 (9-10).
Khan, M., Saddiq, F.S., Khan, S., Islam, S., & Ahmad, F. 2014. Application of homotopy perturbation method to an SIR epidemic model. Journal of Applied Environmental and Biological Sciences, 49-54; 4.
Noor, M.A., & Khan, W.A. 2012. New iterative methods for solving nonlinear equation by using homotopy perturbation method. Applied Mathematics and Computation, 219(8), pp. 3565-3574. doi:10.1016/j.amc.2012.09.040
Stojanović, V., Kevkić, T., Jelić, G., & Randjelović, D. 2018. Determination of invariant measures: An approach based on homotopy perturbations. UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, in press, accepted manuscript.
Tripathi, R., & Mishra, H.K. 2016. Homotopy perturbation method with Laplace Transform (LT-HPM) for solving Lane–Emden type differential equations (LETDEs). SpringerPlus, 5(1). doi:10.1186/s40064-016-3487-4
van Langevelde, R., & Klaassen, F.M. 2000. An explicit surface-potential-based MOSFET model for circuit simulation. Solid-State Electronics, 44(3), pp. 409-418. doi:10.1016/s0038-1101(99)00219-1
Zeb, M., Haroon, T., & Siddiqui, A. 2014. Homotopy perturbation solution for flow of a third-grade fluid in helical screw rheometer. UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 76 (4), pp. 179–190.
Zhang, M., Liu, Y., & Zhou, X. 2015. Efficient homotopy perturbation method for fractional non-linear equations using Sumudu transform. Thermal Science, 19(4), pp. 1167-1171. doi:10.2298/tsci1504167z
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.