Abstract
We suggested a proposal for widening the usage of He's homotopy perturbation methodology to solve complex algebraic formulae by utilizing the first seven elements of Taylor's series. This approach would expand the use of He's original method. The fundamental goal of our research is to come up with a solution that is approximative yet still manages to attain an elevated degree of accuracy. We are seeking a method that will allow us to answer non-linear algebraic questions. To arrive at an analytical function within the framework of the suggested hybrid scheme, we used Taylor's series and the homotopy technique. To accomplish this, the initial terms of Taylor's series were trimmed, and the variational iteration technique was then employed. We were able to demonstrate that the recommended technique is successful by locating a rough response that was precise to a significant degree. This allowed us to locate an approximate solution. This exact answer can be obtained by looking at the spot on the graph where the horizontal axis intersects with the graph. In particular, we devised a brand new iterative equation that, at each phase of the procedure, generates a solution estimation that gets increasingly nearer to the accurate one, while concurrently decreasing the amount of errors that occur. This was accomplished by combining the two concepts described above. Examples are supplied to illustrate the accuracy of the process that has been described so far.
Keywords: Iterative Scheme, Taylor's Series, Homotopy Perturbation Method, Nonlinear Algebraic Equations.
References
- M. Javidi, A. Golbabi, "Modified Homotopy perturbation method for solving a non-integral equation, "Choas, Solutions and Fractals Vol. 40, pp. 1408-1412, 2009.
- J. H. He, "A coupling method of a homotopy technique and a perturbation technique for non-linear problems," Int. J. Nonlinear Mechanics, Vol. 35, No. 1, pp. 37-43, 2000
- B. N. Kharrat, G. Toma, "A New Hybrid Sumudu Transform with Homotopy Perturbation Method for Solving Boundary Value Problems," Middle-East Journal of Scientific Research , Vol. 28, No. 2, pp. 142-149, 2020.
- B. N. Kharrat, G. Toma, "Modified Homotopy perturbation method by using Sumudu Transform for Solving Initial value problems Represented By a system of Nonlinear Partial Differential Equations," World Applied Science Journal, Vol. 36, No. 7, pp. 844-849, 2018.