Analytical Algorithms: Some Modifications and Applications

Abstract

Initial and boundary value problems arise in different fields of mathematics and engineering. They are a result of mathematical modeling of various real-life phenomena. Some of these models are of higher nonlinearity. Thus, an exact solution of such a problem is very less likely. For those kinds of problems, we see different approximation methods, both analytical and numerical. The subject of this study is to work out some analytical algorithms that can be used to obtained solutions of nonlinear problems. A major issue with series solution algorithms is the convergence of these methods, especially for the cases of semi-infinite domain. We have tried to address this issue and have developed some modified algorithms that can work even where the traditional ones fail. Techniques like variational iteration method (VIM), variation of parameters method (VPM) and homotopy perturbation method (HPM) have been improved. This study is supposed to help the research community to remove some inbuilt deficiencies (such as divergent results, small parameter assumptions, need of perturbation, huge computational work and very limited convergence) of these traditional techniques. Efforts have been made to modify these analytical techniques. The modified schemes so obtained are free from these deficiencies. The modified schemes, like variation of parameters method with auxiliary parameter (VPMAP), optimal variation of parameters method with Adomian’s polynomials (OVPMAP), optimal variational iteration method (OVIM), optimal variational iteration method with Adomian’s polynomials (OVIMAP) and optimal homotopy perturbation method (OHPM) have been implemented on many problems arising in different fields of sciences such as, mathematical biology, fluid flow through different geometries, heat transfer equations related to chemical engineering etc. Convergent solutions are obtained for both bounded and unbounded domains by making an appropriate use of the developed modified versions. A brand new analytical algorithm, namely generalized iterative scheme (GIS), has also been introduced. Once can see its effectiveness for certain types of problems. Accuracy of the results is verified by comparing approximate solution with exact solutions, wherever available, or the residual error analysis.