Abstract
Here a new analytical scheme is presented to solve nonlinear boundary value problems (BVPs) of higher order occurring in nonlinear phenomena. This method is called second alternative of Optimal Homotopy Asymptotic Method. It converts a complex nonlinear problem into zeroth order and first order problem. A homotopy and auxiliary functions which are consisted of unknown convergence controlling parameters are used in this technique. The unknown parameters are determined by minimizing the residual. Many methods are used to determine these parameters. Here Galerkin's method is used for this purpose. It is applied to solve non-linear BVPs of order four, five, six, and seven. The Consequences are compared with other methods e.g., Differential Transform Method (DTM), Adomain Decomposition Method (ADM), Variational Iteration Method (VIM), and Optimal Homotopy Asymptotic Method (OHAM). It gives efficient and accurate first-order approximate solution. The achieved results are compared with the exact solutions as well as with other methods to authenticate the applied technique. This method is very simple and easy but more operative.
Highlights
Article No~e00913Many real world problems can be expressed in terms of boundary value problems (BVPs)
Various Investigators have created some other techniques based on Homotopy are HPM [14, 15, 10, 16], Optimal Homotopy Asymptotic Method (OHAM) [17, 18, 19, 20, 21, 22, 23, 24, 25, 26], and Optimal Homotopy Perturbation Method (OHPM) [27, 28, 29, 30] to achieve the solution of BVPs
Four nonlinear BVPs are solved by the new form of OHAM forming in different nonlinear phenomena
Summary
Many real world problems can be expressed in terms of BVPs. They have a significant contribution in almost every field. Asymptotic and perturbation schemes are used to gain the solutions of BVPs but unluckily these techniques are suitable for weak nonlinear problems and in general they dependent upon physical parameters. Various Investigators have created some other techniques based on Homotopy are HPM [14, 15, 10, 16], OHAM [17, 18, 19, 20, 21, 22, 23, 24, 25, 26], and Optimal Homotopy Perturbation Method (OHPM) [27, 28, 29, 30] to achieve the solution of BVPs. The new relevant work can be seen in [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. Inspired and aggravated by the continuing research in this area, we apply a new form of OHAM (OHAM-2) explained below for solving the nonlinear BVPs of order four, five, six, and seven as given in [54, 55, 26, 25, 56, 11]
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