Advancing nonlinear problem solutions: Newton-like iterative techniques for varied convergence orders using homotopy perturbation

Semi-exact solution of elastic non-uniform thickness and density rotating disks by homotopy perturbation and Adomian's decomposition methods. Part I: Elastic solution

Comparison between homotopy analysis method and homotopy renormalization method in fluid mechanics

In this study, we applied the homotopy perturbation (HP) method for solving linear and nonlinear fourth-order boundary value problems. The analytical results of the boundary value problems have been obtained in terms of a convergent series with easily computable components. Comparisons between the results of the HP method and the analytical solution showed that this method gives very precise results with a few terms. In the implied HP method, some unknown parameters in the initial guess are introduced, which are identified after applying boundary conditions. This improvement results in higher accuracy.

This article is aimed to introduce a new hybrid analytical-digital technique for solving a wide range of problems in fluid mechanics. This method is according to the Different Transform Method (DTM) and Newton’s Iterative Method (NIM). In the Boundary Value Problems (BVP), the system and the boundary conditions converted to an algebraic equation set, and the Taylor series of the solution are subsequently calculated. By finding Jacobian matrix, the unknown parameters of the solution may be calculated using the multi-variable iterative Newton's method. The techniques are employed to determine a proximate solution for the problem. To expound upon the application of the new hybrid method illustratively, two nonlinear problems in fluid mechanics are considered: condensation film on the inclined rotating disk and the rotating MHD flow on a porous shrinking sheet. Using comparing the present results obtained with the numerical solutions and results presented in the literature, an excellent accuracy is observed. Quick convergence of the solution is another important merit of the proposed method.

In this paper we present the development of numerical methods for two problems in chemical engineering: adsorption of carbon dioxide into phenyl glycidyl ether and diffusion and reaction processes in a porous catalyst. Both problems are modelled by systems of two coupled nonlinear ordinary differential equations. Dirichlet and Neumann type boundary conditions are considered. We develop the standard homotopy analysis method and the boundary-domain element method to solve for the unknown steady-state concentrations of reactants. The validity of the proposed methods is checked and demonstrated by numerical examples. Convergence rates are compared and the advantages and drawbacks of the proposed methods are discussed and compared to the Mathematica NDSOLVE solver. We show, that the proposed homotopy analysis method features exponential convergence rate and is highly accurate and efficient. As low as 6 terms are needed to reach error norms of 10 − 5 . Furthermore, the boundary-domain integral approach is also capable of solving the same problem very efficiently, using a very coarse computational mesh (21 nodes).

In this article, the homotopy analysis method (HAM) has been applied to solve coupled nonlinear evolution equations in physics. The validity of this method has been successfully demonstrated by applying it to two nonlinear evolution equations, namely coupled nonlinear diffusion reaction equations and the (2+1)-dimensional Nizhnik–Novikov Veselov system. The results obtained by this method show good agreement with the ones obtained by other methods.The proposed method is a powerful and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The HAM solutions contain an auxiliary parameter that provides a convenient way of controlling the convergence region of series solutions. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear evolution equations. The basic ideas of this approach can be widely employed to solve other strongly nonlinear problems.

In this paper, the Homotopy Analysis Method (HAM) proposed by Liao (1992a, 1992b, 1992c, 1992e, 1995a, 1997a) is greatly improved by introducing a nonzero variable ℏ. Based on the HAM, a new numerical approach for strongly non-linear problems is proposed and applied to solve, as an example, a non-linear heat transfer problem, i.e. microwave heating of an unit plate, so as to verify its validity and great potential. Our numerical experiments show that, by the proposed approach, iteration is not absolutely necessary for solving non-linear problems. This fact may deepen our understanding about numerical techniques for non-linear problems and widen our field of vision. Moreover, the basic ideas proposed in this paper may afford us a great possibility to greatly improve our current numerical techniques.

NONLINEAR PROBLEMS in physics are difficult to discuss for several reasons. First, what are nonlinear problems? Practically every problem in theoretical physics is governed by nonlinear mathematical equations, except perhaps quantum theory, and even in quantum theory it is a rather controversial question whether it will finally be a linear or nonlinear theory. Therefore by far the largest part of theoretical physics is devoted to nonlinear problems.

Since analytical perturbation methods depend on a small parameter and finding this small parameter is difficult, two powerful analytical methods are introduced to be applied to solve nonlinear heat transfer problems in this article. One is He's variational iteration method ( VIM ) and the other is the homotopy-perturbation method ( HPM ). The VIM is used to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory. The HPM converts a difficult problem into a simple problem which can be easily solved. In this article, the VIM is used to solve some nonlinear heat transfer equations with variable heat transfer coefficient. The results are then compared with those obtained by the HPM and the exact solution.

The Homotopy Analysis Method (HAM) has emerged as a powerful analytical technique for solving nonlinear ordinary and partial differential equations across diverse scientific and engineering fields. Unlike traditional perturbation methods, HAM offers flexibility in handling strongly nonlinear problems without requiring small parameters. This paper provides a comprehensive review of the theoretical foundation, computational implementations, and applications of HAM, its development, and its wide range of applications. The discussion includes the mathematical framework, convergence control, comparison with other methods, and a detailed survey of its applications in fluid mechanics, heat transfer, quantum mechanics, and beyond. Furthermore, we highlight recent advancements and potential future directions for HAM.

A new free derivative iterative method is presented in this article. The method is developed by combining Newton’s method and Euler’s method. Deriva- tives in this method are approximated by forward difference, hyperbola and divided difference. The order of convergence is proven analytically to be of sixth order. Numerical results exhibit that the new method is comparable to other methods. Basins of attraction are also provided to support the proposed method.

In this article, we first construct a family of optimal 2-step iterative methods for finding a single root of the nonlinear equation using the procedure of weight function. We then extend these methods for determining all roots simultaneously. Convergence analysis is presented for both cases to show that the order of convergence is 4 in case of the single-root finding method and is 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The dynamical behavior is presented to analyze the stability of fixed and critical points of the rational operator of one-point iterative methods. The computational cost, basins of attraction, efficiency, log of the residual, and numerical test examples show that the newly constructed methods are more efficient as compared with the existing methods in the literature.

In this research paper, a semi-analytical method,i.e.,homotopy analysis method, is implemented for finding the solution of non-linear partial differential equations. Homotopy analysis method is very effective, and easy to evaluate as compared to other numerical methods. The results through HAM on illustrative examples are compared with two well-known methods, namely, Variational Iteration Method(VIM) and Adomian Decomposition Method(ADM). The comparisons of obtained solutions results in the high accuracy through HAM when compared to the other competing methods. Hence, the solution obtained using HAM has much faster convergence to the exact solution.KeywordsNon-linear partial differential equationNumerical resultsHomotopy analysis method

Homotopy method of fundamental solutions for solving nonlinear heat conduction problems

In this work, the electrohydrodynamic flow equation of a fluid in an ion drag configuration in a circular cylindrical conduit is considered. The differential equation of the problem is a highly nonlinear and singular boundary value problem (BVP). An analytical solution of the problem is presented using the differential transform method (DTM). In this method, the differential equation and related boundary conditions transformed into a recurrence set of equations and at the end, the coefficients of Taylor series are calculated based on the solution of this set of equations. DTM results are compared with the numerical solution of the problem and an excellent accuracy is observed. It showed that the proposed method overcame the nonlinearity and singularity of the problem without any need to restrict assumptions or linearization. Most of the scientific problems and engineering phenomena are usually defined based on the nonlinear differential equations of both initial value problems (IVP) and boundary value problems (BVP). It is important to remember that except a limited number of these problems, finding exact analytical solution is difficult. Therefore, researchers use semi-analytical techniques and numerical methods. The semi-analytical methods such as the perturbation method (PM), Adomian decomposition method (ADM), variational iteration method (VIM), homotopy perturbation method (HPM), homotopy analysis method (HAM), and differential transform method (DTM) are useful mathematical ones, which present the analytical solutions for nonlinear differential equations. In this paper, solution of electrohydrodynamic flow equation of a fluid in an ion drag configuration in a circular cylindrical conduit is presented using DTM. Already, the electrohydrodynamic flow of a fluid and its governing equations is considered (1). The electrohydrodynamic flow is important in analysis of the flow meters, accelerators, pumps and magnetohydrodynamic generators. The differential equation of the problem is the nonlinear singular boundary value problem. 2 2