A Study on Homotopy Analysis Method and Clique Polynomial Method

Comparative study of Adomian decomposition method and Clique polynomial method

Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods

A high-precision and space-time fully decoupled numerical method is developed for a class of nonlinear initial boundary value problems. It is established based on a proposed Coiflet-based approximation scheme with an adjustable high order for the functions over a bounded interval, which allows the expansion coefficients to be explicitly expressed by the function values at a series of single points. When the solution method is used, the nonlinear initial boundary value problems are first spatially discretized into a series of nonlinear initial value problems by combining the proposed wavelet approximation and the conventional Galerkin method, and a novel high-order step-by-step time integrating approach is then developed for the resulting nonlinear initial value problems with the same function approximation scheme based on the wavelet theory. The solution method is shown to have the N th-order accuracy, as long as the Coiflet with[0, 3 N -1] compact support is adopted, where N can be any positive even number. Typical examples in mechanics are considered to justify the accuracy and efficiency of the method.

This paper generates a novel approach called the clique polynomial method (CPM) using the clique polynomials raised in graph theory and used for solving the fractional order PDE. The fractional derivative is defined in terms of the Caputo fractional sense and the fractional partial differential equations (FPDE) are converted into nonlinear algebraic equations and collocated with suitable grid points in the current approach. The convergence analysis for the proposed scheme is constructed and the technique proved to be uniformly convegant. We applied the method for solving four problems to justify the proposed technique. Tables and graphs reveal that this new approach yield better results. Some theorems are discussed with proof.

In this paper, a class of linear and nonlinear nth-order initial value problems (IVPs) is considered. The solutions of these IVPs are obtained by the homotopy-perturbation method (HPM). The HPM can be considered as one of the new methods belonging to the general classification of perturbation methods. Generally, the HPM deals with exact solvers for linear differential equations and approximative solvers for nonlinear equations. Several test cases are chosen to demonstrate the efficiency of HPM.

An iterative technique based on HPM for a class of one dimensional Bratu’s type problem

Derivation of the Adomian decomposition method using the homotopy analysis method

This article proposes a novel version of the popular method from the ever-so-famous homotopy analysis methods, namely Quotient Homotopy Analysis Method or simply QHAM, in order to provide numerical solutions for nonlinear initial value problems. The basic approach of this method relies firstly on constructing a suitable homotopy equation so that its solution will be then established in the form of a quotient of two formulated power series which will, in its turn, be the basis stone of the general numerical solution for the nonlinear problems. Two numerical comparisons via two examples are performed between the exact solution and the approximate numerical solution to reveal the importance and potential role of the established method in this work.

This paper investigates an effective computational method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems appeared in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica®12. The improved effective computational methods (I-ECMs) have been implemented to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the proposed methods has been presented. Furthermore, the maximum error remainder () has been computed to prove the proposed methods' accuracy. The results convincingly prove that ECM and I-ECMs are effective and accurate in obtaining novel approximate solutions to the problems.

In this article a mathematical analysis of steady free convection and mass transfer flow of a conducting micropolar fluid between two vertical walls in the presence of transverse magnetic field is discussed. The approximate analytical expressions of the dimensionless velocity profile, dimensionless micro rotation profile, dimensionless temperature and the dimensionless concentration are derived analytically and graphically by using the Homotopy analysis method. The effect of vortex viscosity, material parameter, chemical reaction, heat source/sink parameter are discussed and presented through graphs. The Homotopy analysis method contains the convergence control parameter h, so it can be easily extended to solve the other non-linear initial and boundary value problems in other mathematical sciences.

In this paper, several new iterative methods for solving nonlinear algebraic equations are presented. The iterative formulas are based on the He's homotopy perturbation method (HPM). It is shown that the new methods lead to eight algorithms which are of fifth, seventh, tenth and fourteenth order convergence. These methods result in real or complex simple roots of certain nonlinear equations. The merit of the new algorithms is that, in case the nonlinear equation have complex roots, it can give complex solutions even when the initial approximation is chosen real. Several examples are presented and compared to other methods, showing the accuracy and fast convergence of the presented method. Key words: Iterative methods, homotopy perturbation method, nonlinear algebraic equations, efficency index, convergence order.

In this paper Homotopy perturbation method (HPM) is implemented to give an approximate analytical solution to the system of non-linear differential equation corresponding to S-SEIR model. The S-SEIR model is constructed for information dissemination characteristics on social network. Our analytical results are compared with the numerical simulation and a satisfactory agreement is noted. The graphical results are shown the effect of information value and user behavior on information dissemination. Using the Homotopy perturbation method we can easily solve other strongly non-linear initial and boundary value problems in engineering and sciences.

In this study, the concept of Homotopy analysis method (HAM) is briefly introduced. Furthermore some non-linear problems are handled and the solutions of these problems are given using by HAM, DTM, ADM methods and the convergence of the solution is shown to the exact solution. Additionally, the three methods are compared and it is observed that the HAM is more-less efficient and effective than the ADM and DTM in according to the exact solution. In the end, some of the numerical solutions of two examples are presented and the results are shown in graphs and figures.

In this paper, an efficient modification of homotopy perturbation method, namely optimal homotopy perturbation method, is introduced for solving linear and nonlinear partial differential equations with large solution domain based on a new homotopy perturbation method and Pade approximation method. We compare the performance of the method with those of new homotopy perturbation and optimal variational iteration methods via three partial differential equations with large solution domain. Numerical results explicitly reveal that the suggested technique is highly capable to control the convergence region of approximate solution. Key words: New homotopy perturbation method, Pade´ approximation, optimal variational iteration method.

In this paper, a new spectral algorithm based on employing ultraspherical wavelets along with the spectral collocation method is developed. The proposed algorithm is utilized to solve linear and nonlinear even-order initial and boundary value problems. This algorithm is supported by studying the convergence analysis of the used ultraspherical wavelets expansion. The principle idea for obtaining the proposed spectral numerical solutions for the above-mentioned problems is actually based on using wavelets collocation method to reduce the linear or nonlinear differential equations with their initial or boundary conditions into systems of linear or nonlinear algebraic equations in the unknown expansion coefficients. Some specific important problems such as Lane–Emden and Burger’s type equations can be solved efficiently with the suggested algorithm. Some numerical examples are given for the sake of testing the efficiency and the applicability of the proposed algorithm.