Homotopy Analysis Transform Method Research Articles

Comparative study of a cubic autocatalytic reaction via different analysis methods

In this paper we discuss an approximate solutions of the space-time fractional cubic autocatalytic chemical system (STFCACS) equations. The main objective is to find and compare approximate solutions of these equations found using Optimal q-Homotopy Analysis Method (Oq-HAM), Homotopy Analysis Transform Method (HATM), Varitional Iteration Method (VIM) and Adomian Decomposition Method (ADM).

Coupled reaction-diffusion waves in a chemical system via fractional derivatives in Liouville-Caputo sense

In this paper, we have generalized the fractional cubic isothermal auto-catalytic chemical system (FCIACS) with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional time derivatives, respectively. We apply the Homotopy Analysis Transform Method (HATM) to compute the approximate solutions of FCIACS using these fractional derivatives. We study the convergence analysis of HATM by computing the residual error function. Also, we find the optimal values of h so we assure the convergence of the approximate solutions. Finally we show the behavior of the approximate solutions graphically. The results obtained are very effectiveness and accuracy.

New fractional derivatives with non-singular kernel applied to the Burgers equation.

In this paper, we extend the model of the Burgers (B) to the new model of time fractional Burgers (TFB) based on Liouville-Caputo (LC), Caputo-Fabrizio (CF), and Mittag-Leffler (ML) fractional time derivatives, respectively. We utilize the Homotopy Analysis Transform Method (HATM) to compute the approximate solutions of TFB using LC, CF, and ML in the Liouville-Caputo sense. We study the convergence analysis of HATM by computing the interval of the convergence, the residual error function (REF), and the average residual error (ARE), respectively. The results are very effective and accurate.

A novel technique to solve the modified epidemiological model of computer viruses

The aim of this paper is to present a simple and accurate method to estimate the approximate solution of non-linear epidemiological model of computer viruses. For this reason, the variational iteration method (VIM) is applied. Also, in order to show the efficiency of presented method, we compare the numerical results with the differential transform method (DTM) and the homotopy analysis transform method (HATM). Several graphs of residual error functions for various iterations are demonstrated. By applying these graphs we show the results of VIM are accurate in comparison with the other methods.

Population Balances Involving Aggregation and Breakage Through Homotopy Approaches

Abstract Homotopy techniques in nonlinear problems are getting increasingly popular in engineering practice. The main reason is because the homotopy method deforms continuously a difficult problem under study into a simple problem, which then can be easy to solve. This study explores several homotopy approaches to obtain semi- or approximate analytical solutions for various cases involving mechanistic phenomena such as aggregation and breakage. The well-established approximate analytical methods namely, the Homotopy Perturbation Method (HPM), the Homotopy Analysis Method (HAM), and the more recent forms of homotopy approaches such as the Optimal Homotopy Asymptotic Method (OHAM) and the Homotopy Analysis Transform Method (HATM) have been used to solve using a general mathematical framework based on population balances. In this study, several test cases have been discussed such as conditions in which the aggregation kernel is not only constant, but also sum or product dependent. Furthermore cases involving pure breakage, pure aggregation and a combined aggregation-breakage have been studied to understand the sensitivity of these homotopy-based methods in solving PBM. In all these cases, the solutions have been analytically studied and compared with literature. Using symbolic computation and carefully chosen perturbation parameters, the approximate analytical solutions are compared with each other and with the available analytical solution. A convergence analysis of the solution methods is made in comparison to the available solution. The case studies indicate that OHAM performs slightly better than both HATM and HPM in solving nonlinear equations such as the PBEs.

A hybrid computational approach for Jeffery–Hamel flow in non-parallel walls

The key goal of this article is to present an efficient hybrid computational technique, namely homotopy analysis transform method (HATM), to investigate Jeffery–Hamel flow. The HATM is an innovative and efficient amalgamation of homotopy analysis technique, standard Laplace transform scheme and homotopy polynomials. The effect of Reynolds number on velocity profile is studied graphically. The obtained results are compared with existing results and it is noticed that the outcomes are in an excellent agreement. The outcomes of the suggested method reveal that the technique is easy to handle and computationally very fantastic.

A computational approach for fractional convection-diffusion equation via integral transforms

Abstract In this paper, two efficient analytic techniques namely the homotopy analysis transform method (HATM) and homotopy perturbation Sumudu transform method (HPSTM) are implemented to give a series solution of fractional convection-diffusion equation which describes the flow of heat. These proposed techniques introduce significance in the field over the existing techniques that make them computationally very attractive for applications. Numerical solutions clearly demonstrate the reliability and efficiency of HATM and HPSTM to solve strongly nonlinear fractional problems.

SOLVING FRACTIONAL INTEGRO DIFFERENTIAL EQUATIONS BY HOMOTOPY ANALYSIS TRANSFORM METHOD

In this paper, we introduce an analytical method, which so called the homotopy analysis transform method (HATM) which is a combination of HAM and Laplace decomposition method (LDM). This scheme is simple to apply linear and nonlinear fractional integro-differential equation and having less computational work in comparison of other exiting methods. The fractional derivatives are described in the Caputo sense. The most useful advantage of this method is to solve the fractional integro-differential equation without using Adomian polynomials and He’s polynomials for the computation of nonlinear terms. AMS Subject Classification: 00A71, 45A05, 34A12, 45E10, 7Q10

Analytic approximate solution for some integral equations by optimal homotopy analysis transform method

The main aim of this paper is to propose a new and simple algorithm namely homotopy analysis transform method (HATM), to obtain approximate analytical solutions of integral equations. Integral equation occurs in the mathematical modeling of several models in physics, astrophysics, solid mechanics and applied sciences. The numerical solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. Finally, several numerical examples are given to illustrate the accuracy and stability of this method. Comparison of the approximate solution with the exact solutions also we show that the proposed method is very efficient and computationally attractive. A new efficient approach is proposed to obtain the optimal value of convergence controller parameter $\hslash$ to guarantee the convergence of the obtained series solution.

Numerical computation of fractional Lotka-Volterra equation arising in biological systems

AbstractIn this paper, we present the homotopy analysis transform method (HATM) to solve fractional Lotka- Volterra equation, which describes the long term servival of species. The HATM solutions, denotes less error compare with their respective exact solution for alpha = 1. In addition to non-proposed techniques, HATMis valid for both small and large parameters, it also provides us with a simpleway to adjust and control the parameter hbar and auxiliary function H(t), which play effective role for convergence solutions of fractional differential-difference equations (FDDEs).

Numerical computation of Klein–Gordon equations arising in quantum field theory by using homotopy analysis transform method

In this paper, we present a reliable algorithm based on the homotopy analysis transform method (HATM) to solve the linear and nonlinear Klein–Gordon equations. The Klein–Gordon equation is the equation of motion of a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. It describes the quantum amplitude for finding a point particle in various places, the relativistic wave function, but the particle propagates both forwards and backwards in time. The HATM is a combined form of the Laplace transform method and homotopy analysis method. The method provides the solution in the form of a rapidly convergent series. Some numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. The results show that the HATM is very efficient, simple and can be applied to other nonlinear problems.

A new analytical modelling for fractional telegraph equation via Laplace transform

The main aim of the present work is to propose a new and simple algorithm for space-fractional telegraph equation, namely new fractional homotopy analysis transform method (HATM). The fractional homotopy analysis transform method is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method (HPTM). The beauty of the paper is error analysis which shows that our solution obtained by proposed method converges very rapidly to the known exact solution. The numerical solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. Finally, several numerical examples are given to illustrate the accuracy and stability of this method.

A new comparative study between homotopy analysis transform method and homotopy perturbation transform method on a semi infinite domain

The aim of this article is to introduce a new approximate method, namely homotopy analysis transform method (HATM) which is a combination of homotopy analysis method (HAM) and Laplace decomposition method (LDM) to provide an analytical approximate solution to Blasius flow equations. The proposed scheme finds the solutions of nonlinear problems without any discretization, restrictive assumptions and avoids the rounding-off errors. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method (HPTM). A good agreement between the obtained solution and some well-known results has been demonstrated.