Search results

The purpose of this paper is to obtain analytic solutions of the (1+1) and (2+1)‐dimensional dispersive long wave equations by the homotopy analysis and the homotopy Padé methods.

The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series.

The approximation solutions by [m,m] homotopy Padé technique is often independent of auxiliary parameter ℏ and this technique accelerates the convergence of the related series.

In this paper, analytic solutions of the (1+1) and (2+1)‐dimensional dispersive long wave equations are obtained by the homotopy analysis and the homotopy Padé methods. The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m,m] homotopy Padé technique are often independent of auxiliary parameter ℏ and this technique accelerates the convergence of the related series.

The aim of the study is to show that merging two areas of mathematics – topology and discrete optimization – could result in a viable option to solve classical or specialized integer problems.

In the paper, discrete topology concepts are applied to propose a metaheuristic algorithm that is capable to solve binary programming problems. Particularly, some of the homotopy for paths principles are used to explore the solution space associated with four well-known NP-hard problems herein considered as follows: knapsack, set covering, bi-level single plant location with order and one-max.

Computational experimentation confirms that the proposed algorithm performs in an effective manner, and it is able to efficiently solve the sets of instances used for the benchmark. Moreover, the performance of the proposed algorithm is compared with a standard genetic algorithm (GA), a scatter search (SS) method and a memetic algorithm (MA). Acceptable results are obtained for all four implemented metaheuristics, but the path homotopy algorithm stands out.

A novel metaheuristic is proposed for the first time. It uses topology concepts to design an algorithmic framework to solve binary programming problems in an effective and efficient manner.

The purpose of this paper is to obtain analytic solutions of telegraph equation by the homotopy Padé method.

The authors used Maple Package to calculate the solutions obtained from the homotopy Padé method.

The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m, m] homotopy Padé technique are often independent of auxiliary parameter h and this technique accelerates the convergence of the related series. Finally, numerical results for some test problems with known solutions are presented and the numerical results are given to show the efficiency of the proposed techniques.

The paper is shown that homotopy Padé technique is a promising tool with accelerated convergence for complicated nonlinear differential equations.

The purpose of this paper is to present a weighted algorithm based on the homotopy perturbation method for solving the heat transfer equation in the cast‐mould heterogeneous domain.

A weighted algorithm based on the homotopy perturbation method is used to minimize the volume of computations. The authors show that this technique yields the analytical solution of the desired problem in the form of a rapidly convergent series with easily computable components.

The authors illustrate that the proposed method produces satisfactory results with respect to Adomian decomposition method and standard homotopy perturbation method. The reliability of the method and the reduction in the size of computational domain give this method a wider applicability.

This research presents, for the first time, a new modification of the proposed technique, for aforementioned problems and some interesting results are obtained.

The purpose of this paper is to determine the fin efficiency of convective straight fins with temperature dependent thermal conductivity by using Homotopy Perturbation Method.

Most engineering problems, especially heat transfer equations are in nonlinear form. Homotopy Perturbation Method (HPM) has been applied to solve a wide series of nonlinear differential equations. In this paper, HPM is used for obtaining the fin efficiency of convective straight fins with temperature‐dependent thermal conductivity. Comparison of the results with those of Homotopy Perturbation Method, exact solution, numerical results and Adomian's decomposition method (ADM) were been done by Cihat Arslanturk.

Results show that both Homotopy Perturbation Method and ADM applied to the nonlinear equations were capable of solving them with successive rapidly convergent approximations without any restrictive assumptions or transformations causing changes in the physical properties of the problem. Moreover, adding up the number of iterations leads to explicit solution for the problem. The results are just obtained with two iterations. This shows the accuracy and great potential of this method. Finally, it can be seen that, with increase of thermo‐geometric fin parameter (v), the fin efficiency increases too.

The results demonstrate good validity and great potential of the HPM for Heat Transfer equations in engineering problems.

The purpose of this paper is to present a general framework of Homotopy perturbation method (HPM) for analytic inverse heat source problems.

The proposed numerical technique is based on HPM to determine a heat source in the parabolic heat equation using the usual conditions. Then this shows the pertinent features of the technique in inverse problems.

Using this HPM, a rapid convergent sequence which tends to the exact solution of the problem can be obtained. And the HPM does not require the discretization of the inverse problems. So HPM is a powerful and efficient technique in finding exact and approximate solutions without dispersing the inverse problems.

The essential idea of this method is to introduce a homotopy parameter p which takes values from 0 to 1. When p=0, the system of equations usually reduces to a sufficiently simplified form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.

The purpose of this paper is to use the generalized differential transform method (GDTM) and homotopy perturbation method (HPM) for solving time‐fractional Burgers and coupled Burgers equations. The fractional derivatives are described in the Caputo sense.

In these schemes, the solutions takes the form of a convergent series. In GDTM, the differential equation and related initial conditions are transformed into a recurrence relation that finally leads to the solution of a system of algebraic equations as coefficients of a power series solution. HPM requires a homotopy with an embedding parameter which is considered as a small parameter.

The paper extends the application and numerical comparison of the GDTM and HPM to obtain analytic and approximate solutions to the time‐fractional Burgers and coupled Burgers equations.

Burgers and coupled Burgers equations with time‐fractional derivative used.

The implications include traffic flow, acoustic transmission, shocks, boundary layer, the steepening of the waves and fluids, thermal radiation, chemical reaction, gas dynamics and many other phenomena.

The numerical results demonstrate the significant features, efficiency and reliability of the two approaches. The results show that HPM is more promising, convenient, and computationally attractive than GDTM.

This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense.

Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation and fractional Klein‐Gordon equation are investigated to show the pertinent features of the technique.

HPM is a powerful and efficient technique in finding exact and approximate solutions for fractional partial differential equations in fluid mechanics. The implementation of the noise terms, if they exist, is a powerful tool to accelerate the convergence of the solution. The results so obtained reinforce the conclusions made by many researchers that the efficiency of the HPM and related phenomena gives it much wider applicability.

The essential idea of this method is to introduce a homotopy parameter, say p, which takes values from 0 to 1. When p = 0, the system of equations usually reduces to a sufficiently simplied form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.

This purpose of this paper is to find the approximate analytical solutions of a multidimensional partial differential equation such as Helmholtz equation with space fractional derivatives α,β,γ (1<α,β,γ≤2). The fractional derivatives are described in the Caputo sense.

By using initial values, the explicit solutions of the equation are solved with powerful mathematical tools such as He's homotopy perturbation method (HPM).

This result reveals that the HPM demonstrates the effectiveness, validity, potentiality and reliability of the method in reality and gives the exact solution.

The most important part of this method is to introduce a homotopy parameter (p), which takes values from [0,1]. When p=0, the equation usually reduces to a sufficiently initial form, which normally admits a rather simple solution. When p→1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation. Here, we also discuss the approximate analytical solution of multidimensional fractional Helmholtz equation.

The purpose of this paper is to provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function of time. Combination of the Laplace transform and homotopy perturbation methods (LTHPM) are considered as an algorithm which converges rapidly to the exact solution of the nonlinear Boussinesq equation.

The authors present the solution of nonlinear Boussinesq equation by combination of Laplace transform and new homotopy perturbation methods. An important property of the proposed method, which is clearly demonstrated in example, is that spectral accuracy is accessible in solving specific nonlinear nonlinear Boussinesq equation which has analytic solution functions.

The authors proposed a combination of Laplace transform method and homotopy perturbation method to solve the one-dimensional Boussinesq equation. The results are found to be in excellent agreement. The results show that the LTNHPM is an effective mathematical tool which can play a very important role in nonlinear sciences.

The authors provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function of time. In this work combination of Laplace transform and new homotopy perturbation methods (LTNHPM) are considered as an algorithm which converges rapidly to the exact solution of the nonlinear Boussinesq equation.