A comparison between the modified homotopy perturbation method and adomian decomposition method for solving nonlinear heat transfer equations

In this study, we applied a new algorithm based on Homotopy Perturbation Method (HPM) to evaluate the temperature distribution of a straight rectangular fin with temperature dependent surface heat flux for all possible types of heat transfer. The local heat transfer coefficient is considered to vary with a power-law function of temperature. The time interval is divided into several subintervals and the HPM solutions are applied successively over these reduced time intervals. Comparisons between the 13-term Adomian decomposition solution and 6-term modified HPM solution are made. Comparison of the results obtained by modified HPM with that obtained by the Adomian Decomposition Method (ADM) reveals that the obtained modified HPM solution is quite accurate when only the six terms are used in the series expansion

Accurate approximations of the nonlinear vibration of couple-mass-spring systems with linear and nonlinear stiffnesses

An analytical technique has been developed based on the harmonic balance method to obtain approximate angular frequencies. This technique also offers the periodic solutions to the nonlinear free vibration of a conservative, couple-mass-spring system having linear and nonlinear stiffnesses with cubic nonlinearity. Two real-world cases of these systems are analysed and introduced. After applying the harmonic balance method, a set of complicated higher-order nonlinear algebraic equations are obtained. Analytical investigation of the complicated higher-order nonlinear algebraic equations is cumbersome, especially in the case when the vibration amplitude of the oscillation is large. The proposed technique overcomes this limitation to utilize the iterative method based on the homotopy perturbation method. This produces desired results for small as well as large values of vibration amplitude of the oscillation. In addition, a new suitable truncation principle has been used in which the solution achieves better results than existing solutions. Comparing with published results and the exact ones, the approximated angular frequencies and corresponding periodic solutions show excellent agreement. This proposed technique provides results of high accuracy and a simple solution procedure. It could be widely applicable to other nonlinear oscillatory problems arising in science and engineering

Solutions of time-dependent Emden–Fowler type equations by homotopy-perturbation method

In this Letter, we apply the homotopy-perturbation method (HPM) to obtain approximate analytical solutions of the time-dependent Emden– Fowler type equations. We also present a reliable new algorithm based on HPM to overcome the difficulty of the singular point at x = 0. The analysis is accompanied by some linear and nonlinear time-dependent singular initial value problems. The results prove that HPM is very effective and simple

Taylor-newton homotopy method for computing the depth of flow rate for a channel

Homotopy approximation methods (HAM) can be considered as one of the new methods belong to the general classification of the computational methods which can be used to find the numerical solution of many types of the problems in science and engineering. The general problem relates to the flow and the depth of water in open channels such as rivers and canals is a nonlinear algebraic equation which is known as continuity equation. The solution of this equation is the depth of the water. This paper represents attempt to solve the equation of depth and flow using Newton homotopy based on Taylor series. Numerical example is given to show the effectiveness of the purposed method using MATLAB language

On the numerical solution of linear stiff IVPs by modified homotopy perturbation method

In this paper, we introduce a method to solve linear sti® IVPs. The sug-gested method, which we call modi¯ed homotopy perturbation method, can be considered as an extension of the homotopy perturbation method (HPM) which is very efficient in solving a varety of di®erential and algebraic equations. In this work, a class of linear stiff initial value problems (IVPs) are solved by the classical homotopy per-turbation method (HPM), modified homotopy perturbation method and an explicit Runge-Kutta-type method (RK). Numerical comparisons demonstrate the limitations of HPM and promising capability of the MHPM for solving stiff IVPs. The results prove that the modified HPM is a powerful tool for the solution of linear stiff IVPs

Solving linear and non-linear stiff system of ordinary differential equations by multistage adomian decomposition method

In this paper, linear and non-linear stiff systems of ordinary differential equations are solved by the classical Adomian decomposition method (ADM) and the multistage Adomian decomposition method (MADM). The MADM is a technique adapted from the standard Adomian decomposition method (ADM) where standard ADM is converted into a hybrid numeric-analytic method called the multistage ADM (MADM). The MADM is tested for several examples. Comparisons with an explicit Runge-Kutta-type method (RK) and the classical ADM demonstrate the limitations of ADM and promising capability of the MADM for solving stiff initial value problems (IVPs)

Comparison of homotopy analysis method and homotopy-perturbation method for purely nonlinear fin-type problems

In this paper, the homotopy analysis method (HAM) is compared with the homotopy-perturbation method (HPM) and the Adomian decomposition method (ADM) to determine the temperature distribution of a straight rectangular fin with power-law temperature dependent surface heat flux. Comparisons of the results obtained by the HAM with that obtained by the ADM and HPM suggest that both the HPM and ADM are special case of the HAM

Solution of cubic-quintic Duffing oscillators using harmonic balance method

In this study, Harmonic balance Method (HBM) is applied to approximate analytic solutions of strongly nonlinear Duffing oscillators with cubic-quintic nonlinear restoring force. In this paper, the HBM is mainly illustrated by the cubic-quintic Duffing oscillator. Numerical comparisons between the HBM and the exact solutions reveal that the HBM is a promising tool for strongly nonlinear oscillator’s problems