Here, an analytic technique, namely the homotopy analysis method (HAM), is applied to solve a generalized Hirota–Satsuma coupled KdV equation. HAM is a strong and easy-to-use analytic tool for ...nonlinear problems and dose not need small parameters in the equations. Comparison of the results with those of Adomian's decomposition method (ADM) and homotopy perturbation method (HPM), has led us to significant consequences. The homotopy analysis method contains the auxiliary parameter
ℏ, which provides us with a simple way to adjust and control the convergence region of solution series.
In this paper, the homotopy analysis method (HAM), one of the most effective method, is applied to obtain the approximate solution of the nonlinear model of diffusion and reaction in catalyst pellets ...for the case of
nth-order reactions. The approximate analytical solution obtained by HAM logically contains the solution obtained with Adomian decomposition method. The homotopy analysis method contains the auxiliary parameter
ℏ
, which provides us with a simple way to adjust and control the convergence region of solution series. By suitable choice of the auxiliary parameter
ℏ
, we can obtain reasonable solution for large Thiele modulus.
In this paper, the quadratic Riccati differential equation is solved by He's variational iteration method with considering Adomian's polynomials. Comparisons were made between Adomian's decomposition ...method (ADM), homotopy perturbation method (HPM) and the exact solution. In this application, we do not have secular terms, and if
λ
, Lagrange multiplier, is equal
-
1
then the Adomian's decomposition method is obtained. The results reveal that the proposed method is very effective and simple and can be applied for other nonlinear problems.
In this article, a general formulation for the fractional-order Legendre functions (FLFs) is constructed to obtain the solution of the fractional-order differential equations. Fractional calculus has ...been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, an efficient and reliable technique for the solution of them is too important. For the concept of fractional derivative we will adopt Caputo’s definition by using Riemann–Liouville fractional integral operator. Our main aim is to generalize the new orthogonal functions based on Legendre polynomials to the fractional calculus. Also a general formulation for FLFs fractional derivatives and product operational matrices is driven. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.
An analytic technique, the homotopy analysis method (HAM), is applied to obtain the soliton solution of the Fitzhugh–Nagumo equation. The homotopy analysis method (HAM) is one of the most effective ...method to obtain the exact solution and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter
ℏ, which provides us with a simple way to adjust and control the convergence region of series solution.
Here, the homotopy analysis method (HAM), one of the newest analytical methods which is powerful and easy-to-use, is applied to solve heat transfer problems with high nonlinearity order. Also, the ...results are compared with the perturbation and numerical Runge–Kutta methods and homotopy perturbation method (HPM). Here, homotopy analysis method is used to solve an unsteady nonlinear convective–radiative equation containing two small parameters of
ϵ
1 and
ϵ
2. The homotopy analysis method contains the auxiliary parameter
h¯, which provides us with a simple way to adjust and control the convergence region of solution series.
In this paper, we present some efficient numerical algorithms for solving nonlinear equations based on Newton–Raphson method. The modified Adomian decomposition method is applied to construct the ...numerical algorithms. Some numerical illustrations are given to show the efficiency of algorithms.
In this paper, a homotopy perturbation method is proposed to solve non-singular integral equations. Comparisons are made between Adomian’s decomposition method and the proposed method. It is shown, ...Adomian’s decomposition method is a homotopy, only. Finally, by using homotopy perturbation method, a new iterative scheme, like Adomian’s decomposition method, is proposed for solving the non-singular integral equations of the first kind. The results reveal that the proposed method is very effective and simple.
In this paper, a homotopy perturbation method is proposed to solve quadratic Riccati differential equation. Comparisons are made between Adomian’s decomposition method (ADM) and the exact solution ...and the proposed method. The results reveal that the proposed method is very effective and simple.
The homotopy analysis method (HAM) is used to find a family of travelling-wave solutions of the Kawahara equation. This approximate solution, which is obtained as a series of exponentials, has a ...reasonable residual error. The homotopy analysis method contains the auxiliary parameter
ħ
, which provides us with a simple way to adjust and control the convergence region of series solution. This method is reliable and manageable.
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