Search

Empirical investigations of solute fate and carrying in streams and rivers often contain inventive liberate of solutes at an upstream perimeter for a finite interval of time. An analysis of various worth references on surface-water-grade mathematical formulation reveals that the logical solution to the continual-parameter advection- dispersion problem for this type of boundary state has been generally missed. In this work, we study the q-fractional homotopy analysis transform method (q-FHATM) to find the analytical and approximate solutions of space-time arbitrary order advection-dispersion equations with nonlocal effects. The diagrammatical representation is done by using Maple package, which enhance the discretion and stability of family of q-FHATM series solutions of fractional advection-dispersion equations. The efficiency of the applied technique is demonstrated by using three numerical examples of space- and time-fractional advection-dispersion equations.

In this work, we aim to apply a reliable analytic algorithm based on homotopy perturbation Sumudu transform method (HPSTM) to examine the nonlinear time-fractional coupled Burger’s equations. The approximate analytical solution and some numerical examples show the accuracy and efficiency of the proposed method, which is simple and accurate in comparison to the Adomain decomposition method (ADM), homotopy perturbation method (HPM) and generalized differential transform method (GDTM).

In this paper, we study a dynamical Brusselator reaction-diffusion system arising in triple collision and enzymatic reactions with time fractional Caputo derivative. The present article involves a more generalized effective approach, proposed for the Brusselator system say q-homotopy analysis transform method (q-HATM), providing the family of series solutions with nonlocal generalized effects. The convergence of the q-HATM series solution is adjusted and controlled by auxiliary parameter ℏ and asymptotic parameter n. The numerical results are demonstrated graphically. The outcomes of the study show that the q-HATM is computationally very effective and accurate to analyze nonlinear fractional differential equations.

The aim of the present work is to propose a user friendly approach based on homotopy analysis method combined with Sumudu transform method to drive analytical and numerical solutions of the fractional Newell-Whitehead-Segel amplitude equation which describes the appearance of the stripe patterns in 2-dimensional systems. The coupling of homotopy analysis method with Sumudu transform algorithm makes the calculation very easy. The proposed technique gives an analytic solution in the form of series which converge very fastly. The analytical and numerical results reveal that the coupling of homotopy analysis technique with Sumudu transform algorithm is very easy to apply and highly accuratewhen apply to non-linear differential equation of fractional order.

The aim of the present work is to propose a simple and reliable algorithm namely homotopy perturbation transform method (HPTM) for KdV equations in warm plasma. The homotopy perturbation transform method is a combined form of the Laplace transform method with the homotopy perturbation method. In this method, the solution is calculated in the form of a convergent series with an easily computable compact. To illustrate the simplicity and reliability of the method, several examples are provided. In this paper, the homotopy perturbation transform method has been applied to obtain the solution of the KdV, mKdV, K(2, 2) and K(3,3) equations. We compared our solutions with the exact solutions. Results illustrate the applicability, efficiency and accuracy of HPTM to solve nonlinear equations despite needlessness to any linearization or perturbation. It is predicted that the proposed algorithm can be widely applied to other nonlinear problems in science and engineering.

The aim of the present paper is to present analytical solution of fractional differential equations. The fractional derivative is considered in Caputo sense. The results are derived by the application of Sumudu transform in term of Mittag-Leffler function, which are suitable for numerical computation. The results obtain by the Sumudu transform method indicate that the approach is easy to implement and computationally very attractive.

In this paper, we present a numerical algorithm based on new homotopy perturbation transform method (HPTM) to solve a time-fractional nonlinear shock wave equation which describes the flow of gases. The fractional derivative is considered in the Caputo sense. The HPTM is combined form of Laplace transform, homotopy perturbation method and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The technique finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The obtained results are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, efficient, easy to implement and computationally very attractive.