In this paper, a new approximate analytical method is proposed for solving the fractional biological population model, the fractional derivative is described in the Caputo sense. This method is based upon the Aboodh transform method and the new iterative method, the Aboodh transform is a modification of the Laplace transform. Illustrative cases are considered and the comparison between exact solutions and numerical solutions are considered for different values of alpha. Furthermore, the surface plots are provided in order to understand the effect of the fractional order. The advantage of this method is that it is efficient, precise, and easy to implement with less computational effort.

Motivated by the simplicity, natural and efficient nature of the new fractional derivative introduced by R Khalil et al in J. Comput. Appl. Math. 264, 65 (2014), analytical solution of space-time fractional Fornberg-Whitham equation is obtained in series form using the relatively new method called q-homotopy analysis method (q-HAM). The new fractional derivative makes it possible to introduce fractional order in space to the Fornberg-Whitham equation and be able to obtain its solution. This work displays the elegant nature of the application of q-HAM to solve strongly nonlinear fractional differential equations. The presence of the auxiliary parameter h helps in an effective way to obtain better approximation comparable to exact solutions. The fraction-factor in this method gives it an edge over other existing analytical methods for nonlinear differential equations. Comparisons are made on the existence of exact solutions to these models. The analysis shows that our analytical solutions converge very rapidly to the exact solutions.

A relatively new and efficient approach based on a new iterative method and the Aboodh transform called the Aboodh transform iterative method is proposed to solve space-time fractional differential equations, the fractional order is considered in the Caputo sense. This method is a combination of the Aboodh transform and the new iterative method and gives the solution in series form with easily computable components. The nonlinear term is easily handled by the new iterative method, to affirm the simplicity and performance of the proposed method, five examples were considered, and the solution plots were presented to show the effect of the fractional order. The outcome reveals that the approach is accurate and easy to implement.

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