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This paper aims to solve the Black-Scholes (B-S) model for the European options pricing problem using a hybrid method called fractional generalized homotopy analysis method (FGHAM). The convergence region of the B-S model solutions are clearly identified using h-curve and the closed form series solutions are produced using FGHAM. To verify the convergence of the proposed series solutions, sequence of errors are obtained by estimating the deviation between the exact solution and the series solution, which is increased in number of terms in the series. The convergence of sequence of errors is verified using the convergence criteria and the results are graphically illustrated. Moreover, the FGHAM approach has overcome the difficulties of applying multiple integration and differentiation procedures while obtaining the solution using well-established methods such as homotopy analysis method and homotopy perturbation method. The computational efficiency of the proposed method is analyzed using a comparative study. The advantage of the proposed method is shown with a numerical example using the comparative study between FGHAM and Monte Carlo simulation. Using the numerical example, analytical expression for the implied volatility is derived and the non-local behavior is studied for the various values of the fractional parameter. The results of FGHAM are statistically validated with the exact solution and the other existing computational methods.

This paper presents a novel hybrid technique which is developed by incorporating the fractional derivatives in the generalised integral transform method. Homotopy analysis method is combined with fractional generalised integral transform method to solve the fractional order nonlinear differential equations. The performance of the proposed method is analysed by solving various categories of nonlinear fractional differential equations like Navier Stokes's model and Riccatti equations, etc. Unlike the other analytical methods, the hybrid method provides a better way to control the convergence region of the obtained series solution through an auxiliary parameter h. Furthermore, as proposed in this paper, the 'Fractional Generalised Homotopy Analysis Method' along with the several examples reveal that this method can be effectively used as a tool for solving various kinds of nonlinear fractional differential equations.

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This paper proposes an efficient hybrid analytical method named the Generalized Homotopy Analysis Method (GHAM) for solving nonlinear differential equations. The proposed hybrid method consists of topology based Homotopy Analysis Method (HAM) and Laplace typed integral transform (G-transform). Compared with HAM, GHAM does not require the effort to perform repeated integration and differentiation, when it is applied to solve higher order differential equations. Compared with G-transform, GHAM serves as an effective approach to tackle higher-order nonlinear differential equations. The effectiveness of GHAM is illustrated through the analysis of numerical examples, and the obtained results are graphically depicted. Also, GHAM is effectively utilized to analyze the density of the forest cover incorporating various parameters, which include the seed reproduction, seed deposition, seed establishment rates, old trees due to aging and the coefficients of mortality due to space variables.