A Novel Hybrid Approach for Obtaining Approximate Series and Exact Solutions to the Caputo Fractional Black-Scholes Equations

This study employs a novel approach to derive both approximate series and exact solutions to the fractional Black-Scholes model, utilizing the Elzaki transform and residual functions. This method is called the Elzaki Residual Method (ERM). By applying the basic limit principle at zero, the ERM demonstrates a superior ability to determine the coefficients of terms in the fractional power series, whereas other well-known methods such as the Adomian decomposition method, the variational iteration method, and the Homotopy perturbation method require integration, and the residual power series method relies on differentiation, both of which are challenging in fractional contexts. Moreover, the ERM outperforms series solution techniques that rely on Adomian and He’s polynomials for solving nonlinear problems, as it eliminates the need for such polynomials. To verify the reliability of our approach, we perform recurrence and absolute error analyses. Additionally, comparative assessments with the projected method, the Adomian method, and the Aboodh decomposition method demonstrate strong agreement. Finally, graphical illustrations further confirm the accuracy of our solutions. Therefore, the ERM can serve as a valuable alternative for solving both linear and nonlinear fractional systems. Maple software is used to calculate the numerical and symbolic quantities in the paper.