The purpose of this study is to develop and apply the Elzaki-Homotopy Analysis Method (EHAM) as an efficient analytical technique for solving fractional-order partial differential equations (PDEs). By integrating the Elzaki transformation with the homotopy analysis framework, the study aims to obtain accurate and convergent analytical solutions for nonlinear and random fractional PDEs. Additionally, it seeks to evaluate the influence of stochastic parameters, where coefficients or initial conditions follow various probability distributions. The research further aims to compute the variance, expected value, and confidence intervals of the solutions and visualize the outcomes using MATLAB software.
In this study, the Elzaki-Homotopy Analysis Method (EHAM) is employed by combining the Elzaki transformation with the homotopy analysis method to construct analytical solutions for fractional-order partial differential equations. The local fractional derivative is utilized to handle non-integer order behavior effectively. The proposed approach is applied to both linear and nonlinear equations under random conditions, where initial parameters and coefficients follow uniform, beta, normal, and gamma distributions. Statistical measures such as variance, expected value, and confidence intervals are calculated to analyze solution behavior. All computations and graphical representations are performed using MATLAB (2013a) to verify the method's accuracy and convergence.
The findings of this study demonstrate that the Elzaki-Homotopy Analysis Method (EHAM) provides highly accurate and rapidly convergent analytical solutions for fractional-order partial differential equations. The method effectively handles both linear and nonlinear problems involving random parameters. Statistical analysis of the obtained solutions reveals that the expected values and variances remain stable across different probability distributions, confirming the robustness of the approach. The convergence region can be easily controlled through the auxiliary parameters of the homotopy method. Comparisons with numerical results validate the reliability and efficiency of EHAM, establishing it as a powerful alternative to conventional numerical and perturbation techniques.
This study presents an innovative analytical framework for solving fractional-order partial differential equations using the Elzaki-Homotopy Analysis Method (EHAM). By integrating the Elzaki transformation with the homotopy analysis approach, the method efficiently addresses the complexities of nonlinear and stochastic systems. The inclusion of random variables with different probability distributions enhances the model's applicability to real-world phenomena. Analytical solutions are systematically derived, and their statistical properties—such as variance, expected value, and confidence intervals—are examined. Implemented through MATLAB (2013a), the results confirm EHAM's strong convergence, reliability, and computational efficiency, positioning it as a superior alternative to traditional numerical techniques.
