Merdan, Mehmet; Aydemir, Hilal

PurposeThe 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.Design/methodology/approachIn 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.FindingsThe 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.Originality/valueThis 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.

Şahin, Yasin; Merdan, Mehmet; Açıkgöz, Pınar

Purpose – The aim of the paper titled “Development of a hybrid technique for solving the Newell–Whitehead–Segel (NWS) equation using the Caputo-Fabrizio derivative” is to develop a new numerical method for solving the NWS equation using the Caputo-Fabrizio fractional derivative. This method involves the combination of the Caputo-Fabrizio Elzaki transformation and the q-homotopy analysis transformation method. The study aims to analyze the NWS equation that models the reaction-diffusion dynamics in biological and chemical systems. The convergence analysis of the proposed method is performed, and the obtained solutions are visualized with two- and three-dimensional graphics using the Maple software. The results show that the proposed method is more powerful and effective than the existing methods. Design/methodology/approach – In this study, a hybrid analytical-numerical method is developed to solve the NWS equation formulated using the Caputo-Fabrizio fractional derivative. The method is a combination of: q-homotopy analysis method (q-HATM) and Shehu transformation technique (ST). By combining these two powerful methods, a solution approach that does not require linearization or complex calculations for nonlinear fractional partial differential equations, converges quickly and gives high accuracy results, is presented. The obtained solutions are tested for numerical accuracy and are supported graphically. Findings – In the article titled “Development of a hybrid technique for solving the Newell–Whitehead–Segel equation using the Caputo–Fabrizio derivative”, the authors propose a new hybrid method for solving the time-fractional NWS equation. This method is a combination of the q-HATM and the ST. The proposed q-HSATM provides an efficient and lightweight approach for solving nonlinear fractional partial differential equations without requiring linearization, discretization or complex polynomials. This hybrid technique obtains the approximate analytical solution of the NWS equation using the Caputo–Fabrizio derivative. The method provides fast convergence and high accuracy with less memory and processing power requirements. It also stands out as a reliable tool for modeling complex systems in applied sciences such as physics, biology and engineering. In conclusion, this study provides an effective and applicable method for solving nonlinear differential equations formulated with the Caputo-Fabrizio derivative, making a significant contribution to the field of fractional calculus. Originality/value – This section of the article titled “Development of a hybrid technique for solving the Newell-Whitehead-Segel equation using the Caputo-Fabrizio derivative” can be summarized as follows: This study is one of the first hybrid q-HATM and ST-based methods developed to solve the NWS equation using the Caputo-Fabrizio fractional derivative operator. Its unique aspects: Unlike traditional methods, it provides models more suitable for physical systems by working with the Caputo-Fabrizio derivative, which includes the memory effect and whose kernel function is not exponential. The presented method presents a new and powerful analytical-numerical technique that can produce approximate solutions with high accuracy without requiring linearization or discretization. The developed technique is valuable not only theoretically but also with its applicability to reaction-diffusion systems in applied sciences.

Merdan, Mehmet; Yildirim, Ahmet; Gökdoǧan, Ahmet

Purpose - The purpose of this paper is to show how an application of fractional two dimensional differential transformation method (DTM) obtained approximate analytical solution of time-fraction modified equal width wave (MEW) equation. Design/methodology/approach - The fractional derivative is described in the Caputo sense. Findings - It is indicated that the solutions obtained by the two dimensional DTM are reliable and that this is an effective method for strongly nonlinear partial equations. Originality/value - The paper shows that exact solutions can also be obtained from the known forms of the series solutions.