İlgi alanları

Sürdürülebilir Kalkınma Amaçları

• 11

• MakaleYalnız künye2026/r/gumushane/16726
  Solving nonlinear random fractional partial differential equations using Elzaki-homotopy analysis method

  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.

  İncele
• MakaleAçık Erişim2026/r/gumushane/19444
  Using the Laplace Transform Technique to Solve Hilfer Fractional Partial Differential Equations

  Merdan, Mehmet; Şişman, Şeyma

  This work derives analytical solutions to fractional partial differential equations utilizing the Hilfer fractional derivative through the use of the Laplace transform method. The suggested method offers a cohesive solution framework that concurrently retrieves the Liouville-Caputo and Riemann-Liouville formulations as particular instances for designated selections of the Hilfer parameters. Exact novel solution representations are derived for the fractional KdV, modified KdV, K(2,2), and Klein-Gordon equations under appropriate initial conditions, emphasizing the impact of fractional order and type parameters on wave propagation characteristics. The approach circumvents linearization or perturbation assumptions and demonstrates a rapid convergence rate. Numerical simulations and graphical representations produced with the Maple program corroborate the analytical findings and illustrate the efficacy and universality of the proposed method.

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• MakaleYalnız künye2026/r/gumushane/13656
  Development of a hybrid technique for solving the Newell–Whitehead–Segel equation using the Caputo–Fabrizio derivative

  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.

  İncele
• MakaleAçık Erişim2026/r/gumushane/13437
  Approximate Analytical Solutions of the Time-Fractional Swift–Hohenberg Equation via Yang Transform Adomian Decomposition Method

  Ali, Mustafa Ahmed; Merdan, Mehmet

  This study examines approximate analytical solutions of the time-fractional Swift–Hohenberg equation involving the Caputo–Fabrizio fractional derivative. The main objective of this study is to investigate efficient analytical approximation techniques using the Yang transform Adomian decomposition method (YTADM). The proposed approach combines the Yang transform with the Adomian decomposition technique to construct a rapidly convergent series solution for the fractional partial differential equation. The effects of the fractional order and system parameters on the behavior of the solution have been examined via tables and are graphically represented. The results confirm that the proposed method provides accurate and computationally efficient approximate solutions to the fractional Swift–Hohenberg equation. These results show that YTADM is an appropriate and useful analytical tool to solve nonlinear fractional partial differential equations arising in mathematical physics, especially those equations involving nonsingular fractional operators. The principal contribution of this work effort is the development and application of the YTADM framework for solving the fractional Swift–Hohenberg equation; therefore, it serves as a reliable analytic tool for nonlinear fractional partial differential equations arising in mathematical physics.

  İncele
• MakaleAçık Erişim2026/r/gumushane/12288
  Solution of Time-Fractional Coupled Burgers Equations by the Yang Transform Adomian Decomposition Method

  Ali, Mustafa Ahmed; Merdan, Mehmet

  In this paper, the Yang transform Adomian decomposition method (YTADM) is employed in the solution of nonlinear time-fractional coupled Burgers equations. The technique solves the fractional and nonlinear terms successfully via the Adomian decomposition of the Yang transform. Illustrative examples are given, which show the accuracy and ease of implementation of the method. The results show that the YTADM method gives good, reliable results and is very accurate, so that its solutions agree well with the exact solution. Finally, tables and graphs are shown, which illustrate the behavior of the solutions found.

  İncele
• MakaleAçık Erişim2026/r/gumushane/12145
  Enhanced Computational Efficiency in Solving Delay Fractional Partial Differential Equations Through the Yang Decomposition Method

  Ali, Mustafa Ahmed; Merdan, Mehmet

  This study presents the Yang Transform Adomian Decomposition Method (YTADM), a semi-analytical framework for solving one-dimensional linear and nonlinear delay fractional partial differential equations involving the Caputo fractional derivative. The proposed method combines the Yang transform with the Adomian decomposition method to construct recursive solution series while efficiently handling delayed nonlinear terms. The applicability of the proposed framework is demonstrated through several examples, including proportional-delay Burgers-type equations, and its convergence properties are analyzed. The obtained results show that YTADM yields rapidly convergent semi-analytical approximations and provides an effective framework for solving one-dimensional delay fractional partial differential equations.

  İncele
• MakaleAçık Erişim2025/r/gumushane/19766
  The Solutions of Caputo-Fabrizio Random Fractional Ordinary Differential Equations by Aboodh Transform Method

  Merdan, Mehmet; Şahin, Yasin

  The primary objective of the paper is to employ the Aboodh transformation to solve Caputo-Fabrizio ordinary fractional differential equations. Furthermore, the methodology described is used to solve some common fractional differential equations. Analyzed using the Aboodh Transformation Method are random ordinary differential equations derived by randomly picking the coefficients or initial conditions of Caputo-Fabrizio fractional ordinary differential equations. The initial conditions or coefficients of the equations are converted into random variables distributed according to Normal, Uniform, and Exponential distributions. The probability characteristics, including expected value, variance, coefficient of variation, and confidence interval, of the random ordinary differential equations solved are computed using the MATLAB (2013a) package software. The acquired results are then shown and discussed.

  İncele
• MakaleAçık Erişim2025/r/gumushane/19511
  Analysis of Diabetes Complications Model With Time Delays

  Merdan, Mehmet; Şişman, Şeyma

  In this article, diabetes, a disease known to be influenced by genetic and environmental factors, is divided into two types: diabetes without complications (D) and diabetes with complications (C). The stability of the equilibrium points of this two-dimensional diabetes complication (DC) model, the D and C equations are studied by time delay. In addition, the behavior of the model is obtained with the help of the MATLAB package program.

  İncele
• Kitap BölümüYalnız künye2025/r/gumushane/14318
  Investigation of Fractional Order Genesio-Tesi System Using Laplace Transformation

  Bekiryazici, Zafer; Sengul, Suleyman; Merdan, Mehmet

  In this paper the fractional-order Genesio-Tesi chaotic system is investigated with Laplace transformation method. Genesio-Tesi system is a three-dimensional equation system with quadratic nonlinearity and is frequently used for exemplifying chaotic systems. In this study, Caputo fractional derivative is used to obtain the fractional system and Laplace transformation is applied in order to analyze the explicit solutions of the system.

  İncele
• MakaleAçık Erişim2025/r/gumushane/14307
  Mathematical analysis of fractional order Covid 19 epidemic model

  Açikgöz, Pınar; Merdan, Mehmet

  This article aims to examine the dynamics of the fractional epidemic model of covid 19. The model in question incorporates the notion of a compatible derivative. Given their consistent and worldwide characteristics, fractional derivatives are presently employed to address nu-merous practical issues. In addition, we employ two numerical techniques, namely the con-formal differential transform and the variational iteration approach, to provide an approxi-mate solution for the given model. The research closes by providing an in-depth analysis and visually representing the numerical findings. Furthermore, it has been demonstrated that the solution obtained is convergent.

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