Sürdürülebilir Kalkınma Amaçları
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- MakaleAçık Erişim2026Using 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.
- MakaleYalnız künye2026Solving 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.
- MakaleYalnız künye2026Development 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.
- MakaleAçık Erişim2026Approximate 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.
- MakaleAçık Erişim2026Solution 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.
- MakaleAçık Erişim2026Enhanced 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.
- MakaleAçık Erişim2025The 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.
- MakaleAçık Erişim2025Analysis 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.
- Kitap BölümüYalnız künye2025Investigation 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.
- MakaleAçık Erişim2025Mathematical 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.
- MakaleAçık Erişim2025Investigation of fractional order covid-19 model with q-homotopy analysis transform method
Merdan, Mehmet; Açikgöz, Pınar
The primary objective of this paper is to derive approximate analytic solutions for the time fractional-order Covid-19 model by employing the q-homotopy analysis transform method (q-HATM) and the Laplace transform homotopy perturbation method (LT-HPM). The covid 19 model is a system of five-dimensional nonlinear ordinary differential equations. Moreover, this method applies even to more complex partial differential equations originating from mathematical biology, demonstrating computational efficiency and wide application. In this study, Caputo fractional derivative is used to obtain the fractional system and Laplace transformation is applied to analyze the approximate solutions of the system. Graphical results are presented and discussed quantitatively to illustrate the approximate solution. Copyright 2021, Yıldız Technical University.
- MakaleYalnız künye2025A Novel Numeric-analytic Method for Time-fractional Swift–Hohenberg (S–H) Equation with Modified Caputo-Fabrizio Derivative
Şahin, Yasin; Merdan, Mehmet
This work investigates utilizing Adomian decomposition technique (A-ADM) coupled with Aboodh transform as a hybrid approach the solution of Caputo-Fabrizio (CF) fractional order Swift-Hohenberg problem. Effective framework for modeling memory effects and complex system dynamics is provided by CF fractional derivative defined by non-singular exponential kernel. Classical Swift-Hohenberg equation is expanded to account for long-term memory effects, which are crucial in understanding pattern development and chaotic behavior in many physical systems by including CF fractional derivatives. With the suggested Aboodh-Adomian decomposition technique (A-ADM), an approximate analytical solution is provided showing its efficiency and accuracy in solving fractional nonlinear problem. Furthermore shown are convergence and originality of the solution. Graphical depictions of the acquired solutions make use of MATLAB package software. Theoretical results are confirmed by numerical simulations, which further expose the important influence of fractional order on the dynamic features of the system.
- MakaleAçık Erişim2025The Analysis of Three-Dimensional Time-Fractional Helmholtz Model Using a New İterative Method
Şahin, Yasin; Merdan, Mehmet; Açıkgöz, Pınar
This paper proposes a novel analytical method to address the Helmholtz fractional differential equation by combining the Aboodh transform with the Adomian Decomposition Method, resulting in the Aboodh–Adomian Decomposition Method (A-ADM). Fractional differential equations offer a comprehensive framework for describing intricate physical processes, including memory effects and anomalous diffusion. This work employs the Caputo–Fabrizio fractional derivative, defined by a non-singular exponential kernel, to more precisely capture these non-local effects. The classical Helmholtz equation, pivotal in acoustics, electromagnetics, and quantum physics, is extended to the fractional domain. Following the exposition of fundamental concepts and characteristics of fractional calculus and the Aboodh transform, the suggested A-ADM is employed to derive the analytical solution of the fractional Helmholtz equation. The method’s validity and efficiency are evidenced by comparisons of analytical and approximation solutions. The findings validate that A-ADM is a proficient and methodical approach for addressing fractional differential equations that incorporate Caputo–Fabrizio derivatives.
- MakaleAçık Erişim2025Investigation of Nonstandard Finite Difference for Fractional Order Covid-19 Model
Merdan, Mehmet; Acikgoz, Pınar
This article examines a mathematical model of the Covid-19 type. We demonstrate how the population is impacted by immigration, protection, the mortality, exposure, curing, and interactions between sick and healthy individuals. There are five classifications in our model: exposed, susceptible, infected, quarantined, and recovered. The model is subjected to numerical and fractional analysis in this instance. The numerical analysis is performed using the fractional order non-standard finite difference (NSFD) scheme. The Grunwald-Letnikov numerical approximation technique is used for fractional analysis. The findings are evaluated by simulations using the Matlab tool.
- MakaleAçık Erişim2025A new approach to solving local fractional Riccati differential equations using the Adomian-Elzaki method
Aydemir, Hilal; Merdan, Mehmet; Demir, Ümit
The Elzaki-Adomian decomposition method (EADM) is intended to serve as an efficient analytical method for the resolution of these original fractional-order Riccati differential equations. This can be accomplished permanently by incorporating the Adomian decomposition method with Elzaki. The local fractional derivative is implemented in this format. Particularly in the context of nonlinear differential equations (ODE), this approach is preferred over digital gaps. Additionally, the method’s convergence. Random individuals with uniform, beta, normal, and gamma distributions are used to select the initial conditions or coefficients of the equations. The variance, confidence interval, and expected value of the solutions that are obtained will be determined. MATLAB (2013a) package software will be employed to display the individuals that were brought together, and the results will be analyzed randomly.
- MakaleAçık Erişim2025New Numerical Solutions for Caputo–Fabrizio Fractional Differential Multidimensional Diffusion Problems
Şahin, Yasin; Merdan, Mehmet
This study examined the multidimensional fractional diffusion equations that characterize the density dynamics in a diffusing medium utilizing the exact and analytical Aboodh transform decomposition method (ATDM). The Caputo–Fabrizio (CF) derivative is utilized to account for the fractional derivative that is being employed here. The suggested method handles nonlinear terms by combining the Adomian decomposition method (ADM) with the Aboodh transform (AT) and Adomian polynomials. Since the AT can only be applied to linear equations, ADM is a helpful technique for estimating solutions to nonlinear differential equations. In nonlinear systems, multidimensional diffusion problems which account for the production of stripes in two-dimensional systems are important. Additionally, we used MATLAB to compute numerical and graphical data that illustrate how the close-form analytical solution compares to the precise solution. Promising and appropriate for addressing multidimensional diffusion problems with fractional derivatives are the obtained results. The key benefit is that our devised approach does not depend on presumptions or constraints on variables that skew the actual issue. Overcoming fluctuating constraints that can make it difficult to discover the answer and represent the problem depends much on this technique. In the domains of science and technology, the approach presented in this work offers exceptional computational accuracy and convenience of use for analyzing and resolving intricate phenomena associated with CF fractional nonlinear partial differential equations.
- MakaleAçık Erişim2024Investigation of solution behavior of Differential Equations by Sumudu methods of random complex Partial Differential Equations
Merdan, Mehmet; Merdan, Merve; Şahin, Rıdvan
In this study, solutions of random complex partial differential equations were found using the two-dimensional Sumudu transformation method(STM). The initial conditions of a deterministic equation or the non-homogeneous part of the equation are transformed into random variables to obtain a random complex partial differential equation. With the help of the properties of two-dimensional Sumudu and inverse Sumudu transformation, an approximate analytical solution of a complex partial differential equation with random constant coefficients was obtained by selecting a random variable with an initial condition of Normal and Gamma distribution. The probability characteristics of the resulting solutions, such as expected value and variance, were obtained and graphically shown with the help of the Maple package program.
- MakaleAçık Erişim2024Analysis of Random Difference Equations Using the Differential Transformation Method
Şişman, Şeyma; Merdan, Mehmet
The differential transformation method (DTM) is one of the best methods easily applied to linear and nonlinear difference equations with random coefficients. In this study, we apply the theorems related to the DTM to the given examples and investigate the behaviour of the approximate analytical solutions. The expected value, variance, coefficient of variation, and confidence intervals of the solutions of random difference equations obtained from discrete probability distributions such as uniform, geometric, Poisson, and binomial distributions will be calculated. Maple and MATLAB software packages are used to plot the solution graphs and also to interpret the solution behaviour.
- MakaleAçık Erişim2024Approximate Solutions of Fractional Differential Equations Using Optimal q-Homotopy Analysis Method: A Case Study of Abel Differential Equations
Şengül, Süleyman; Bekiryazici, Zafer; Merdan, Mehmet
In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Riccati equations, are given with ordinary derivatives and fractional derivatives in the Caputo sense to present the application of the method. The optimal q-HAM is an improved version of the Homotopy Analysis Method (HAM) and its modification q-HAM and focuses on finding the optimal value of the convergence parameters for a better approximation. Numerical applications are given where optimal values of the convergence control parameters are found. Additionally, the correspondence of the approximate solutions obtained for these optimal values and the exact or numerical solutions are shown with figures and tables. The results show that the optimal q-HAM improves the convergence of the approximate solutions obtained with the q-HAM. Approximate solutions obtained with the fractional Differential Transform Method, q-HAM and predictor–corrector method are also used to highlight the superiority of the optimal q-HAM. Analysis of the results from various methods points out that optimal q-HAM is a strong tool for the analysis of the approximate analytical solution in Abel-type differential equations. This approach can be used to analyze other fractional differential equations arising in mathematical investigations.
- MakaleAçık Erişim2024On solutions of time fractional order random HIV/AIDS modelling
Merdan, Mehmet; Öktem, Selami
In this study, The fractional random HIV/AIDS model approximate analytical solutions were produced using the differential transformation method. The approximate analytical solution of the fractional order Random HIV/AIDS model was obtained with the help of the differential transformation method. For the fractional random HIV/AIDS model, which was created by choosing the initial conditions from the exponential, beta, and normal distributions, graphic simulations of the expected value, variance, and confidence intervals of the most commonly used probability characteristics were obtained with the help of the MATLAB package program. Results obtained are interpreted.