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- 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.
- 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 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.
- MakaleAçık Erişim2023On the solution of random linear difference equations with Laplace transform method
Şişman, Şeyma; Merdan, Mehmet
In this study, Laplace transformation, which is very important for solutions to initial value problems, is examined. To solve the initial value problem of a discrete-time equation, Laplace implements the conversion method. Here, Laplace transformation is used to obtain an approach to the solutions of random difference equations formed by randomizing components of deterministic difference equations. For random behavior of linear difference equations under random effects, uniform, geometric, binomial, Poisson, and Bernouilli distributions are used, and approximate expected value, variance, standard deviation, and confidence interval of equations obtained by Laplace transformation are calculated. The results were obtained through the Maple package program.
- MakaleAçık Erişim2022Homotopy Perturbation Elzaki Transform Method for Obtaining the Approximate Solutions of the Random Partial Differential Equations
Anac, Halil; Merdan, Mehmet; Kesemen, Tulay
The series solutions of the random nonlinear partial differential equations have been examined by a hybrid method. The random nonlinear partial differential equations are studied by both normal and uniform distributions. Two initial-value problems are indicated to exemplify the influence of the solutions acquired by the hybrid method. Also, the functions for the first and second moments of the approximate solutions of random nonlinear partial differential equations are acquired in the MAPLE software. The hybrid method is implemented to analyze the solutions of the random nonlinear partial differential equations. MAPLE software is used to find the solutions. Besides, MAPLE software is used for the drawing the figures.
- MakaleYalnız künye2020APPLICATION OF CRANK-NICOLSON METHOD TO A RANDOM COMPONENT HEAT EQUATION
Anaç, Halil; Merdan, Mehmet; Kesemen, Tülay
In this study, the solution of a random component heat equation is obtained by using Crank-Nicolson Method. The initial condition of this equation is examined by Normal distribution. The expected value and variance of solution of this equation are obtained. Crank-Nicolson method is applied to analyze the solution of this equation. Also, the solution and the graphs of the expected value and variance are obtained by using MATLAB software. The results of the heat equation are compared with random characteristics of this equation. Firstly, a random component heat equation is solved by this method.
- MakaleAçık Erişim2020Investigation of linear difference equations with random effects
Merdan, Mehmet; Şişman, Şeyma
In this study, random linear difference equations obtained by transforming the components of deterministic difference equations to random variables are investigated. Uniform, Bernoulli, binomial, negative binomial (or Pascal), geometric, hypergeometric and Poisson distributions have been used for the random effects for obtaining the random behavior of linear difference equations. The random version of the Z-transform, the RZ-transform, has been used to obtain an approximation for the random linear difference equation. Approximate expected values and variances are calculated by using the RZ-transform. The results have been obtained with Maple and are shown in graphs. It is shown that the random Z-transform is an effective tool for the investigation of random linear difference equations.
- MakaleYalnız künye2020ANALYSIS OF RANDOM DISCRETE TIME LOGISTIC MODEL
Merdan, Mehmet; Şişman, Şeyma
In this study, the behavior of the logistic difference model is investigated under random conditions using discrete probability distributions. The logistic difference model consists of parameters that depend on the population models to be used. For the study of random difference equation population models, the parameters are treated as random variables which constitutes the basis of the study. Random models were created using Uniform, Bernouilli, Binom, Negative Binomial (or Pascal), Geometric, Hypergeometric, Poisson distributions and their numerical characteristics are obtained through their simulations. Then, the results showing random numerical characteristics such as expected value, variance, standard deviation, coefficient of variation and confidence intervals were obtained with MATLAB package program. Analysis of random logistic difference model is given with the help of graphics and tables.
- MakaleYalnız künye2019THE NEW SUMUDU TRANSFORM ITERATIVE METHOD FOR STUDYING THE RANDOM COMPONENT TIME-FRACTIONAL KLEIN-GORDON EQUATION
Merdan, Mehmet; Anaç, Halil; Kesemen, Tülay
In this study, the solutions of the random component time-fractional Klein-Gordon equation is obtained as approximately or exactly. The initial condition of this Klein-Gordon equation is studied by Gamma distribution. The fractional derivatives are defined in the Caputo sense. An example is shown to illustrate the influence of the solutions obtained by the new Sumudu transform iterative method (NSTIM). The expected value and variance of these solutions of this Klein-Gordon equation are obtained. The approximate analytical solution of this equation obtained by NSTIM and VIM are compared. NSTIM is applied to analyze the solution of this equation. Solution and figures are obtained by using MAPLE software. The formulas for the expected values and variances and results from the simulations of this Klein-Gordon equation are compared and the efficiency of this method is investigated.