Sürdürülebilir Kalkınma Amaçları

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• 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.

  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
• MakaleAçık Erişim2024/r/gumushane/12812
  Analysis of a discrete time fractional-order Vallis system

  ŞİŞMAN, Şeyma; MERDAN, Mehmet

  Vallis system is a model describing nonlinear interactions of the atmosphere and temperature fluctuations with a strong influence in the equatorial part of the Pacific Ocean. As the model approaches the fractional order from the integer order, numerical simulations for different situations arise. To see the behavior of the simulations, several cases involving integer analysis with different non-integer values of the Vallis systems were applied. In this work, a fractional mathematical model is constructed using the Caputo derivative. The local asymptotic stability of the equilibrium points of the fractional-order model is obtained from the fundamental production number. The chaotic behavior of this system is studied using the Caputo derivative and Lyapunov stability theory. Hopf bifurcation is used to vary the oscillation of the system in steady and unsteady states. In order to perform these numerical simulations, we apply Grünwald–Letnikov tactics with Binomial coefficients to obtain the effects on the non-integer fractional degree and discrete time vallis system and plot the phase diagrams and phase portraits with the help of MATLAB and MAPLE packages.

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• MakaleAçık Erişim2024/r/gumushane/12767
  Analysis 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.

  İncele
• MakaleAçık Erişim2023/r/gumushane/12797
  On 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.

  İncele
• MakaleAçık Erişim2022/r/gumushane/17223
  A Mathematical Model of Susceptible Diabetes Complication (SDC) Model in Discrete Time Fuzzy and Crisp Environment

  Merdan, Mehmet; Şişman, Şeyma

  In this study, we examined the mathematical model of the discrete-time equation system with susceptible diabetes complication (SDC), which is known to be caused by environmental and genetic factors in a fuzzy environment. From the diabetes complication (DC) model, the susceptible diabetes complication (SDC) model is being developed. It was obtained using definitions of how the behavior of this model changes in a fuzzy environment. A nonlinear differential equation system transforms the sensitive diabetes complication (SDC) model into a discrete time equation system. Stability analysis of the model with jury criterion was examined. In addition, numerical solutions and graphics of the analysis of the discrete model in fuzzy environment are obtained by using the MATLAB package program.

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  İncele
• MakaleYalnız künye2022/r/gumushane/14591
  DYNAMICAL ANALYSIS OF THE CONFORMABLE FRACTIONAL ORDER HOST-PARASITE MODEL

  Şişman, Şeyma; Merdan, Mehmet

  In this study, the differential equation system with a mathematical model of parasites is examined in cases where infection does not depend on transmission and defense, but on the level of infectivity and defense of the parasite and host. When discretization is applied to the differential equation, a two-dimensional discrete system is obtained in the range of t ∈ [n, n + 1] then the stability of the Neimark-Sacker bifurcation of the positive equilibrium point of this discrete system is investigated. Finally, MAPLE and MATLAB package program are used to show the accuracy of the results obtained.

  İncele
• MakaleAçık Erişim2021/r/gumushane/14851
  Global stability of Susceptible Diabetes Complication (SDC) model in discrete time

  Şi̇Şman, Şeyma; Merdan, Mehmet

  In this study, the mathematical model (DC) of diabetes disease is discussed. This model divides people into (D) uncomplicated and (C) complex diabetics two. In addition, diabetes is a disease known to be caused by genetic and environmental factors, and this factor is one of the main causes of genetic disorder at birth. Considering these two factors, the diabetes complication (SDC) model, which is sensitive from the diabetes complication (DC) model, is being developed. In this model, the responsive diabetes complication (SDC) model of a nonlinear system of differential equations is transformed into a discrete-time system of equations. The positivity and limitation of Model solutions were examined R0 the basic increment number is calculated. If R0 < 1, it has a global asymptotically stable balance for the situation where there is no genetic disorder at birth, and for R0 < 1, the system has an unstable balance. In addition, random behavior of the discrete model was examined for different probability distributions.

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  İncele
• Yüksek Lisans TeziAçık Erişim2020/r/gumushane/21914
  Ayrık zamanlı duyarlı diyabet komplikasyon modelinin deterministik ve rastgele analizi

  Şeyma Şişman

  Bu çalışmada, diyabet hastalığının matematiksel modeli (DC) ele alınmaktadır. Bu modelde (D) insanları komplike olmayan, (C) ise karmaşık diyabetliler olmak üzere ikiye ayırmaktadır. Bunun yanı sıra diyabet, genetik ve çevresel faktörlerin neden olduğu bilinen bir hastalıktır ve bu faktör doğumdaki genetik bozukluğun ana nedenlerden bir tanesidir. Bu her iki faktör göz önünde bulundurarak, diyabet komplikasyonu (DC) modelinden duyarlı diyabet komplikasyonu olan (SDC) modeli geliştirilmektedir. Bu modelde, doğrusal olmayan bir diferansiyel denklem sistemi duyarlı diyabet komplikasyonu (SDC) modeli ayrık zamanlı bir denklem sistemine dönüştürülmektedir. Model çözümlerinin pozitifliği ve sınırlılığı incelenmiştir. temel artış sayısı hesaplanmaktadır. Eğer R_0<1 ise doğumda genetik bozukluğun olmadığı durum için global asimptotik olarak kararlı bir dengeye sahiptir ve R_0>1 için ise sistem kararsız bir dengeye sahip olur. Buna ilaveten ayrık modelin rastgele davranışları için Düzgün, Binom, Geometrik, Hipergeometrik ve Poisson gibi kesikli zamanlı olasılık dağılımları kullanılarak incelendi. Ayrıca stokastik etki sonucunda elde edilen modelin sonuçları yorumlanmıştı. Anahtar Kelimeler: Denge noktaları, Duyarlı diyabet komplikasyon modeli, Global kararlılık, Kesikli zamanlı olasılık dağılımları

  İncele
• MakaleAçık Erişim2020/r/gumushane/14791
  Investigation 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.

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