Sürdürülebilir Kalkınma Amaçları

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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
• Doktora TeziAçık Erişim2025/r/gumushane/21720
  Dynamic analysis of COVID-19 model

  Pınar Açıkgöz

  In this thesis, the spread dynamics of the Covid-19 pandemic were investigated using mathematical modeling methods. The study first addressed the SIR and its derivatives, which are classical deterministic models, and conducted analyses of the fundamental reproduction number (R_0 ), equilibrium points, and stability of these models. Subsequently, mathematical models developed for Covid-19 were analyzed, investigating the positivity, boundedness, and invariant regions of the solution. The memory effect and long-term dependency structure of the epidemic were investigated using fractional-order differential equations. Fractional models based on Caputo, Conformal, and Grünwald-Letnikov derivatives were solved using analytical and numerical methods. In this context, the Homotopy Analysis Method (HAM), q-HATM, Variational Iteration Method (VIM), and Non-Standard Finite Difference (NSFD) schemes were applied, and the resulting solutions were comparatively evaluated. Additionally, the uncertain spread of Covid-19 was modeled using stochastic differential equations, and numerical solutions were obtained using the Euler–Maruyama, Milstein, and Runge–Kutta methods. Furthermore, time-delay differential equations were used to incorporate effects such as immunity and incubation period into the model. The results demonstrate that the spread of Covid-19 can be more realistically explained not only with deterministic models but also with fractional-order and stochastic models. In this respect, the thesis contributes to both the epidemiological modeling literature and the applications of fractional analysis and numerical methods.

  RDL DOI10.71008/gumushane.thesis.2025.239indekslendi
  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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• MakaleAçık Erişim2025/r/gumushane/14060
  Investigation 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.

  İncele
• MakaleAçık Erişim2025/r/gumushane/12495
  The 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.

  İncele
• MakaleAçık Erişim2025/r/gumushane/12464
  Investigation 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.

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  İncele
• Yüksek Lisans TeziAçık Erişim2018/r/gumushane/20802
  Zeeman modelinin deterministik ve stokastik analizi

  Pınar Açıkgöz

  ÖZET Bu çalışmada iki ve üç boyutlu Zeeman modeli (kalp atış modeli ) incelenmiştir. Söz konusu modelde kararlılık analizi yapılacaktır. Tipik gevşemiş lif uzunluğunun durumu için analizler yapılmaktadır. Zeeman modellerinin yaklaşık analitik çözümleri elde edilecektir. Başlangıç olarak iki boyutlu Zeeman modeline gecikme (delay) terimi ilave edilerek kararlılık analizi gerçekleştirilecektir. Daha sonra bu çalışmada Zeeman tarafından geliştirilen kalp atış modeli olarak da bilinen modele, rastgele stokastik efektler eklenerek çözümün davranışını, p tansiyon normal dağılıma sahip bir rastgele değişken olması durumunda Zeeman modeli ele alınmış, x kalp kas lifinin ilk dört momentleri hesaplanarak, beklenen değer, varyans, standart sapma ve güven aralığı hesaplanmaktadır. Ayrıca sayısal yöntemlerden yararlanarak kalp atış modeli için grafik similasyonu oluşturulmaktadır. Kesirli (fractional) mertebeden türevli Zeeman modelinin kararlılık analizi yapılmıştır,ayrıca farklı mertebeden türevler için çözüm eğrileri elde edilmektedir. İki ve üç boyutlu Zeeman modeline wiener süreci (beyaz gürültü) terimleri eklenerek stokastik Zeeman modeli elde edilmiştir. Elde edilen bu stokastik model, Milstein ve Euler-Maruyama yöntemleriyle çözülmektedir. Deterministik, rastgele efektli ve stokastik Zeeman modelinin çözümleri karşılaştırılmaktadır. Rastgele Zeeman Kalp Atış Modelinin çözümü için Random Diferansiyel Dönüşüm Metodu (RDDM) kullanılmıştır. Parametreler ve başlangıç şartları sırasıyla Beta ve Normal dağılıma sahip rastgele değişkenler olarak alınmıştır. Rastgele Zeeman modelinin yaklaşık analitik çözümü yapılarak sistemin yaklaşık beklenen değer ve varyansı hesaplanmıştır. Random Diferansiyel Dönüşüm Metodu, bu metoddan elde edilen kesik seri çözümünün yakınsaklık oranı ve doğruluğunu artırmak için Pade yaklaşımı ve Laplace dönüşümünün kullanımını içeren Modified Diferansiyel Random Dönüşümü Metodu kullanılmıştır. Rastgele modelin çözümünden elde edilen sonuçlar ve bileşenlerin rastgele davranışları grafikler ve tablolar yardımıyla verilmiştir. Beta ve normal dağılımlı rastgele etkileri içeren rastgele modellerin çözüm davranışları karşılaştırılmıştır. Anahtar Kelimeler:Zeeman kalp atış modeli, kararlılık analizi, wiener süreç,delay (gecikme), Euler-Maruyama,Milstein,beklenen değer, varyans, Laplace-Pade yaklaşımı, Random Diferansiyel Dönüşüm Yöntemi.

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