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    Comparative study of fractional Newell-Whitehead-Segel equation using optimal auxiliary function method and a novel iterative approach
    (Aip Publishing, 2024) Xin, Xiao; Khan, Ibrar; Ganie, Abdul Hamid; Akgul, Ali; Bonyah, Ebenezer; Fathima, Dowlath; Yousif, Badria Almaz Ali
    This research explores the solution of the time-fractional Newell-Whitehead-Segel equation using two separate methods: the optimal auxiliary function method and a new iterative method. The Newell-Whitehead-Segel equation holds significance in modeling nonlinear systems, particularly in delineating stripe patterns within two-dimensional systems. Employing the Caputo fractional derivative operator, we address two case study problems pertaining to this equation through our proposed methods. Comparative analysis between the numerical results obtained from our techniques and an exact solution reveals a strong alignment. Graphs and tables illustrate this alignment, showcasing the effectiveness of our methods. Notably, as the fractional orders vary, the results achieved at different fractional orders are compared, highlighting their convergence toward the exact solution as the fractional order approaches an integer. Demonstrating both interest and simplicity, our proposed methods exhibit high accuracy in resolving diverse nonlinear fractional order partial differential equations. (c) 2024 Author(s).
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    On solutions of an obesity model in the light of new type fractional derivatives
    (Pergamon-Elsevier Science Ltd, 2021) Bonyah, Ebenezer; Akgul, Ali
    Obesity has become a serious problem in both developed and developing countries. In this work, we analyse the obesity model that involves three operators Caputo, Atangana-Baleanu in Caputo sense and fractal-fractional derivatives. The existence and uniqueness of the solution have been examined. The various numerical schemes have been presented for each operator. Many numerical analysis are carried out based on associated scheme to make meanings for the analytical analysis. Different fractional order values and fractal order values have been simulated. The fractal-fractional operator presents a unique dynamics of the Obesity model. (c) 2021 Elsevier Ltd. All rights reserved.
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    Reproducing kernel Hilbert space method for the solutions of generalized Kuramoto-Sivashinsky equation
    (Taylor & Francis Ltd, 2019) Akgul, Ali; Bonyah, Ebenezer
    Reproducing kernel Hilbert space method is given for the solution of generalized Kuramoto-Sivashinsky equation. Reproducing kernel functions are obtained to get the solution of the generalized Kuramoto-Sivashinsky equation. Two examples have been introduced to prove the accuracy of the method. The obtained results show that the reproducing kernel Hilbert space method gives approximate analytical solutions which are very close to the exact solution of the generalized Kuramoto-Sivashinsky equation, which demonstrates the power of the proposed technique. We prove the efficiency of the reproducing kernel Hilbert space method in this paper.