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    Comparative analysis of the fractional order Cahn-Allen equation
    (Elsevier B.V., 2023) Khan, Ibrar; Nawaz, Rashid; Ali, Ali Hasan; Akgul, Ali; Lone, Showkat Ahmad
    This current work presents a comparative study of the fractional-order Cahn-Allen (CA) equation, where the non-integer derivative is taken in the Caputo sense.The Cahn-Allen equation is an equation that assists in the comprehension of phase transitions and pattern formation in physical systems. This equation describes how different phases of matter, such as solids and liquids, change and interact throughout time. We employ two analytical methods: the Laplace Residual Power Series Method (LRPSM) and the New Iterative Method (NIM), to solve the proposed model. The LRPSM is a combination of the Laplace Transform and the Residual Power Series Method, while the New Iterative Method is a modified form of the Adomian Decomposition Method that does not require any type of polynomial or digitization. For the purpose of accuracy and reliability, we compare our findings with other methods and the exact solution used in the literature. Additionally, 2D and 3D plots are generated for various fractional order values denoted as p. These plots illustrate that as the fractional order p approaches 1, the graph of the approximate solution gradually coincides with the graph of the exact solution. © 2023 The Author(s)
  • [ X ]
    Öğe
    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).