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    A Comparative Analysis of the Fractional-Order Coupled Korteweg-De Vries Equations with the Mittag-Leffler Law
    (Hindawi Ltd, 2022) Aljahdaly, Noufe H.; Akgul, Ali; Shah, Rasool; Mahariq, Ibrahim; Kafle, Jeevan
    This article applies efficient methods, namely, modified decomposition method and new iterative transformation method, to analyze a nonlinear system of Korteweg-de Vries equations with the Atangana-Baleanu fractional derivative. The nonlinear fractional coupled systems investigated in this current analysis are the system of Korteweg-de Vries and the modified system of Korteweg-de Vries equations applied as a model in nonlinear physical phenomena arising in chemistry, biology, physics, and applied sciences. Approximate analytical results are represented in the form of a series with straightforward components, and some aspects showed an appropriate dependence on the values of the fractional-order derivatives. The convergence and uniqueness analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. The series result achieved applying this technique is proved to be accurate and reliable with minimal calculations. The numerical simulations for obtained solutions are discussed for different values of the fractional order.
  • [ X ]
    Öğe
    Analysis of Fractional-Order Regularized Long-Wave Models via a Novel Transform
    (Hindawi Ltd, 2022) Shah, Nehad Ali; El-Zahar, Essam R.; Akgul, Ali; Khan, Adnan; Kafle, Jeevan
    A new integral transform method for regularized long-wave (RLW) models having fractional-order is presented in this study. Although analytical approaches are challenging to apply to such models, semianalytical or numerical techniques have received much attention in the literature. We propose a new technique combining integral transformation, the Elzaki transform (ET), and apply it to regularized long-wave equations in this study. The RLW equations describe ion-acoustic waves in plasma and shallow water waves in seas. The results obtained are extremely important and necessary for describing various physical phenomena. This work considers an up-to-date approach and fractional operators in this context to obtain satisfactory approximate solutions to the proposed problems. We first define the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD) and implement them for solving RLW equations. We can readily obtain numerical results that provide us with improved approximations after only a few iterations. The derived solutions were found to be in close contact with the exact solutions. Furthermore, the suggested procedure has attained the best level of accuracy. In fact, when compared to other analytical techniques for solving nonlinear fractional partial differential equations, the present method might be considered one of the finest.