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    Öğe
    Bright Soliton Behaviours of Fractal Fractional Nonlinear Good Boussinesq Equation with Nonsingular Kernels
    (Mdpi, 2022) Sadiq, Gulaly; Ali, Amir; Ahmad, Shabir; Nonlaopon, Kamsing; Akgul, Ali
    In this manuscript, we investigate the nonlinear Boussinesq equation (BEQ) under fractal-fractional derivatives in the sense of the Caputo-Fabrizio and Atangana-Baleanu operators. We use the double modified Laplace transform (LT) method to determine the general series solution of the Boussinesq equation. We study the convergence, existence, uniqueness, boundedness, and stability of the solution of the considered good BEQ under the aforementioned derivatives. The obtained solutions are presented with numerical illustrations considering a particular example by two cases based on both derivatives with suitable initial conditions. The results are illustrated graphically where good agreements are obtained. Our results show that fractal-fractional derivatives are a very effective tool for studying nonlinear systems. Furthermore, when t increases, the solitary waves of the system oscillate. As the fractional order a or fractal dimension beta increases, the soliton solutions become coherently close to the exact solution. For compactness, an error analysis is performed. The absolute error reveals an approximate linear evolution in the soliton solutions as time increases and that the system does not blow up nonlinearly.
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    Öğe
    Complex dynamics of multi strain TB model under nonlocal and nonsingular fractal fractional operator
    (Elsevier, 2021) Adnan; Ahmad, Shabir; Ullah, Aman; Riaz, Muhammad Bilal; Ali, Amir; Akgul, Ali; Partohaghighi, Mohammad
    Researchers have recently begun to use fractal fractional operators in the Atangana-Baleanu sense to analyze complicated dynamics of various models in applied sciences, as the Atangana-Baleanu operator generalizes the integer and fractional order operators. To analyze the complex dynamics of the multi-strain TB model, we use the AB-fractal fractional operator. We use the Banach fixed point theorem to ensure that at most one solution exists to the model. Further, the Ulam-Hyers type stability of the model is investigated with the help of functional analysis. The Adams-Bashforth approach is used to get numerical results for the proposed model. The analysis of the chaotic behavior of the proposed TB model was missing in the literature. Therefore, for different values of fractional and fractal order, we study the nonlinear dynamics and chaotic behavior of the obtained results of the proposed model.
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    Öğe
    Nonlinear Schrodinger equation under non-singular fractional operators: A computational study
    (Elsevier, 2022) Khan, Asif; Ali, Amir; Ahmad, Shabir; Saifullah, Sayed; Nonlaopon, Kamsing; Akgul, Ali
    In this article, we present study on time fractional nonlinear Schrodinger equation. We investigate the behaviour of the aforesaid equation in two numerous types of operators having non-singular kernels, which are Atangana-Baleanu and Caputo-Fabrizio operators both considered in Caputo's sense. The considered operators are very useful as they present tremendous dynamics of the suggested equation. We obtain numerical and analytical solutions of the proposed equation under the aforementioned fractional operators by modified double Laplace transform. We present the error analysis of the suggested scheme, where we observed that the considered system primarily depend on time. When time is small, we obtain very small error between the exact and approximate solutions. For the efficiency of our considered scheme, we present some examples. Further, we present the graphical and numerical analysis of the scheme used for the solution.