The main features of scientific effort in physics and engineering are the development of models for various physical issues and the development of solutions. In this paper, we investigate the numerical solution of time-fractional non-linear dispersive K(m, n, 1) type equations using two innovative approaches: the homotopy perturbation transform method and Yang transform decomposition method. Our suggested approaches elegantly combine Yang transform, homotopy perturbation method (HPM) and adomian decomposition method (ADM). With the help of the Yang transform, we first convert the problem into its differential partner before using HPM to get the He's polynomials and ADM to get the Adomian polynomials, both of which are extremely effective supports for non-linear issues. In this case, Caputo sense is used for defining the fractional derivative. The derived solutions are shown in series form and converge quickly. To ensure the effectiveness and applicability of the proposed approaches, the examined problems were analyzed using various fractional orders. We analyze and demonstrate the validity and applicability of the solution approaches under consideration with given initial conditions. Two and three dimensional graphs reflect the outcomes that were attained. To verify the effectiveness of the strategies, numerical simulations are presented. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. The results of this study demonstrate that the studied methods are effective and strong in solving nonlinear differential equations that appear in science and technology.
Citation: Abdul Hamid Ganie, Fatemah Mofarreh, Adnan Khan. A novel analysis of the time-fractional nonlinear dispersive K(m, n, 1) equations using the homotopy perturbation transform method and Yang transform decomposition method[J]. AIMS Mathematics, 2024, 9(1): 1877-1898. doi: 10.3934/math.2024092
The main features of scientific effort in physics and engineering are the development of models for various physical issues and the development of solutions. In this paper, we investigate the numerical solution of time-fractional non-linear dispersive K(m, n, 1) type equations using two innovative approaches: the homotopy perturbation transform method and Yang transform decomposition method. Our suggested approaches elegantly combine Yang transform, homotopy perturbation method (HPM) and adomian decomposition method (ADM). With the help of the Yang transform, we first convert the problem into its differential partner before using HPM to get the He's polynomials and ADM to get the Adomian polynomials, both of which are extremely effective supports for non-linear issues. In this case, Caputo sense is used for defining the fractional derivative. The derived solutions are shown in series form and converge quickly. To ensure the effectiveness and applicability of the proposed approaches, the examined problems were analyzed using various fractional orders. We analyze and demonstrate the validity and applicability of the solution approaches under consideration with given initial conditions. Two and three dimensional graphs reflect the outcomes that were attained. To verify the effectiveness of the strategies, numerical simulations are presented. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. The results of this study demonstrate that the studied methods are effective and strong in solving nonlinear differential equations that appear in science and technology.
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Abdul Hamid Ganie, Fatemah Mofarreh, Adnan Khan. A novel analysis of the time-fractional nonlinear dispersive K(m, n, 1) equations using the homotopy perturbation transform method and Yang transform decomposition method[J]. AIMS Mathematics, 2024, 9(1): 1877-1898. doi: 10.3934/math.2024092
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