A new solution of the nonlinear fractional logistic differential equations utilizing efficient techniques

Abstract

The formulation of models and solutions for various physical problems are the primary goals of scientific achievements in engineering and physics. Our paper focuses on using the Caputo fractional derivative operator to solve nonlinear fractional logistic differential equations. In order to solve general nonlinear fractional differential equations, we first introduce a novel numerical methodology termed the Homotopy perturbation transform method. The perturbation approach and the Yang transform method are combined to create the suggested strategy. Second, we introduce a new hybrid method that uses the time-fractional Caputo derivative to approximate and analytically solve nonlinear fractional logistic differential equations. This method combines the Yang transform with the decomposition method. To validate the analysis, we offer three numerical cases of nonlinear fractional logistic differential equations employing the Caputo fractional derivative operator. The resulting solutions exhibit rapid convergence and are presented in series form. In order to verify the efficacy and relevance of the suggested methodologies, the investigated issues were assessed through the implementation of different fractional orders. We examine and show that, under the specified initial conditions, the solution approaches under evaluation are accurate and effective. Graphs in two and three dimensions show the results that were obtained. Numerical simulations are presented to confirm the efficacy of the strategies. The numerical results show that an accurate, reliable, and efficient approximation can be obtained with a minimal number of terms. The results obtained demonstrate that the new analytical solution method is easy to apply and very successful in solving difficult fractional problems that occur in relevant engineering and scientific domains.