In this paper, the Caputo-based fractional derivative optimal control model is looked at to learn more about how the human respiratory syncytial virus (RSV) spreads. Model solution properties such as boundedness and non-negativity are checked and found to be true. The fundamental reproduction number is calculated by using the next-generation matrix’s spectral radius. The fractional optimal control model includes the control functions of vaccination and treatment to illustrate the impact of these interventions on the dynamics of virus transmission. In addition, the order of the derivative in the fractional optimal control problem indicates that encouraging vaccination and treatment early on can slow the spread of RSV. The overall analysis and the simulated behavior of the fractional optimum control model are in good agreement, and this is due in large part to the use of the MATLAB platform.

Nearly every area of mathematics, natural, social, and engineering now includes research into finding exact answers to nonlinear fractional differential equations (NFDES). In order to discover the exact solutions to the higher order Sasa-Satsuma equation in the sense of the beta derivative, the paper will discuss the modified simple equation (MSE) and exponential rational function (ERF) approaches. In general, symmetry and travelling wave solutions of the Sasa-Satsuma equation have a common correlation with each other, thus we reduce equations from wave transformations to ordinary differential equations with the help of Lie symmetries. Actually, we can say that wave moves are symmetrical. The considered procedures are effective, accurate, simple, and straightforward to compute. In order to highlight the physical characteristics of the solutions, we also provide 2D and 3D plots of the results.

This paper studied the existence and uniqueness of the solution of the fractional logistic differential equation using Hadamard derivative and integral. Previous work has shown that there is not an exact solution to this fractional model. Hence several numerical approaches, such as generalized Euler’s method (GEM), power series expansion (PSE) method, and Caputo–Fabrizio (CF) method, were used to compute the solution. The classical solution obtained from the first order non-linear differential equation was also considered to enable the comparison of error levels.

This article studies a pharmacokinetics problem, which is the mathematical modeling of a drug concentration variation in human blood, starting from the injection time. Theories and applications of fractional calculus are the main tools through which we establish main results. The psi‐Caputo fractional derivative plays a substantial role in the study. We prove the existence and uniqueness of the solution to the problem using the psi‐Caputo fractional derivative. The application of the theoretical results on two data sets shows the following results. For the first data set, a psi‐Caputo with the kernel ψ = x + 1 is the best approach as it yields a mean square error (MSE) of 0.04065. The second best is the simple fractional method whose MSE is 0.05814; finally, the classical approach is in the third position with an MSE of 0.07299. For the second data set, a psi‐Caputo with the kernel ψ = x + 1 is the best approach as it yields an MSE of 0.03482. The second best is the simple fractional method whose MSE is 0.04116 and, finally, the classical approach with an MSE of 0.048640.

This article develops the existence theory for sequential fractional differential equations involving Caputo fractional derivative of order \(1<\alpha<2\) with nonlocal integral boundary conditions. Examples are given to demonstrate applications of our results.

Navier–Stokes (NS) equation, in fluid mechanics, is a partial differential equation that describes the flow of incompressible fluids. We study the fractional derivative by using fractional differential equation by using a mild solution. In this work, anomaly diffusion in fractal media is simulated using the Navier–Stokes equations (NSEs) with time-fractional derivatives of order $\beta \in (0,1)$. In ${\mathbf{H}}^{\gamma ,\wp }$, we prove the existence and uniqueness of local and global mild solutions by using fuzzy techniques. Meanwhile, we provide a local moderate solution in Banach space. We further show that classical solutions to such equations exist and are regular in Banach space.