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Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trigonometric and hyperbolic solutions to the time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation have been constructed in this paper. Notably, the definition of our fractional derivative is based on the Jumarie’s modified Riemann–Liouville derivative, which offers a strong basis for our mathematical explorations. This equation is widely utilized to report a variety of fascinating physical events in the domains of classical mechanics, plasma physics, fluid dynamics, heat transfer, and acoustics. It is presumed that the acquired outcomes have not been documented in earlier research. Numerous standard wave profiles, such as kink, smooth bell-shaped and anti-bell-shaped soliton, W-shaped, M-shaped, multi-wave, periodic, bright singular and dark singular soliton, and combined dark and bright soliton, are illustrated in order to thoroughly analyze the wave nature of the solutions. Painlevé analysis of the proposed study is also part of this work. To illustrate how the fractional derivative affects the precise solutions of the equation via 2D and 3D plots.

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions using definitions of the β-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions.

Tuberculosis (TB), caused by Mycobacterium tuberculosis is one of the treacherous infectious diseases of global concern. In this paper, we consider a deterministic model of TB infection with the public health education and hospital treatment impact. The effective reproductive number, Rph, that measures the potential spread of TB is presented by employing the next generation matrix approach. We investigate local and global stability of the TB-free equilibrium point, endemic equilibrium point, and sensitivity analysis. The analyses of the proposed model show that the model undergoes the phenomenon of backward bifurcation when the effective reproduction number (Rph) is less than one, where two stable equilibria, namely, the DFE and an EEP coexist. Further, we compute the sensitivity of the impact of each parameter on the effective reproductive number of the model by employing a normalized sensitivity index formula. Numerical simulation of the proposed model was conducted using Maple 2016 and MatLab R2020b software and compared with the theoretical results for illustration purposes. The investigation results can be useful in providing information to policy makers and public health authorities in mitigating the spread of TB infection by public health education and hospital treatment.

Fractional differential equations describe nature adequately because of the symmetry properties that describe physical and biological processes. In this paper, a new approximation is found for the variable-order (VO) Riemann–Liouville fractional derivative (RLFD) operator; on that basis, an efficient numerical approach is formulated for VO time-fractional modified subdiffusion equations (TFMSDE). Complete theoretical analysis is performed, such as stability by the Fourier series, consistency, and convergence, and the feasibility of the proposed approach is also discussed. A numerical example illustrates that the proposed scheme demonstrates high accuracy, and that the obtained results are more feasible and accurate.

This study extends a deterministic mathematical model for the dynamics of tuberculosis transmission to examine the impact of an imperfect vaccine and other exogenous factors, such as re-infection among treated individuals and exogenous re-infection. The qualitative behaviors of the model are investigated, covering many distinct aspects of the transmission of the disease. The proposed model is observed to show a backward bifurcation, even when Rv<1. As such, we assume that diminishing Rv to less than unity is not effective for the elimination of tuberculosis. Furthermore, the results reveal that an imperfect tuberculosis vaccine is always effective at reducing the spread of infectious diseases within the population, though the general effect increases with the increase in effectiveness and coverage. In particular, it is shown that a limited portion of people being vaccinated at steady-state and vaccine efficacy assume a equivalent role in decreasing disease burden. From the numerical simulation, it is shown that using an imperfect vaccine lead to effective control of tuberculosis in a population, provided that the efficacy of the vaccine and its coverage are reasonably high.

The variable-order (VO) fractional calculus can be seen as a natural extension of the constant-order, which can be utilized in physical and biological applications. In this study, we derive a new numerical approximation for the VO fractional Riemann–Liouville integral formula and developed an implicit difference scheme (IDS) for the variable-order fractional sub-diffusion equation (VO-FSDE). The derived approximation used in the VO time fractional derivative with the central difference approximation for the space derivative. Investigated the unconditional stability by the van Neumann method, consistency, and convergence analysis of the proposed scheme. Finally, a numerical example is presented to verify the theoretical analysis and effectiveness of the proposed scheme.

Fractional differential equations describe nature adequately because of the symmetry properties which describe physical and biological processes. In this article, a fourth-order new implicit difference scheme is formulated and applied to solve the two-dimensional time-fractional modified sub-diffusion equation involving two times Riemann–Liouville fractional derivatives. The stability of the fourth-order implicit difference scheme is investigated using the von Neumann technique. The proposed scheme is shown to be unconditionally stable. Numerical examples are given to illustrate the feasibility of the proposed scheme.