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In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear smoking model concerning the Caputo derivative. The outcomes of the proposed technique exhibit good agreement with the Laplace decomposition method, demonstrating that our technique is an excellent alternative to various series solution methods. Our approach utilizes the simple limit principle at zero, making it the easiest way to extract series solutions, while variational iteration, Adomian decomposition, and homotopy perturbation methods require integration. Moreover, our technique is also superior to the residual method by eliminating the need for derivatives, as fractional integration and differentiation are particularly challenging in fractional contexts. Significantly, our technique is simpler than other series solution techniques by not relying on Adomian's and He's polynomials, thereby offering a more efficient way of solving nonlinear problems. [ABSTRACT FROM AUTHOR]

Piecewise fractional-order differential operators have received more attention in recent years because they can be used to describe various evolutionary dynamical problems to investigate crossover behaviors. In this manuscript, we use the aforementioned operators to investigate a mathematical model of COVID-19. By utilizing fractional calculus, our approach aims to capture the crossover dynamics of disease spread, considering heterogeneity and transitions between epidemic phases. This research seeks to develop a framework using specialized mathematical techniques, such as the Caputo fractional derivative, with the potential to investigate the crossover dynamical behaviors of the considered epidemic model. The anticipated contribution lies in bridging fractional calculus and epidemiology, offering insights for both theoretical advancements and practical public health interventions. In order to improve our understanding of epidemic dynamics and support, we used MATLAB to simulate numerical results for a visual representation of our findings. For this interpretation, we used various fractional-order values. In addition, we also compare our simulated results with some reported results for infected and death classes to demonstrate the efficiency of our numerical method. [ABSTRACT FROM AUTHOR]

In the present research, we consider a biological model of serum hepatitis disease. We carry out a detailed analysis of the mentioned model along with a class with asymptomatic carriers to explore its theoretical and numerical aspects. We initiate the study by using the piecewise fractal–fractional derivative (FFD) by which the crossover effects within the model are examined. We split the time interval into subintervals. In one subinterval, FFD with a power law kernel is taken, while in the second one, FFD with an exponential decay kernel of the proposed model is considered. This model is then studied for its disease-free equilibrium point, existence, and Hyers–Ulam (H-U) stability results. For numerical results of the model and a visual presentation, we apply the Lagrange interpolation method and an extended Adams–Bashforth–Moulton (ABM) method, respectively. [ABSTRACT FROM AUTHOR]

This manuscript is devoted to using Bernoulli polynomials to establish a new spectral method for computing the approximate solutions of initial and boundary value problems of variable-order fractional differential equations. With the help of the aforementioned method, some operational matrices of variable-order integration and differentiation are developed. With the aid of these operational matrices, the considered problems are converted to algebraic-type equations, which can be easily solved using computational software. Various examples are solved by applying the method described above, and their graphical presentation and accuracy performance are provided. [ABSTRACT FROM AUTHOR]

We construct a subclass of Copson's integral inequality in this article. In order to achieve this goal, we attempt to use the Steklov operator for generalizing different inequalities of the Copson type relevant to the situations r > 1 as well as r < 1. We demonstrate the inequalities with the guidance of basic comparison, Holder's inequality, and the integration by parts approach. Moreover, some new variations of Hardy's integral inequality are also presented with the utilization of Steklov operator. We also formulate many remarks and two examples to show the novelty and authenticity of our results. [ABSTRACT FROM AUTHOR]

In this work, we use the idea of interval-valued convex functions of Center-Radius (cr)-order to give fractional versions of Hermite–Hadamard inequality. The results are supported by some numerical estimations and graphical representations considering some suitable examples. The results are novel in the context of cr -convex interval-valued functions and deal with differintegrals of the p + s 2 type. We believe this will be an important contribution to spurring additional research. [ABSTRACT FROM AUTHOR]

Many authors have established various integral and differential formulas involving different special functions in recent years. In continuation, we explore some image formulas associated with the product of Srivastava's polynomials and extended Wright function by using Marichev–Saigo–Maeda fractional integral and differential operators, Lavoie–Trottier and Oberhettinger integral operators. The obtained outcomes are in the form of the Fox–Wright function. It is worth mentioning that some interesting special cases are also discussed. [ABSTRACT FROM AUTHOR]