Narrow Results

This paper investigates the dynamical semi-analysis of the delayed food chain model under the considered fractional order. The food chain model is composed of three compartments, namely, population of the prey, intermediate predator and a top predator. By using the fixed point theorem approach, we exploit some conditions for existence results and stability for the considered system via Atangana–Baleanu–Caputo derivative with fractional order. Also, using the well-known Adam–Bashforth technique for numerics, we simulate the concerning results for the interference between the prey and intermediate predator. Graphical results are discussed for different fractional-order values for the considered model.

The scaling exponent of a hierarchy of cities used to be regarded as a fractional. This paper investigates a newly constructed system of equation for Hepatitis B disease in sense of Atanganaa–Baleanu Caputo (ABC) fractional order derivative. The proposed approach has five distinctive quantities, namely, susceptible, acute infections, chronic infection, immunized and vaccinated populace. By applying some well-known results of fixed point theory, we find the Ulam–Hyers type stability and qualitative analysis of the candidate solution. The deterministic stability for the proposed system is also computed. We apply well-known transform due to Laplace and decomposition techniques (LADM) and Adomian polynomial for nonlinear terms for computing the series solution for the proposed model. Graphical results show that LADM is an efficient and robust method for solving nonlinear problems.

In this paper, we investigated some essential provisions for the existence and stability of the solution to integral boundary value problems with proportional delay of fractional order Atangana–Baleanu–Caputo (ABC) derivative. By the guidance of fixed point theory, we acquire the deserted results. Moreover, different types of Ullam–Hyers stabilities are investigated for the proposed problem. We also provide an appropriate example for illustrative purposes.

This work is devoted to studying the transmission dynamics of CoV-2 under the effect of vaccination. The aforesaid model is considered under fractional derivative with variable order of nonsingular kernel type known as Atangan–Baleanue–Caputo (ABC). Fundamental properties of the proposed model including equilibrium points and $R_0$ are obtained by using nonlinear analysis. The existence and uniqueness of solution to the considered model are investigated via fixed point theorems due to Banach and Krasnoselskii. Also, the Ulam–Hyers (UH) approach of stability is used for the said model. Further numerical analysis is investigated by using fundamental theorems of AB fractional calculus and the iterative numerical techniques due to Adams–Bashforth. Numerical simulations are performed by using different values of fractional-variable order $\varrho (𝜗)$ for the model. The respective results are demonstrated by using real data from Saudi Arabia for graphical presentation.

This paper is devoted to investigate Korteweg–De Vries equations (KDVEs) with third order under fractional Atangana–Baleanu–Caputo (ABC) derivative. We use an analytical method due to the Laplace transform and the Adomian decomposition technique. We establish the convergence analysis results for the considered problem by using nonlinear functional analysis. First of all, a general algorithm is developed for the computation of the required analytical approximate solution by using the mentioned method. By applying the developed algorithm, we then investigate the approximate solutions for some examples. The computed solution has also been presented graphically using Matlab. Laplace Adomian decomposition method (LADM) is a powerful full technique that is easy to implement. In addition, the said method has also a fast convergent quality.