In this paper, we find the solution and analyse the behaviour of the obtained results for the nonlinear Schrödinger-Boussinesq equations using q-homotopy analysis transform method (q-HATM) within the frame of fractional order. The considered system describes the interfaces between intermediate long and short waves. The projected fractional operator is proposed with the help of Mittag-Leffler function to incorporate the nonsingular kernel to the system. The projected algorithm is a modified and accurate method with the help of Laplace transform. The convergence analysis is presented with the help of the fixed point theorem in the form existence and uniqueness. To validate and illustrate the effectiveness of the algorithm considered, we exemplified considered system with respect of arbitrary order. Further, the behaviour of achieved results is captured in contour and 3D plots for distinct arbitrary order. The results show that the projected scheme is very effective, highly methodical and easy to apply for complex and nonlinear systems and help us to captured associated behaviour diverse classes of the phenomenon.

The fundamental aim of the present study is to analyse and find the solution for the system of nonlinear ordinary differential equations describing the deadly and most dangerous virus from the lost three months called coronavirus. The mathematical model consisting of six nonlinear ordinary differential equations are exemplified and the corresponding solution is studied within the frame of 𝑞-homotopy analysis transform method (𝑞-HATM). Moreover, a newly defined fractional operator is employed in order to understand more effectively, known as Atangana-Baleanu (AB) operator. For the obtained results, the fixed point theorem is hired to present the exactness as well as uniqueness. For diverse arbitrary order, the behaviour of the outcomes is presented in terms of plots. Finally, the present study may help to examine the wild class of real-world models and also aid to predict their behaviour with respect to parameters considered in the models.

In this article, an efficient novel technique, namely the q‐homotopy analysis transform method (q‐HATM) is applied to find the solution for the time‐fractional Kaup‐Kupershmidt (KK) equation. The KK equation plays a vital role while studying the capillary gravity waves and nonlinear dispersive waves. To check the effectiveness and pertinency of the projected method, we consider three distinct cases of the fractional nonlinear KK equation. The q‐HATM provides the auxiliary parameter ℏ, called convergence control parameter, with the help of that we can manipulate and adjust the area of convergence of the series solution. Moreover, to authenticate the accuracy and reliability of the considered technique the numerical simulations have been presented. The retrieved results ensure that the projected scheme is effortless to carry out and analyze the highly nonlinear problems arising in science and technology.

In the present work, we find and analyze the approximated analytical solution for the vector-borne diseases model of fractional order with the help of q -homotopy analysis transform method ( q -HATM). Many novel definitions of fractional derivatives have been suggested and utilized in recent years to build mathematical models for a wide range of complex problems with nonlocal effects, memory, or history. The primary goal of this work is to create and assess a Caputo–Fabrizio fractional derivative model for Vector-borne diseases. In this investigation, we looked at a system of six equations that explain how vector-borne diseases evolve in a population and how they affect community public health. With the influence of the fixed-point theorem, we establish the existence and uniqueness of the models system of solutions. Conditions for the presence of the equilibrium point and its local asymptotic stability are derived. We discover novel approximate solutions that swiftly converge. Furthermore, the future technique includes auxiliary parameters that are both trustworthy and practical for managing the convergence of the solution found. The current study reveals that the investigated model is notably dependent on the time chronology and also the time instant, which can be effectively studied with the help of the arbitrary order calculus idea.

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