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In this manuscript, we focus on the application of recently developed analytical scheme, namely, the rational sine-Gordon expansion method (SGEM). Some new exact solutions of Riemann wave system and the nonlinear Gross−Pitaevskii equation (GPE) by using this method are extracted. This method is based on the general properties of the SGEM which uses the fundamental properties of trigonometric functions. Many novel analytical solutions such as dark, bright, mixed dark–bright, hyperbolic, and periodic wave solutions are successfully extracted. Physical meanings of solutions are simulated by the various figures in 2D and 3D along with the contour graphs. Strain conditions of the existence are also reported in detail. Finally, modulation instability analysis of the nonlinear GPE is studied in detail. [ABSTRACT FROM AUTHOR]

In this paper, we study the extraction of some analytical solutions to the nonlinear perturbed sine-Gordon equation with the long Josephson junction properties. The model studied was formed to observe the long Josephson junction properties separated by two superconductors. Moreover, it is also used to explain the Josephson effect arising in the highly nonlinear nature of the Josephson junctions. This provides the shunt inductances to realize a Josephson left-handed transmission line. A powerful scheme is used to extract the complex function solutions. These complex results are used to explain deeper properties of Josephson effects in the frame of impedance. Various simulations of solutions obtained in this paper are also reported. [ABSTRACT FROM AUTHOR]

This paper constructs approximate solutions of the epidemic system of HIV/AIDS transmission model with the help of the conformable fractional differential transform method. The fractional derivatives considered in this paper are described in the sense of the conformable operator. This method provides the approximate solutions speedily in the form of a convergent series. The results show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The stomach is usually considered as a hollow muscular sac, which initiates the second segment of digestion. It is the most sophisticated endocrine structure having unique biochemistry, physiology, microbiology, and immunology. The pivotal aim of the present study is to propose the nonlinear mathematical model of the nervous stomach system based on three compartments namely, tension (T), food (F), and medicine (M). The detailed description of each compartment is provided along with the mathematical form and different rates/factors, such as sleep factor, food rate, tension rate, medicine term, and death rate. The solution of the designed model is presented numerically by using the well-known differential transformation technique. The behavior of the obtained solution has been captured with respect to time as well as presentations of the numerical simulations. [ABSTRACT FROM AUTHOR]

The analytical solution of the longitudinal wave equation in the MEE circular rod is analyzed by the powerful sine-Gordon expansion method. Sine - Gordon expansion is based on the well-known wave transformation and sine - Gordon equation. In the longitudinal wave equation in mathematical physics, the transverse Poisson MEE circular rod is caused by the dispersion. Some solutions with complex, hyperbolic and trigonometric functions have been successfully implemented. Numerical simulations of all solutions are given by selecting the appropriate parameter values. The physical meaning of the analytical solution explaining some practical physical problems is given. [ABSTRACT FROM AUTHOR]

An editorial is presented on mathematical models in the form of numerical solutions, analytical solutions, approximate solutions, and periodic properties. Topics include Sumudu decomposition method (SDM) finding approximate solutions to two-dimensional fractional partial differential equations; and applying an integral transform-based technique for solving a fractional order Volterra-type integro-differential equation (FVIDE).

The pivotal aim of the present work is to obtain an approximated analytical solution for the fractional smoking epidemic model with the aid of a novel technique called q-homotopy analysis transform method (q-HATM). The considered nonlinear mathematical model has been effectively employed to elucidate the evolution of smoking in a population and its impact on public health in a community. We find some new approximate solutions in a series form, which converges rapidly, and the proposed algorithm provides auxiliary parameters, which are very reliable and feasible in controlling the convergence of obtained approximate solutions. Further, we present novel simulations for all cases of results to validate the applicability and effectiveness of proposed scheme. The outcomes of the study reveal that the q-HATM is computationally very effective to analyse nonlinear fractional differential equations arises in daily life problems. [ABSTRACT FROM AUTHOR]

In this study, we have applied the modified exp(-Ω(ξ))-expansion function method to the (2 + 1)-dimensional Boussinesq water equation. We have obtained some new analytical solutions such as exponential function, complex function and hyperbolic function solutions. It has been observed that all analytical solutions have been verified to the (2 + 1)-dimensional Boussinesq water equation by using Wolfram Mathematica 9. We have constructed the two- and three-dimensional surfaces for all analytical solutions obtained in this paper using the same computer program. [ABSTRACT FROM AUTHOR]

In this study, we investigate some new analytical solutions to the (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin-Bona-Mahony equation and the (2 + 1)-dimensional cubic Klein-Gordon equation by using the generalized Kudryashov method. After we submitted the general properties of the generalized Kudryashov method in Section 2, we applied this method to these problems to obtain some new analytical solutions, such as rational function solutions, exponential function solutions and hyperbolic function solutions in Section 3. Afterwards, we draw two- and three-dimensional surfaces of analytical solutions by using Wolfram Mathematica 9. [ABSTRACT FROM AUTHOR]

In the present investigation, the q-homotopy analysis transform method (q-HATM) is applied to find approximated analytical solution for the system of fractional differential equations describing the unsteady flow of a polytropic gas. Numerical simulation has been conducted to prove that the proposed technique is reliable and accurate, and the outcomes are revealed using plots and tables. The comparison between the obtained solutions and the exact solutions shows that the proposed method is efficient and effective in solving nonlinear complex problems. Moreover, the proposed algorithm controls and manipulates the obtained series solution in a huge acceptable region in an extreme manner and it provides us a simple procedure to control and adjust the convergence region of the series solution. [ABSTRACT FROM AUTHOR]

The modified exp - Ω ζ -expansion function method is applied to the Wu–Zhang system with conformable time-fractional derivative to construct new analytical solutions. We have obtained some soliton-type solutions such as dark, singular and combined soliton solutions. We have seen that all solutions have provided the mentioned equation system. For the suitable value of the solutions, the 2D–3D and contour surfaces have been plotted. [ABSTRACT FROM AUTHOR]