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The fundamental objective of this work is focused to achieve a class of advanced and impressive exact estimations to the Zoomeron equation and the time-fraction biological population model through contrivance by a couple of important and magnificent techniques, namely, the modified extended tanh-function method which depend on the balance theory and the Ricatti–Bernoulli sub-ODE method which is independent of the balancing principle. The suggested model is one of the major concerns for studying population distribution dynamics as well as the quantum field theory which is an important discipline for the description of interactions between light and electrons. The two suggested reliable, effective techniques are considered famous among ths ansatz methods that have various visions to realize the exact solutions to the non linear partial differential equation that reduce the volume of calculations examined before and usually give good results. It is solicited for this contrivance finding new exact solutions for two models in terms of some variable. The models are significant in quantum field theory, description of interactions between light and electron, quantum electrodynamics, demographic model, important to bring it into line with the reasonable distribution of wealth, resources, income, etc. The achieved results predict many types of solutions as trigonometric functions, hyperbolic functions, perfect periodic soliton solutions, singular periodic soliton solutions, and other rational solitons solutions. The efficiency of the techniques is demonstrated by the satisfactory results obtained through the derivation of closed-form soliton solutions from the exact solution by assigning definite values to the variables present in it.

In this article, we employ the Black Scholes model which plays a vital role in economic operation and financial market management. The Paul-Painlevé approach is used for the first time to achieve the exact wave solution to this equation. Furthermore, the numerical solution to this equation has been constructed by using the variational iteration method. [ABSTRACT FROM AUTHOR]

In this article, new variety types of exact solution to the Fujimoto- Watanable- equation (FWE) that equivalent to the modified Korteweg- de Vries- equation have been derived. These new types of solutions which weren't realized before by any other technique have been established in the framework of the Ricatti-Bernolli Sub-ODE method (RBSODM). Also, the identical numerical solutions whose initial conditions are emerged from the achieved exact solutions have been constructed by using the famous numerical variational iteration method (VIM). [ABSTRACT FROM AUTHOR]

In this article, we will implement the(G'=G)-expansion method which is used for the first time to obtain new optical soliton solutions of the thin-film ferroelectric materials equation (TFFME). Also, the numerical solutions of the suggested equation according to the variational iteration method (VIM) are demonstrated effectively. A comparison between the achieved exact and numerical solutions has been established successfully. [ABSTRACT FROM AUTHOR]

In this paper, new exact traveling wave solutions for the coupling BoitiLeon-Pempinelli system are obtained by using two important different methods. The first is the modified extended tanh function methods which depend on the balance rule and the second is the Ricatti-Bernoulli Sub-ODE method which doesn’t depend on the balance rule. The solitary waves solutions can be derived from the exact wave solutions by give the parameters a special value. The consistent and inconsistent of the obtained solutions are studied not only between these two methods but also with that relisted by the other methods. [ABSTRACT FROM AUTHOR]

In this article, we employed the 3D-fractional Wazwaz-Benjamin-Bona- Mahony (3D-FWBBM) equation with its spatial and temporal variables which is stretch for the Korteweg-de-Vries equation that represent the unidirectional propagation of small amplitude long waves on the surface of hydro magnetic and acoustic waves in channel specially for shallow water. New exact soliton solution has been realized using the (G' / G) -- expansion method. Furthermore, the numerical solution of the suggested equation according to the variational iteration method (VIM) is listed effectively. A good comparison between the obtained exact and numerical solution are successfully demonstrated. [ABSTRACT FROM AUTHOR]