By modifying the extended mapping method, we get traveling wave solutions of nonlinear complex physical models, arising in various fields of applied sciences. This method is applied to the (1 + 1)-dimensional Hirota equation. Consequently, different kinds of family of exact traveling wave solutions are fruitfully surveyed. The obtained novel exact traveling wave solutions represent in different forms such as bright and dark solitary wave, periodic solitary wave and dark and bright soliton. These solutions are represented in the form of trigonometric, hyperbolic, exponential and rational functions. The properties of some of the novel traveling wave solutions are shown in figures. The obtained results exhibit the effectiveness, power and exactness of the method that can be used for many other nonlinear problems.

The propagations are generally described through nonlinear Schrödinger equation (NLSE) in the optical solitons. In the NLSEs, the higher order NLSE with derivative non-Kerr nonlinear terms is a model that depicts propagation of pulses beyond ultra-short range in optical communication system. Several novel exact solutions of different kinds such as solitons, solitary waves and Jacobi elliptic function solutions are achieved via using modified extended mapping technique. Different kinds of exact results have prestigious exertions in engineering and physics. Structures of solitons different kinds are shown graphically by giving suitable values to parameters. The physical interpretations of solutions can be understand through structures. Several exact solutions and computing work confirm the supremacy and usefulness of the current technique.

The viral infection of hepatitis B virus (HBV) is a dangerous problem for health around the globe and counted in the top leading causes of death. To explore the viral dynamics of this infection, an HBV epidemic model has been developed by dividing the infected compartment into three subclasses, acute, chronically infected and carrier individuals with both vertical as well as horizontal transmission. After formulating the model, we prove that the positive solution of the model exists. The next generation matrix approach has been used to investigate the threshold quantity known as basic reproduction number. The global stability conditions at endemic equilibria (EE) and disease-free equilibrium (DFE) are established by using the method of geometrical approach and Castillo-Chavez, respectively. We use the optimization theory and the three time-dependent control variables to establish the control program. By the help of this control policy, we reduce the number of susceptible, acute, chronically hepatitis B infected and carrier persons, while the numbers of recovered and vaccinated populations are maximized. Finally, numerical results will be found out for the support and feasibility of the analytical results.

Exact traveling wave solutions of the fifth order space-time fractional Sawada-Kotera equation are derived by generalised exp(-Phi(xi))-expansion and an improved fractional sub-equation method. Among the solutions obtained there are hyperbolic, trigonometric, exponential and rational ones. The methods are simple, efficient and can be applied to other nonlinear problems.

In the present paper, we utilized fractional reduced differential transform method (FRDTM) to get the better optimal solution of space-time fractional order heat-like and wave-like partial differential equations (PDEs). One of the distinguishing practical features of this technique is its implementation on space-time fractional order linear and nonlinear PDEs without using discretization and linearization. The advantage of this method to other existing methods such as HAM, HPM etc. is that no prescribed assumptions are required, and the huge computational exertion is reduced and round-off errors are also evaded. To show the capability, usefulness and consistency of the proposed scheme, a few examples have been given to enlightening its efficiency, accuracy and simplicity. Hence, it is a powerful method for solving general space-time fractional order PDEs used in the field of engineering, physics and other disciplines.