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This review is an elementary introduction to the concepts of the recently developed asymptotic methods and new developments. Particular attention is paid throughout the paper to giving an intuitive grasp for Lagrange multiplier, calculus of variations, optimization, variational iteration method, parameter-expansion method, exp-function method, homotopy perturbation method, and ancient Chinese mathematics as well. Subsequently, nanomechanics in textile engineering and E-infinity theory in high energy physics, Kleiber's 3/4 law in biology, possible mechanism in spider-spinning process and fractal approach to carbon nanotube are briefly introduced. Bubble-electrospinning for mass production of nanofibers is illustrated. There are in total more than 280 references.

The amplitude of nonlinear excitations in BECs with inhomogeneities is governed by a generalized variable-coefficient Korteweg–de Vries model. With symbolic computation, the Exp-function method is modified to obtain analytical nontraveling solitary-wave and periodic-wave solutions. Through the qualitative analysis and graphical illustration, the inhomogeneous propagation features of solitary waves are discussed, and some observable effects for BEC dynamic in the presence of external potentials are provided. The modified Exp-function method is also applicable to other variable-coefficient nonlinear evolution equations.

Based on the rational solutions to a generalized Riccati equation, a new method which is called as rational function method is proposed. We apply this method to solve the coupled mKdV equations and derive a set of rational solutions. This method is also available for seeking their solutions to other NLPDEs. It shows that the rational function method is more universal and powerful than the auxiliary Riccati equation method.

In this work, a new method has been applied for finding solutions to the Kundu–Mukherjee–Naskar (KMN) equation in (2+1) dimensions. These optical solutions were derived with the aid of a new method named the Exp-Function method. The method is used for solving the nonlinear governing equation, which contains high nonlinear term. This method is considered as an alternative technique for obtaining both analytical and approximate solutions for a different type of PDE with several applications in science and engineering. The efficiency of the proposed technique has been shown by a few test problems. By comparing the results obtained in this paper with other researchers it can be noticed that this method is a promising technique for finding solutions to other similar problems.

The modeling of nonlinear dynamical models and their solutions have imperative values, whereas the soliton mathematical methods are favorable tools for analyzing such highly nonlinear practical problems. Herein, time-fractional coupled Burger’s model is analyzed. Some solitary wave solutions, containing one-soliton, two-soliton, three-soliton and $N$-soliton, have been fetched using a modified sort of the well-known exp-function approach. Dominant effects of the fractional parameters have been illustrated and it is noted that the higher values of the fractional number cause a significant impact on the wave propagation. The one wave pattern for the $u$-component is quite identical, while an increase in the fractional parameter is producing an increment in the wave structure while a similar but opposite pattern is noted for the $v$-component. An increment in the wave structure is noted for higher values of the time-fractional parameter for both $u$- and $v$-components in the case of a two-wave solution. The benefits of the proposed computational code of the exp-function scheme are consistent, a general form of exact solutions, reduced computational complexity and stable and efficiently useful. The obtained solitary waves solutions will be a useful extension to the preceding results and greatly assist in clarifying the features of nonlinear waves in viscous flow.

In this study, we apply relatively analytical techniques, the multiple $exp$-function method, $exp$-function method and $\frac{G^{\prime }}{G^2}$-expansion method to get approximate and analytic solutions of some nonlinear partial differential equations (PDEs), i.e., the nonlinear space–time fractional partial differential symmetric regularized long wave equation, an impressive model to characterize ion-acoustic and space change waves, the nonlinear $(4+1)$-dimensional Fokas PDE, a meaningful multi-dimensional extension of the Kadomtsev–Petviashvili (KP) and Davey–Stewartson (DS) equations, $(1+1)$-dimensional Bateman–Burgers equation, a simplification of a more complex and sophisticated model, and the $(1+1)$-dimensional Benjamin–Ono equation, a model for the propagation of unidirectional internal waves in stratified fluids. Finally, we propose the numerical results in tables and discuss advantages and disadvantages of the mentioned methods.

In this paper, we explore how to generate solitary, peakon, periodic, cuspon and kink wave solution of the well-known partial differential equation Korteweg–de Vries (KdV) by using exp-function and modified exp-function methods. The presented methods construct more efficiently almost all types of soliton solution of KdV equation that can be rarely seen in the history. These methods appear to be straightforward and symbolic calculations are used to solve the problem. All resulting answers are verified for accuracy using the symbolic computation program with $Maple$. To show the physical appearance of the model, 3D plots of all the generated solutions are then displayed. The obtained solutions revealed the compatibility of the proposed techniques which provide the general solution with some free parameters. This is the key benefit of these methods over the other methods.

The exact traveling wave solutions of (4 + 1)-dimensional nonlinear Fokas equation is obtained by using three distinct methods with symbolic computation. The modified tanh–coth method is implemented to obtain single soliton solutions whereas the extended Jacobi elliptic function method is applied to derive doubly periodic wave solutions for this higher-dimensional integrable equation. The Exp-function method gives generalized wave solutions with some free parameters. It is shown that soliton solutions and triangular solutions can be established as the limits of the Jacobi doubly periodic wave solutions.

This paper, studies the long-short-wave interaction (LS) equation. An optical soliton solution is obtained by the exp-function method and the ansatz method. Subsequently, we formally derive the dark (topological) soliton solutions for this equation. By using the exp-function method, some additional solutions will be derived. The physical parameters in the soliton solutions of ansatz method: amplitude, inverse width, and velocity are obtained as functions of the dependent model coefficients.

In this paper, we investigate the solitary wave solutions for the two-dimensional modified Korteweg–de Vries–Burgers (mKdV-B) equation in shallow water model. Despite that Painlevé test fails to prove the integrability of the mKdV-B equation by using the WTC-Kruskal algorithm, the Bäcklund transformation is obtained via the truncation expansion. The exact solutions of the mKdV-B equation are found using factorization techniques, Exp-function and energy integral approach of the corresponding ordinary differential equation. We found that the dispersion relation of the linearized mKdV-B equation lies on the complex plane yielding a damping character. By keeping the water height relatively small, we illustrate the resulting solutions in several figures showing the shock and solitary wave nature in the flow. The stability for the mKdV-B equation is analyzed by using the phase plane method.

In the last decades Exp-function method has been used for solving fractional differential equations. In this paper, we obtain exact solutions of fractional generalized reaction Duffing model and nonlinear fractional diffusion–reaction equation. The fractional derivatives are described in the modified Riemann–Liouville sense. The fractional complex transform has been suggested to convert fractional-order differential equations with modified Riemann–Liouville derivatives into integer-order differential equations, and the reduced equations can be solved by symbolic computation.

This paper is the spectator of the arrangement of an efficient transformation and exp-function technique to build up generalized exact solutions of the biological population model equation. Computational work and subsequent numerical results re-identify the effectiveness of proposed algorithm. It is pragmatic that recommended plan is greatly consistent and may be comprehensive to other nonlinear differential equations of fractional order.

In this paper, a kind of analytical technique, called Exp-Function method is implemented to find the solitary wave solution of the Kuramoto-Sivashinsky equation and some of the most useful equations in physics, the Camassa-Holm equation and the reduced Ostrovsky equation (ROE). This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations applied in engineering mathematics. Some numerical examples are presented to illustrate the efficiency and reliability of the Exp-Function method. It is predicted that Exp-Function method can be found widely applicable in engineering.