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The dynamic stability of orthotropic cylindrical thin shells of exponentially variable geometric and mechanical parameters is studied by using Galerkin and Ritz type variotional methods. The qualitative and quantitative effects of the external geometry, material properties, and design features on the critical loads, corresponding wave numbers, and the dynamic factor are evaluated. Comparing results with those in the literature validates the present analysis.

The purpose of this paper is to show how an application of fractional two dimensional differential transformation method (DTM) obtained approximate analytical solution of time‐fraction modified equal width wave (MEW) equation.

The fractional derivative is described in the Caputo sense.

It is indicated that the solutions obtained by the two dimensional DTM are reliable and that this is an effective method for strongly nonlinear partial equations.

The paper shows that exact solutions can also be obtained from the known forms of the series solutions.

This paper aims to suggest a novel modified Laplace decomposition method (MLDM) for MHD flow over a non-linear stretching sheet with slip condition by suitable choice of an initial solution.

The governing partial differential equations are converted into dimensionless non-linear ordinary differential equation by similarity transformation, which is solved by MLDM. The method is based on the application of Laplace transform to boundary layers in fluid mechanics. The non-linear term can be easily handled by the use of He's polynomials.

The series solution of the MHD flow of an incompressible viscous fluid over a non-linear stretching sheet subject to slip condition is obtained. An excellent agreement between the MLDM and HPM is achieved. Convergence of the obtained series solution is properly checked by using the ratio test.

Stretching surface is an important type of flow occurring in a number of engineering processes such as heat-treated materials travelling between a feed roll and a wind up roll, aerodynamic extrusion of plastic sheets, glass fiber and paper production, cooling of an infinite metallic plate in a cooling path, manufacturing of polymeric sheets are few examples of flow due to stretching surfaces. This work provides a very useful source of information for researchers on this subject.

Such flow analysis is even not available yet for the hydrodynamic fluid. The series solution for MHD boundary layer problem with slip condition by means of MLDM is yet not available in the literature.

The purpose of this paper is to develop a numerical method based on quintic B‐spline to solve the linear and nonlinear Fredholm and Volterra integral equations.

The solution is collocated by quintic B‐spline and then the integral equation is approximated by the Gauss‐Kronrod‐Legendre quadrature formula.

The arising system of linear or nonlinear algebraic equations can solve the linear combination coefficients appearing in the representation of the solution in spline basic functions.

The error analysis of proposed numerical method is studied theoretically. Numerical results are given to illustrate the efficiency of the proposed method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.

The paper provides new method to solve the linear and nonlinear Fredholm and Volterra integral equations.

The purpose of this paper is to apply exp‐function method to construct generalized solitary and periodic solutions of Fitzhugh‐Nagumo equation, which plays a very important role in mathematical physics and engineering sciences.

The authors apply exp‐function method to construct generalized solitary and periodic solutions of Fitzhugh‐Nagumo equation.

Numerical results clearly indicate the reliability and efficiency of the proposed exp‐function method. The suggested algorithm is quite efficient and is practically well suited for use in these problems.

In this paper, the authors applied the exp‐function method to obtain solutions of the Fitzhugh‐Nagumo equation and show that the exp‐function method gives more realistic solutions without disturbing the basic physics of the physical problems.

The purpose of this paper is to apply the variational iterations method to solve two difference types such as the modified Boussinesq (MB) and seven‐order Sawada‐Kotara (sSK) equations and to compare this method with that obtained previously by Adomian decomposition.

The variational iteration method is used for finding the solution of the MB and sSK equations. The solution obtained is an infinite power series for appropriate initial condition. The numerical results obtain for nth approximation and compare with the known analytical solutions; the results show that an excellent approximation to the actual solution of the equations was achieved by using only three iterations.

The comparison demonstrates that the two obtained solutions are an excellent agreement. The numerical results calculated show that this method, variational iteration method, can be readily implemented to this type of nonlinear equation and excellent accuracy can be achieved. The results of variation iteration method confirm the correctness of those obtained by means of Adomian decomposition method.

The results presented in this paper show that the variational iteration method is a powerful mathematical tool for solving the MB and the sSK equations; it is also a promising method for solving other nonlinear equations.

The purpose of this paper is to discuss the integrability studies to the long-short wave equation that is studied in the context of shallow water waves. There are several integration tools that are applied to obtain the soliton and other solutions to the equation. The integration techniques are traveling waves, exp-function method, G′/G-expansion method and several others.

The design of the paper is structured with an introduction to the model. First the traveling wave hypothesis approach leads to the waves of permanent form. This eventually leads to the formulation of other approaches that conforms to the expected results.

The findings are a spectrum of solutions that lead to the clearer understanding of the physical phenomena of long-short waves. There are several constraint conditions that fall out naturally from the solutions. These poses the restrictions for the existence of the soliton solutions.

The results are new and are sharp with Lie symmetry analysis and other advanced integration techniques in place. These lead to the connection between these integration approaches.

The purpose of this paper is to apply the exp‐function method to construct exact solutions of nonlinear wave equations. The proposed technique is tested on the (2+1) and (3+1) dimensional extended shallow water wave equations. These equations play a very important role in mathematical physics and engineering sciences.

In this paper, the authors apply the exp‐function method to construct exact solutions of nonlinear wave equations.

In total, four forms of the extended shallow water wave equation have been studied, from the point of view of its exact solutions using computational method. Exp‐function method was employed to achieve the goal set for this work. The applied method will be used in further works to establish more entirely new solutions for other kinds of nonlinear wave equations. Finally, it is worthwhile to mention that the proposed method is straightforward, concise, and it is a promising and powerful new method for other nonlinear wave equations in mathematical physics.

The algorithm suggested in the paper is quite efficient and is practically well suited for use in these problems. The method is straightforward and concise, and its applications are promising.

The purpose of this paper is to use He's Exp‐function method (EFM) to construct solitary and soliton solutions of the nonlinear evolution equation.

This technique is straightforward and simple to use and is a powerful method to overcome some difficulties in the nonlinear problems.

This method is developed for searching exact traveling wave solutions of the nonlinear partial differential equations. The EFM presents a wider applicability for handling nonlinear wave equations.

The paper shows that EFM, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations. Application of EFM to Fitzhugh‐Nagumo equation illustrates its effectiveness.