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Rosenau‐Hyman equation was discovered as a simplified model to study the role of nonlinear dispersion on pattern formation in liquid drops. Also, this equation has important roles in the modelling of various problems in physics and engineering. The purpose of this paper is to present the solution of Rosenau‐Hyman equation.

This paper aims to present the solution of the Rosenau‐Hyman equation by means of semi‐analytical approaches which are based on the homotopy perturbation method (HPM), variational iteration method (VIM) and Adomian decomposition method (ADM).

These techniques reduce the volume of calculations by not requiring discretization of the variables, linearization or small perturbations. Numerical solutions obtained by these methods are compared with the exact solutions, revealing that the obtained solutions are of high accuracy. These results reveal that the proposed methods are very effective and simple to perform.

Efficient techniques are developed to find the solution of an important equation.

The soliton solutions are obtained by using extended rational sin/cos and sinh-cosh method. The methods are powerful and have ease of use. Applying wave transformation to the nonlinear partial differential equations (NLPDEs) and the considered equation turns into a nonlinear differential equation (NODE). According to the methods, the solution sets of the NODE are supposed to the form of the rational terms as sinh/cosh and sin/cos and the trial solutions are substituted into the NODE. Collecting the same power of the trigonometric functions, a set of algebraic equations is derived.

The main purpose of this paper is to obtain soliton solutions of the modified equal width (MEW) equation. MEW is a form of regularized-long-wave (RLW) equation that represents one-dimensional wave propagation in nonlinear media with dispersion processes. This is also used to simulate the undular bore in a long shallow water canal.

Thus, the solution of the main PDE is reduced to the solution of a set of algebraic equations. In this paper, the kink, singular and singular periodic solitons have been successfully obtained.

Illustrative plots of the solutions have been presented for physical interpretation of the obtained solutions. The methods are powerful and might be used to solve a broad class of differential equations in real-life problems.

The current paper aims to develop a reduced order discontinuous Galerkin method for solving the generalized Swift–Hohenberg equation with application in biological science and mechanical engineering. The generalized Swift–Hohenberg equation is a fourth-order PDE; thus, this paper uses the local discontinuous Galerkin (LDG) method for it.

At first, the spatial direction has been discretized by the LDG technique, as this process results in a nonlinear system of equations based on the time variable. Thus, to achieve more accurate outcomes, this paper uses an exponential time differencing scheme for solving the obtained system of ordinary differential equations. Finally, to decrease the used CPU time, this study combines the proper orthogonal decomposition approach with the LDG method and obtains a reduced order LDG method. The circular and rectangular computational domains have been selected to solve the generalized Swift–Hohenberg equation. Furthermore, the energy stability for the semi-discrete LDG scheme has been discussed.

The results show that the new numerical procedure has not only suitable and acceptable accuracy but also less computational cost compared to the local DG without the proper orthogonal decomposition (POD) approach.

The local DG technique is an efficient numerical procedure for solving models in the fluid flow. The current paper combines the POD approach and the local LDG technique to solve the generalized Swift–Hohenberg equation with application in the fluid mechanics. In the new technique, the computational cost and the used CPU time of the local DG have been reduced.

This study aims to apply a numerical meshless method, namely, the boundary knot method (BKM) combined with the meshless analog equation method (MAEM) in space and use a semi-implicit scheme in time for finding a new numerical solution of the advection–reaction–diffusion and reaction–diffusion systems in two-dimensional spaces, which arise in biology.

First, the BKM is applied to approximate the spatial variables of the studied mathematical models. Then, this study derives fully discrete scheme of the studied models using a semi-implicit scheme based on Crank–Nicolson idea, which gives a linear system of algebraic equations with a non-square matrix per time step that is solved by the singular value decomposition. The proposed approach approximates the solution of a given partial differential equation using particular and homogeneous solutions and without considering the fundamental solutions of the proposed equations.

This study reports some numerical simulations for showing the ability of the presented technique in solving the studied mathematical models arising in biology. The obtained results by the developed numerical scheme are in good agreement with the results reported in the literature. Besides, a simulation of the proposed model is done on buttery shape domain in two-dimensional space.

This study develops the BKM combined with MAEM for solving the coupled systems of (advection) reaction–diffusion equations in two-dimensional spaces. Besides, it does not need the fundamental solution of the mathematical models studied here, which omits any difficulties.

The purpose of this paper is concerned with a reliable treatment of the classical Stephan problem. The Adomian decomposition method (ADM) is used to carry out the analysis, Moreover, the authors extend the work to examine the Stefan problem with variable latent heat. The study confirms the accuracy and efficiency of the employed method.

The new technique, as presented in this paper in extending the applicability of the ADM, has been shown to be very efficient for solving the Stefan problem.

The Stefan problem with variable latent heat was examined as well. The ADM was effectively used for analytic treatment of the Stefan problem with and without variable latent heat.

The paper presents a new solution algorithm for the Stefan problem.

This paper aims to discuss the approximate solution of the nonlinear thin film flow problems. A new analytic approximate technique for addressing nonlinear problems, namely, the optimal homotopy asymptotic method (OHAM), is proposed and used in an application to the nonlinear thin film flow problems.

This approach does not depend upon any small/large parameters. This method provides a convenient way to control the convergence of approximation series and to adjust convergence regions when necessary.

The obtained solutions show that the OHAM is more effective, simpler and easier than other methods. The results reveal that the method is explicit. By applying the method to nonlinear thin film flow problems, it was found to be simpler in applicability, and more convenient to control convergence. Therefore, the method shows its validity and great potential for the solution of nonlinear problems in science and engineering.

The proposed method is tested upon nonlinear thin film flow equation from the literature and the results are compared with the available approximate solutions including Adomian decomposition method (ADM), homotopy perturbation method, modified homotopy perturbation method and HAM. Moreover, the exact solution is compared with the available numerical solutions. The graphical representation of the solution is given by Maple and is physically interpreted.

The purpose of this paper is to analyze the heat transfer effects on the stretched flow of Oldroyd-B fluid in a rotating frame. Cattaneo–Christov heat conduction model is considered, which accounts for the influence of thermal relaxation time.

Based on scale analysis, the usual boundary layer approximations are used to simplify the governing equations. The equations so formed have been reduced to self-similar forms by similarity transformations. A powerful analytic approach, namely, homotopy analysis method (HAM), has been applied to present uniformly convergent solutions for velocity and temperature profiles.

Suitable values of the so-called auxiliary parameter in HAM are obtained by plotting h-curves. The results show that boundary layer thickness has an inverse relation with fluid relaxation time. The rotation parameter gives resistance to the momentum transport and enhances fluid temperature. Thermal boundary layer becomes thinner when larger values of thermal relaxation time are chosen.

To the authors’ knowledge, this is the first attempt to study the three-dimensional rotating flow and heat transfer of Oldroyd-B fluid.