This article presents a new iterative method (NIM) for the investigation of the approximate solution of the Klein-Gordon and sine-Gordon equations. This approach is formulated on the combination of the Mohand transform and the homotopy perturbation method. Mohand transform (MT) is capable to handle the linear terms only, thus we introduce homotopy perturbation method (HPM) to tackle the nonlinear terms. This NIM derives the results in the form of a series solution. The proposed method emphasizes the stability of the derived solutions without any linearization, discretization, or hypothesis. Graphical representation and absolute error demonstrate the efficiency and authenticity of this scheme. Some numerical models are illustrated to show the compactness and reliability of this strategy.

Through this article, we focus on the extension of travelling wave solutions for a prevalent nonlinear pseudo-parabolic physical Oskolkov model for Kevin-Voigt fluids by using two integral techniques. First of all, we explore the bifurcation and phase portraits of the model for different parametric conditions via a dynamical system approach. We derive smooth waves of the bright bell and dark bell, periodic waves, and singular waves of dark and bright cusps, in correspondence to homoclinic, periodic, and open orbits with cusp, respectively. Each orbit of the phase portraits is envisaged through various energy states. Secondly, with the help of a prevalent unified scheme, an inventive version of exact analytic solutions comprising hyperbolic, trigonometric, and rational functions can be invented with some collective parameters. The unified scheme is an excitably auspicious method to procure novel interacting travelling wave solutions and to obtain multipeaked bright and dark solitons, shock waves, bright bell waves with single and double shocks, combo waves of the bright-dark bell and dark-bright bell with a shock, dark bell into a double shock wave, and bright-dark multirogue type wave solutions of the model. The dynamics of the procured nonlinear wave solutions are also presented through 2-D, 3-D, and density plots with specified parameters.

In this paper, we investigate the numerical solution of the coupled fractional massive Thirring equation with the aid of He’s fractional complex transform (FCT). This study plays a significant aspect in the field of quantum physics, weakly nonlinear thrilling waves, and nonlinear optics. The main advantage of FCT is that it converts the fractional differential equation into its traditional parts and is also capable to handle the fractional order, whereas the homotopy perturbation method (HPM) is employed to tackle the nonlinear terms in the coupled fractional massive Thirring equation. An example is illustrated to present the efficiency and validity of the two‐scale theory. The solutions are obtained in the form of series with simple and easy computations which confirm that the present approach is good in agreement and is easy to implement for such type of complex systems in science and engineering.

In nature, symmetry is all around us. The symmetry framework represents integer partial differential equations and their fractional order in the sense of Caputo derivatives. This article suggests a semi-analytical approach based on Aboodh transform ($\mathbb{A}$T) and the homotopy perturbation scheme (HPS) for achieving the approximate solution of time-fractional porous media and heat transfer equations. The $\mathbb{A}$T converts the fractional problems into simple ones and obtains the recurrence relation without any discretization or assumption. This nonlinear recurrence relation can be decomposed via the use of the HPS to obtain the iterations in terms of series solutions. The initial conditions play an important role in determining the successive iterations and yields towards the exact solution. We provide some numerical applications to analyze the accuracy of this proposed scheme and show that the performance of our scheme has strong agreement with the exact results.

In this analysis, we develop a new approach to investigate the semianalytical solution of the delay differential equations. Mohand transform coupled with the homotopy perturbation method is called Mohand homotopy perturbation transform method (MHPTM) and performs the solution results in the form of series. The beauty of this approach is that it does not need to compute the values of the Lagrange multiplier as in the variational iteration method, and also, there is no need to implement the convolution theorem as in the Laplace transform. The main purpose of this scheme is to reduce the less computational work and the error analysis of the problems than others studied in the literature. Some illustrated examples are interpreted to confirm the accuracy of the newly developed scheme.