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This study presents the modified form of the homotopy perturbation method (HPM), and the Yang transform is adopted to simplify the solving process for the Kuramoto–Sivashinsky (KS) problem with fractal derivatives. This scheme is established by combining the two-scale fractal scheme and Yang transform, which is very helpful to evaluate the approximate solution of the fractal KS problem. Initially, we transfer the fractal problem into its partners using the two-scale fractal approach, and then we use the Yang transform ( T) to obtain the recurrent relation. Second, the HPM is then introduced to deal with the nonlinear elements of the fractal model. The numerical example demonstrates how the suggested technique is incredibly straightforward and precise for nonlinear fractal models. In addition, the graphical error of the proposed fractal model is compared with the calculated results of our suggested approach and the exact results. This graphical error displays the strength and authenticity of our proposed scheme. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to introduce a new scheme, known as the Aboodh homotopy integral transform method ( A HITM), for the approximate solution of wave problems in multi-dimensional orders. The Aboodh integral transform ( A IT) removes the restriction of variables in the recurrence relation, whereas the homotopy perturbation method (HPM) derives the successive iterations using the initial conditions. The convergence analysis is provided to study a wave equation with multiple dimensions. Some computational applications are considered to show the efficiency of this scheme. Graphical representation between the approximate and the exact solution predicts the high rate of convergence of this approach. [ABSTRACT FROM AUTHOR]

This article presents an idea of a new approach for the solitary wave solution of the modified Degasperis–Procesi (mDP) and modified Camassa–Holm (mCH) models with a time-fractional derivative. We combine Laplace transform (L T) and homotopy perturbation method (HPM) to formulate the idea of the Laplace transform homotopy perturbation method (L HPTM). This study is considered under the Caputo sense. This proposed strategy does not depend on any assumption and restriction of variables, such as in the classical perturbation method. Some numerical examples are demonstrated and their results are compared graphically in 2D and 3D distribution. This approach presents the iterations in the form of a series solutions. We also compute the absolute error to show the effective performance of this proposed scheme. [ABSTRACT FROM AUTHOR]

The symmetry design of the system contains integer partial differential equations and fractional-order partial differential equations with fractional derivative. In this paper, we develop a scheme to examine fractional-order shock wave equations and wave equations occurring in the motion of gases in the Caputo sense. This scheme is formulated using the Mohand transform (MT) and the homotopy perturbation method (HPM), altogether called Mohand homotopy perturbation transform (MHPT). Our main finding in this paper is the handling of the recurrence relation that produces the series solutions after only a few iterations. This approach presents the approximate and precise solutions in the form of convergent results with certain countable elements, without any discretization or slight perturbation theory. The numerical findings and solution graphs attained using the MHPT confirm that this approach is significant and reliable. [ABSTRACT FROM AUTHOR]

The algebras of the symmetry operators for the Klein–Gordon equation are important for a charged test particle, moving in an external electromagnetic field in a space time manifold on the isotropic hydrosulphate. In this paper, we develop an analytical and numerical approach for providing the solution to a class of linear and nonlinear fractional Klein–Gordon equations arising in classical relativistic and quantum mechanics. We study the Yang homotopy perturbation transform method (Y HPTM), which is associated with the Yang transform (Y T) and the homotopy perturbation method (HPM), where the fractional derivative is taken in a Caputo–Fabrizio (CF) sense. This technique provides the solution very accurately and efficiently in the form of a series with easily computable coefficients. The behavior of the approximate series solution for different fractional-order ℘ values has been shown graphically. Our numerical investigations indicate that Y HPTM is a simple and powerful mathematical tool to deal with the complexity of such problems. [ABSTRACT FROM AUTHOR]

In the present study, a new amendment in Laplace variational iteration method for the solution of fourth-order parabolic partial differential equations with variable coefficients is revealed i.e. modified Laplace variational iteration method (ML-VIM). The proposed modification is made by coupling of two methods: one of them is variational iteration method (VIM) and the other one is Laplace transformation (LT). Our modification has an important beauty that one has no need to compute Lagrange multiplier via integration nor by taking convolution theorem and has a simple way to use in the implementation of our proposed scheme. Moreover, homotopy perturbation method (HPM) with He's polynomials is employed for the computation of nonlinear terms. The main improvement of our proposed scheme is the reduction of the problem to a simple one, which also embraces for solving a nonlinear term. Some illustrated examples are interpreted which unveil the robustness and accuracy of the newly developed scheme. [ABSTRACT FROM AUTHOR]

This study proposes a new method, called the Fractal Yang transform method (F TM), for obtaining the fractal solution of the modified Camassa–Holm (mCH) and Degasperis–Procesi (mDP) models with fractal derivatives. The authors use the two-scale fractal approach to convert the fractal problem into its differential components and implement the Yang transform ( T) to achieve the recurrence iteration. We then apply the homotopy perturbation method (HPM) to overcome the difficulty of nonlinear elements in the recurrence iteration, which makes it simple to acquire further iterations. The most advantage of this fractal approach is that it has no restriction on variables and provides successive iterations. The fractal results are presented in the sense of a series that converges to the exact solution only after a few iteration. Graphical behavior demonstrates that this fractal approach is a very fast and remarkable solution, particularly with fractal derivatives. [ABSTRACT FROM AUTHOR]

This study examines the development of a novel approach known as the fractal Elzaki transform method (F TM) to investigate the approximation solution of the nonlinear fractal Drinfeld–Sokolov–Wilson (NFDSW) model. We adopt He's fractal derivative to change the fractal model into its differential parts and then apply the Elzaki transform to obtain the recurrence relation. We utilize the framework of homotopy perturbation method to handle the nonlinear components of this recurrence relation and thus we can obtain the successive iterations very easily. The derived findings are performed in the form of series and the rate of convergence shows the remarkable solutions due to its fast convergence. The numerical example illustrates that F TM is very easy to implement and a fascinating tool for fractal models. [ABSTRACT FROM AUTHOR]

This study develops a numerical strategy for finding the approximate solution of the nonlinear foam drainage (NFD) equation with a time-fractional derivative. In this paper, we formulate the idea of the Laplace homotopy perturbation transform method (LHPTM) using Laplace transform and the homotopy perturbation method. This approach is free from the heavy calculation of integration and the convolution theorem for the recurrence relation and obtains the solution in the form of a series. Two-dimensional and three-dimensional graphical models are described at various fractional orders. This paper puts forward a practical application to indicate the performance of the proposed method and reveals that all the outputs are in excellent agreement with the exact solutions. [ABSTRACT FROM AUTHOR]