This paper is devoted to investigate and computing the homotopy perturbation method and homotopy perturbation transform method for solving the cubic quintic nonlinear Schrodinger equation. both proposed methods are iterative schemes to find solutions without discretization, or restrictive assumption. The solutions obtained in this research are approximate analytical series solutions to the cubic-quintic nonlinear schrodinger equation. Using the homotopy perturbation method and the homotopy perturbation transform method. These methods generate rapidly converging series that approximate the behavior of the wave function over time. In addition, the current results have been shown graphically and ina table. the results demonstrate that the homotopy perturbation transform method provides more accurate and efficient solutions than the standard homotopy perturbation method.

This paper is devoted to investigating and comparing the variational iteration method(VIM) and the residual power series method(RPSM) for solving the cubic-quintic nolinear Schrodinger equation (CQNLSE) initially developed to elucidate the propagation of pulses in 0ptical fibers. Next, we use the inital conditions to get the numerical solutions of the CQNLSE. We compared the known exact solutions with the approximate results obtained using both the VIM and RPSM. The exact solution of the results from RPSM are evaluated against those from VIM. The findings demonstrated that VIM outperformed RPSM in terms of accuracy, efficiency, and ease of implementation for solving the CQNLSE. In addition, the current results are shown graphically and in the table.

In this paper, we propose two new techniques for solving system of nonlinear partial differential equations numerically, which we first combine Laplace transformation method into a successive approximation method. Second, we combine Padé [2,2] technique into the first proposed technique. To test the efficiency of our techniques, Jaulent-Miodek system was used, which contains partial differential equations and has strongly nonlinear terms. Experimental results revealed that the first proposed technique gives better results when the interval of t is small in terms of error approximation in tabular and graphical manners. Moreover, the results also demonstrated that the second proposed technique gives better results regardless of the given interval of t in terms of the least square errors.

In this work, the residual power series method (RPSM) is used to find the approximate solution of Klien-Gordon Schrodinger (KGS) equation. furthermore, to show the accuracy and the efficiency of the presented method, we compare the obtained approximate solution of Klien-Gordon Schrodinger equation by residual power series method (RPSM) numerically and graphically with exact solution.

This paper is devoted to investigating and computing the successive approximation method (SAM) and variational iteration method (VIM) for solving Klein-Gordon Schrodinger equation (KGS) equation. Furthermore, the approximation solutions that obtained by both methods have been represented numerically and graphically.

This paper is devoted to investigating and comparing the Successive Approximations Method (SAM) and Variational Iteration Method (VIM) for solving Klein-Gordon Schrödinger (KGS) Equation. Furthermore, the approximate solutions that obtained by both methods have been represented numerically and graphically.

In this paper, Modified Variational Iteration Method (MVIM) and Homotopy Analysis Method (HAM) are used to find approximate solutions for the Variable-Coefficient Variant Boussinesq System the (VCVB) system is able to describe the nonlinear and dispersive long gravity waves in shallow water traveling in two horizontal directions with varying depth, as an example we took the Boussinesq-Burgers (B-B) system, (B-B) system arise in the study of fluid flow and describing the long-wave propagation of shallow water waves. The solutions of these equations helpful for the coastal and civil engineering’s

In this paper, we study the (1+1)-dimensional dispersive long wave equations which describe the evolution of horizontal velocity component 𝑢(𝑥, 𝑡) of water waves of height 𝑣(𝑥, 𝑡), and solved it numerically by successive approximation method (SAM) to compare with Adomian’s decomposition method (ADM), we found that SAM is suitable for this kind of problems also its effective and more accure than ADM. Mathematica has been Keywords: used for computation

In this letter, variational homotopy perturbation method (VHPM) has been studied to obtain solitary wave solutions of modified Camassa-Holm and Degasperis-Procesi equations. The results show that the VHPM is suitable for solving nonlinear differential equations with fully nonlinear dispersion term. The travelling wave solution for above equation compared with VIM, HPM, and exact solution. Also, it was shown that the present method is effective, suitable, and reliable for these types of equations.

In this paper, Rayleigh wave equation has been solved numerically for finding an approximate solution by Successive approximation method and Finite difference method. Example showed that Successive approximation method is much faster and effective for this kind of problems than Finite difference method.

In this article, the application of variational homotopy perturbation method is applied to solve Benjamin-Bona-Mahony equation. Then, we obtain the numerical solution of BBM equation using the initial condition. Comparison with Adomian’s decomposition method, homotopy perturbation method, and with the exact solution shows that VHPM is more effective and accurate than ADM and HPM, and is reliable and manageable for this type of equation.

we have studied the numerical solutions for FitzHugh-Nagumo equation (FHN) using Finite Difference Methods (FDM) including explicit method, implicit (Crank-Nicholson) method, fully implicit method, Exponential method. A Comparison was made among all the methods by solving two numerical examples with different time steps.

In this paper, an application of A domain Decomposition method (ADM) is applied for finding the approximate solution of nonlinear partial differential equation. The results reveal that the A domain Decomposition method is very effective, simple and very close to the exact solution.

In this study, the approximate solutions for the Kuramoto-Sivashinsky equation by using the Variational Iteration Method (VIM) are obtained. Comparisons with the exact solutions and the solutions obtained by the Homotopy Perturbation Method (HPM), the numerical example show that the Variational Iteration Method (VIM) is accurate and effective and suitable for this kind of problem.

In this article the homotopy analysis method (HAM) is used to find a numerical solution for the nonlinear diffusion equation with convection term. The numerical results obtained by using this method compared with the exact solution, by solving numerical example shows that (HAM) is accurate and close to the exact solution.

In this paper, an application of homotopy perturbation method (HPM) is applied to finding the approximate solution of nonlinear diffusion equation with convection term, We obtained the numerically solution and compared with the exact solution.The results reveal that the homotopy perturbation method is very effective, simple and very close to the exact solution.

The approximate solutions for the Kuramoto –Sivashinsky Equation are obtained by using the Adomian Decomposition method (ADM). The numerical example show that the approximate solution comparing with the exact solution is accurate and effective and suitable for this kind of problem.

The approximate solutions for the Kuramoto –Sivashinsky Equation are obtained by using the homotopy perturbation method (HPM). The numerical example show that the approximate solution comparing with the exact solution is accurate and effective and suitable for this kind of problem.

Nonlinear diffusion equation with convection term solved numerically using successive approximation method. Numerical example showed that (SAM) can solve this kind of models also comparing with the exact solution showed that SAM accurate and efficient method as shown in table (1) and Figures (1,2).

Klein Gordon equation has been solved numerically by using fully implicit finite difference method (FIFDM) and exponential finite difference method (ExpFDM) and we found that both methods can solve this kind of problems, example showed that fully implicit method is more a accurate than exponential finite difference method.

alternating direction explicit and alternating direction implicit methods (ADE and ADI) were used to solve Schnakenberg model, we were found that alternating direction implicit method is much more accurate and faster than alternating direction explicit in this kind of models.

∅ Klein Gordon Equation has been solved numerically by using two methods: finite difference method (FDM) and Adomain decomposition method (ADM) and we discover that the ADM is much more accurate than FDM in this kind of models as shown in the example(1,2).

The Glycolysis Model Has Been Solved Numerically In Two Dimensions By Using Two Finite Differences Methods: Alternating Direction Explicit And Alternating Direction Implicit Methods (ADE And ADI) And We Were Found That The ADE Method Is Simpler While The ADI Method Is More Accurate. Also, We Found That ADE Method Is Conditionally Stable While ADI Method Is Unconditionally Stable. Keywords: Glycolysis Model, ADE Method, ADI Method.

The Glycolysis model has been solved numerically in one dimension by using two finite dif ferences methods: explicit and C rank Nicolson method and we were found that the explicit method is simpler while the Crank Nicolson is m ore accurate. Also, we found that explicit method is conditionally stable while Crank Nicolson method is unconditionally stablestable.

The numerical stability analysis of Brusselator system has been done in one and two dimensional space. For one dimension we studied the numerical stability for explicit and implicit (Crank- Nicolson) methods and we found that explicit method for solving Brusselator system is stable under the conditionsr1<=(2-k(b+1)/8, r2<=1/2 While the implicit method is unconditionally stable. For two dimensional space we found that ADE method is stable under condition r1<=(2-k(b+1)/8, r2<=1/4, while ADI is unconditionally stable

The numerical stability analysis of Brusselator system has been done in one and two-dimensional space. For one dimension we studied the numerical stability for explicit and implicit (Crank- Nicolson) methods and we found that the explicit method for solving Brusselator system is stable under conditions, While the implicit method is unconditionally stable. For two-dimensional space, we found that ADE method is stable under conditions, while ADI is unconditionally stable.

The Brusselator model has been solved numerically in one and two dimensions by using two finite differences methods: For one dimension we used explicit and crank-Nicolson method and we were found that the explicit method is simpler while the Crank-Nicolson is more accurate. For the two dimensions we used the ADE and the ADI methods and we found that the ADI is more accurate than the ADE.

In this work, we studied the numerical solution of the Brusselator model in one dimension using FDM including explicit and implicit methods; FEM with weighted residual methods and iterative methods. Also, we studied the numerical solution of the Brusselator model in two dimensions using ADI ( Alternating Direction Implicit) and ADE (Alternating Direction Explicit) methods. Besides, we studied the numerical stability of FDM (explicit and implicit methods); the numerical stability analysis of the Brusselator system was done in one-dimensional space and two-dimensional spaces. For one dimensional space, we have studied the numerical stability for explicit and implicit (Crank- Nicolson) methods and we have found the stability condition for explicit method, whereas the implicit method is unconditionally stable. For two dimensional space, we found the stability condition for ADE method, while ADI is unconditionally stable.

In this paper, four types of weighted residual methods (Collocation, Subdomain, Galerkin and least-square methods) are presented for finding an approximate solution of the Brusselator model. We showed the efficiency of the prescribed methods by solving numerical example.

The Brusselator model has been solved numerically in one and two dimensions by using two finite differences methods: For one dimension we used explicit and crank-Nicolson method and we were found that the explicit method is simpler while the Crank-Nicolson is more accurate. For the two dimensions we used the ADE and the ADI methods and we found that the ADI is more accurate than the ADE.

The numerical stability analysis of Brusselator system has been done in one and two dimensional space. For one dimension we studied the numerical stability for explicit and implicit (Crank- Nicolson) methods and we found that explicit method for solving Brusselator system is stable under the conditions 4 2 ( 1) 1 - + r £ k b , and 1/ 2. 2 r £ While the implicit method is unconditionally stable. For two dimensional space we found that ADE method is stable under condition 8 2 ( 1) 1 r £ - k b + , and 1/ 4 2 r < , while ADI is unconditionally stable.