Modified Variational Iteration Method for Solving Nonlinear Partial Differential Equation Using Adomian Polynomials

In this paper, a new modified variational iteration method (MVIM) with a genetic algorithm has been applied for solving nonlinear partial differential equations. Therefore, a new correction function through an auxiliary parameter that makes sure the convergence of the standard method and improved results by using genetic techniques are introduced. The standard variational iteration method (VIM) is first applied to solve numerically the system of two-dimensional coupled Burgers' equations. Then an improvement on this method is done. Numerical experiments have been conducted to demonstrate the efficiency and high-order accuracy of this method. The algorithm converges readily which yields correct solutions. Better accuracy in comparison with other previous methods has been noticed. Moreover, the method can be easily applied to a wide number of linear and nonlinear differential equations with better accuracy.

The Variational Iteration Method (VIM) has proven to be a powerful technique for solving both ordinary and partial differential equations. However, its reliance on Lagrange multipliers for each type of equation has posed significant limitations, complicating its application and reducing its efficiency. This study introduces a Modified Variational Iteration Method (MVIM) that eliminates the need for Lagrange multipliers, addressing these challenges. The MVIM reformulates the correctional functional, simplifying the solution process and enhancing computational efficiency. The method is applied to both linear and nonlinear ordinary and partial differential equations, demonstrating its ability to provide accurate and fast-converging solutions. Numerical examples show that the MVIM outperforms traditional VIM in terms of computational time and convergence speed, and compares favourably with other methods such as the Adomian Decomposition Method (ADM) and New Iteration Method (NIM). The results highlight the potential of MVIM as a versatile and efficient tool for solving complex differential equations in a variety of scientific and engineering applications.

There are various linear and nonlinear one-dimensional partial differential equations that are the focus of this research. There are a large number of these equations that cannot be solved analytically or precisely. The evaluation of nonlinear partial differential equations, even if analytical solutions exist, may be problematic. Therefore, it may be necessary to use approximate analytical methodologies to solve these issues. As a result, a more effective and accurate approach must be investigated and analyzed. It is shown in this study that the Lagrange multiplier may be used to get an ideal value for parameters in a functional form and then used to construct an iterative series solution. Linear and nonlinear partial differential equations may both be solved using the variational iteration method (VIM) method, thanks to its high computing power and high efficiency. Decoding and analyzing possible Korteweg-De-Vries, Benjamin, and Airy equations demonstrates the method’s ability. With just a few iterations, the produced findings are very effective, precise, and convergent to the exact answer. As a result, solving nonlinear equations using VIM is regarded as a viable option.

The role of integer and noninteger order partial differential equations (PDE) is essential in applied sciences and engineering. Exact solutions of these equations are sometimes difficult to find. Therefore, it takes time to develop some numerical techniques to find accurate numerical solutions of these types of differential equations. This work aims to present a novel approach termed as fractional iteration algorithm-I for finding the numerical solution of nonlinear noninteger order partial differential equations. The proposed approach is developed and tested on nonlinear fractional-order Fornberg–Whitham equation and employed without using any transformation, Adomian polynomials, small perturbation, discretization, or linearization. The fractional derivatives are taken in the Caputo sense. To assess the efficiency and precision of the suggested method, the tabulated numerical results are compared with the standard variational iteration method and the exact solution as well. In addition, numerical results for different cases of the fractional-order α are presented graphically, which show the effectiveness of the proposed procedure and revealed that the proposed scheme is very effective, suitable for fractional PDEs, and may be viewed as a generalization of the existing methods for solving integer and noninteger order differential equations.

We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ0, γ1, γ2,… and auxiliary functions H0(x), H1(x), H2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.

Modified variational iteration method (non-homogeneous initial value problem)

A system of nonlinear partial differential equations was solved using a modified variational iteration method (MVIM) combined with a genetic algorithm. The modified method introduced an auxiliary parameter (p) in the correction functional to ensure convergence and improve the outcomes. Before applying the modification, the traditional variational iteration method (VIM) was used firstly. The method was applied to numerically solve the system of Schrödinger-KdV equations. By comparing the two methods in addition to some of the previous approaches, it turns out the new algorithm converges quickly, generates accurate solutions and shows improved accuracy. Additionally, the method can be easily applied to various linear and nonlinear differential equations.

The semidiscrete ordinary differential equation (ODE) system resulting from compact higher-order finite difference spatial discretization of a nonlinear parabolic partial differential equation, for instance, the reaction-diffusion equation, is highly stiff. Therefore numerical time integration methods with stiff stability such as implicit Runge-Kutta methods and implicit multistep methods are required to solve the large-scale stiff ODE system. However those methods are computationally expensive, especially for nonlinear cases. Rosenbrock method is efficient since it is iteration-free; however it suffers from order reduction when it is used for nonlinear parabolic partial differential equation. In this work we construct a new fourth-order Rosenbrock method to solve the nonlinear parabolic partial differential equation supplemented with Dirichlet or Neumann boundary condition. We successfully resolved the phenomena of order reduction, so the new method is fourth-order in time when it is used for nonlinear parabolic partial differential equations. Moreover, it has been shown that the Rosenbrock method is strongly A-stable hence suitable for the stiff ODE system obtained from compact finite difference discretization of the nonlinear parabolic partial differential equation. Several numerical experiments have been conducted to demonstrate the efficiency, stability, and accuracy of the new method.

A modification of the variational iteration method with a genetic algorithm is presented for the numerical results of the nonlinear partial differential equations. Therefore, a new correction function through an auxiliary parameter (p) for making sure the convergence of the standard method and improved results by using genetic techniques will be introduced. The standard variational iteration method is applied first before improving it. This method was studied to solve numerically the system of two-dimensional reaction-diffusion brusselatorequations. Its effectiveness and high-order precision have been demonstrated through numerical experiments. The algorithm converges readily, yields correct solutions and better accuracy in comparison with other previous methods have been noticed. Moreover, with greater accuracy, the approach can be used for a variety of linear and nonlinear differential equations.

The Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM) are used to find solutions of systems of stiff ordinary differential equations for both linear and nonlinear problems. Some examples are given to illustrate the accuracy and effectiveness of these methods. We compare our results with exact results. In some studies related to stiff ordinary differential equations, problems were solved by Adomian Decomposition Method and VIM and Homotopy Perturbation Method. Comparisons with exact solutions reveal that the Variational Iteration Method (VIM) and the Modified Variational Iteration Method (MVIM) are easier to implement. In fact, these methods are promising methods for various systems of linear and nonlinear stiff ordinary differential equations. Furthermore, VIM, or in some cases MVIM, is giving exact solutions in linear cases and very satisfactory solutions when compared to exact solutions for nonlinear cases depending on the stiffness ratio of the stiff system to be solved.

In this paper, we present a comparative study between the modified variational iteration method (MVIM) and a hybrid of Fourier transform and variational iteration method (FTVIM). The study outlines the efficiency and convergence of the two methods. The analysis is illustrated by investigating four singular partial differential equations with variable coefficients. The solution of singular partial differential equations usually needs a coordinate transformation in order to discard the singularity of the partial differential equation. Most often this transformation is not applicable and even does not exist. Therefore in this case the solution for the singular partial differential equation does not exist. In the present study the results of simulation for the singular partial differential equations with variable coefficients using the Fourier transform variational iteration method are compared with the results of simulation using the modified variational iteration method. The comparison shows that the effectiveness and accuracy of Fourier transform variational iteration method is more than that of the modified variational iteration method for the simulation of singular partial differential equations.

So far in this text we have been mainly concerned in applying classic methods, the Adomina decomposition method [3, 4, 5], and the variational iteration method [8, 9, 10] in studying first order and second order linear partial differential equations. In this chapter, we will focus our study on the nonlinear partial differential equations. The nonlinear partial differential equations arise in a wide variety of physical problems such as fluid dynamics, plasma physics, solid mechanics and quantum field theory. Systems of nonlinear partial differential equations have been also noticed to arise in chemical and biological applications. The nonlinear wave equations and the solitons concept have introduced remarkable achievements in the field of applied sciences. The solutions obtained from nonlinear wave equations are different from the solutions of the linear wave equations [1, 2].

Operators with regular singularities. One variable case - Operators with regular singularities. Several variables case - Formal and convergent solutions of singular partial differential equations - Local study of differential equations of the form xy' f(x,y) near x 0 - Holomorphic and singular solutions of non linear singular first order partial differential equations - Maillet's type theorems for non linear singular partial differential equations - Maillet's type theorems for non linear singular partial differential equations without linear part - Holomorphic and singular solutions of non linear singular partial differential equations - On the existence of holomorphic solutions of the Cauchy problem for non linear partial differential equations - Maillet's type theorems for non linear singular integro-differential equations.

The aim of this paper is to study the modification of He’s variational iteration method (VIM), i.e., the modified variational iteration method (MVIM). The study demonstrates the power of the MVIM over the standard VIM. It investigates the exactness of the MVIM by showing that the results obtained by it are far nearer to the exact solutions than those obtained by the VIM. The study also reveals that the MVIM has the capability of reducing the size of calculations. Two-dimensional Brusselator system is solved using the MVIM. The numerical results obtained by these methods are compared with the known closed-form solutions.

In a recent paper, Abassy et al. (J. Comput. Appl. Math. 207:137–147, 2007) proposed a modified variational iteration method (MVIM) for a special kind of nonlinear differential equations. In this paper, we consider variational iteration method (VIM) and MVIM (proposed in Abassy et al., J. Comput. Appl. Math. 207:137–147, 2007) to obtain an approximate series solution to the generalized Fisher’s equation which converges to the exact solution in the region of convergence. It is also shown that the application of VIM to the generalized Fisher’s equation leads to calculation of unneeded terms for series solution. Therefore, we use MVIM to overcome this disadvantage. Comparison of error between VIM and MVIM is made. The results show that the MVIM is more effective than the VIM.