Solutions of 1D Hyperbolic Quasi-linear Partial Differential Equations by Variational Iteration Method (original) (raw)

Variational Iteration Method for 2D Hyperbolic Quasi-linear Pde

In this paper, we have constructed two dimensional (2D ) hyperbolic quasi-linear partial differential equations (HQLPDE) by considering two cases. The Cauchy data is given in the form of exponential curve in the 1st case. The initial value problems (IVP) are mentioned as sin x in the 2nd case where as all the cases are associated with time. We have obtained the solutions by using variational iteration method (VIM). In the both cases, we found that solution represents flat surface during the initial stage. We have observed exponential surface in the 1st case and the curved surface in the 2nd case subject to the increasing values of x and y. Mathematics subject classification: 35Lxx, 35L04

He's Variational Iteration Method for Solving Hyperbolic Differential Equations

International Journal of Nonlinear Sciences and Numerical Simulation, 2007

In this paper, the variational iteration method is applied to obtain analytically approximation solutions for hyperbolic partial differential equations. The results derived by this method are compared with those obtained by the characteristics method, revealing that the method is very effective.

Hyperbolic Partial Differential Equations

We begin our study of finite difference methods for partial differential equations by considering the important class of partial differential equations called hyperbolic equations. In later chapters we consider other classes of partial differential equations, especially parabolic and elliptic equations. For each of these classes of equations we consider prototypical equations, with which we illustrate the important concepts and distinguishing features associated with each class. The reader is referred to other textbooks on partial differential equations for alternate approaches, e.g., Folland [18], Garabedian [22], and Weinberger [68]. After introducing each class of differential equations we consider finite difference methods for the numerical solution of equations in the class.

Numerical scheme methods for solving nonlinear pseudo-hyperbolic partial differential equations

Journal of Applied Mathematics and Computational Mechanics

The numerical solutions to the nonlinear pseudo-hyperbolic partial differential equation with nonlocal conditions are presented in this study. This equation is solved using the homotopy analysis technique (HAM) and the variational iteration method (VIM). Both strategies are compared and contrasted in terms of approximate and accurate solutions. The results show that the HAM technique is more appropriate, effective, and close to the exact solution than the VIM method. Finally, the graphical representations of the obtained results are given.

VARIATIONAL ITERATION METHOD FOR CAUCHY PROBLEMS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATION

In this paper, we consider the numerical treatment of the nonlinear parabolic-hyperbolic partial differential equation of the form (P-H PDE) u t − ∆u tt − ∆ − Fu = 0, where Fu is the nonlinear term, and ∆∈ ℝ 1 is a Laplace operator. Here, the variational iteration method (VIM) was employed to examine the convergence of solution of the nonlinear P-H PDE. Results obtained showed that there is a rapid rate of convergence of the approximate solution to the exact solution as the number of iterations increases. All computational framework of the research were performed using Maple 18 software. Keywords: Variational iteration method, Nonlinear PDE, Lagrange Multiplier, Linear differential operator 1. Introduction Most physical problems that are expressed using more than one variable involve partial derivatives. They are even more significant in modeling real life situations such as shock waves (Burgers equation), gas dynamics (gas dynamic equation), steady state distribution of heat in a two-dimensional plain (Poisson equation), steady state problems involving incompressible fluid in a two-dimensional plain (Poisson equation), the effect of gravitational force on the potential energy of a point in a two-dimensional region (Poisson equation) etc. The parabolic partial differential equation (P-PDE) is very significant in the study of gas diffusion, which is generally called the diffusion equation. Most conventional analytic solvers (such as the integral transform method (ITM), method of characteristic (MOC), separation of variable methods, and the change of variable methods, etc) for partial differential equation over the years proved complex and difficult to handle, and does not have a precise and concise solution that can effectively and sufficiently interpret the external and internal variables of the model in consideration. This defect has attributed to the use of numerical methods by researchers in recent years for the approximation of the analytic solutions of partial differential equations. Popular numerical schemes developed and implemented over the years for partial differential equations include; the finite difference method, the finite element method, the cranck-Nicolsone method, the Bender-Schimdt method etc. The variational iteration method (IVM) was first proposed by the Chinese mathematician, J.H. He in 1998. The method has proved effective in solving both linear and nonlinear problems in differential equations, integral equations, boundary value problems, initial value problems, integro-differential problems etc. For instance, He (2000) seeks the numerical solution of autonomous ordinary

The use of He's variational iteration method for solving the one-dimensional parabolic equation with non-classical boundary conditions

Over the last 15 years, the He's variational iteration method (HVIM) has been applied to obtain formal solutions to a wide class of differential equations. This method leads to computable, efficient, solutions to linear and nonlinear operator equations. The parabolic partial differential equations with non-classical boundary conditions model various physical problems. The aim of this paper is to investigate the application of HVIM for solving the second-order linear parabolic partial differential equation with non-classical boundary conditions. The HVIM provides a reliable technique that requires less work when compared with the traditional techniques such as the Adomian decomposition method (ADM). The present approach can be used and extended for investigating more scientific applications.

Analytical Solutions for Some of the Nonlinear Hyperbolic-Like Equations with Variable Coefficients

In this work Homotopy Analysis Method(HAM) is used for analytic treatment of the nonlinear hyperbolic-like equations with variable coefficients. This method can provide analytical solutions to the problems by just utilizing the initial conditions. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. The proposed method solves nonlinear problems without using Adomain polynomials which is the advantage of this method over Adomain Decomposition method. The results reveal that the HAM is very effective, fast, simple, convenient, flexible and accurate. Outcomes prove that HAM is in very good agreement with ADM,VIM HPM. Abstract-In this work Homotopy Analysis Method(HAM) is used for analytic treatment of the nonlinear hyperbolic-like equations with variable coefficients. This method can provide analytical solutions to the problems by just utilizing the initial conditions. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. The proposed method solves nonlinear problems without using Adomain polynomials which is the advantage of this method over Adomain Decomposition method. The results reveal that the HAM is very effective, fast, simple, convenient, flexible and accurate. Outcomes prove that HAM is in very good agreement with ADM,VIM HPM.