scholarly journals A variational-perturbation method for solving the time-dependent singularly perturbed reaction-diffusion problems

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 801-804 ◽  
Author(s):  
Guanglu Zhou ◽  
Boying Wu
Keyword(s):  
Perturbation Theory ◽  
Boundary Layers ◽  
Iteration Method ◽  
Outer Region ◽  
Reaction Diffusion ◽  
Variational Iteration Method ◽  
Singularly Perturbed ◽  
Time Dependent ◽  
Approximation Solution ◽  
Variational Iteration

In this paper, we combine the variational iteration method and perturbation theory to solve a time-dependent singularly perturbed reaction-diffusion problem. The problem is considered in the boundary layers and outer region. In the boundary layers, the problem is transformed by the variable substitution, and then the variational iteration method is employed to solve the transformed equation. In the outer region, we use the perturbation theory to obtain the approximation equation and the approximation solution. The final numerical experiments show that this method is accurate.

Variational Iteration Method for a Nonlinear Reaction-Diffusion Process

2008 ◽  
Vol 6 (1) ◽  
Author(s):  
Shu-Qiang Wang ◽  
Ji-Huan He
Keyword(s):  
Exact Solution ◽  
Diffusion Process ◽  
Iteration Method ◽  
Unknown Parameter ◽  
Reaction Diffusion ◽  
Variational Iteration Method ◽  
Rigorous Derivation ◽  
Variational Iteration ◽  
Nonlinear Reaction ◽  
The Given

An extremely simple and elementary, but rigorous derivation of temperature distribution of a reaction-diffusion process is given using the variational iteration method. In this method, a trial function (an initial solution) is chosen with some unknown parameter, which is identified after a few iterations according to the given boundary conditions. Comparison with the exact solution shows that the method is very effective and convenient.

The Petrov–Galerkin finite element method for the numerical solution of time-fractional Sharma–Tasso–Olver equation

2019 ◽  
Vol 10 (01) ◽  
pp. 1941007
Author(s):  
A. K. Gupta ◽  
S. Saha Ray
Keyword(s):  
Iteration Method ◽  
Variational Iteration Method ◽  
Galerkin Finite Element Method ◽  
Approximation Solution ◽  
Test Function ◽  
Galerkin Finite Element ◽  
Galerkin Approach ◽  
B Spline ◽  
Variational Iteration ◽  
Appl Math

In this paper, time-fractional Sharma–Tasso–Olver (STO) equation has been solved numerically through the Petrov–Galerkin approach utilizing a quintic B-spline function as the test function and a linear hat function as the trial function. The Petrov–Galerkin technique is effectively implemented to the fractional STO equation for acquiring the approximate solution numerically. The numerical outcomes are observed in adequate compatibility with those obtained from variational iteration method (VIM) and exact solutions. For fractional order, the numerical outcomes of fractional Sharma–Tasso–Olver equations are also compared with those obtained by variational iteration method (VIM) in Song et al. [Song L., Wang Q., Zhang H., Rational approximation solution of the fractional Sharma–Tasso–Olver equation, J. Comput. Appl. Math. 224:210–218, 2009]. Numerical experiments exhibit the accuracy and efficiency of the approach in order to solve nonlinear fractional STO equation.

Identifying a time-dependent heat source with nonlocal boundary and overdetermination conditions by the variational iteration method

2014 ◽  
Vol 24 (7) ◽  
pp. 1545-1552 ◽  
Author(s):  
Guanglu Zhou ◽  
Boying Wu
Keyword(s):  
Exact Solution ◽  
Heat Source ◽  
Iteration Method ◽  
Numerical Technique ◽  
Nonlocal Boundary ◽  
Variational Iteration Method ◽  
Time Dependent ◽  
Successive Approximations ◽  
Content Type ◽  
Variational Iteration

Purpose – The purpose of this paper is to investigate the inverse problem of determining a time-dependent heat source in a parabolic equation with nonlocal boundary and integral overdetermination conditions. Design/methodology/approach – The variational iteration method (VIM) is employed as a numerical technique to develop numerical solution. A numerical example is presented to illustrate the advantages of the method. Findings – Using this method, we obtain the exact solution of this problem. Whether or not there is a noisy overdetermination data, our numerical results are stable. Thus the VIM is suitable for finding the approximation solution of the problem. Originality/value – This method is based on the use of Lagrange multipliers for the identification of optimal values of parameters in a functional and gives rapidly convergent successive approximations of the exact solution if such a solution exists.

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