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.

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.

scholarly journals The solution of nonlinear parabolic equation using variational iteration method

2020 ◽  
Vol 16 (3) ◽  
pp. 287
Author(s):  
Wartono Wartono ◽  
Irma Suryani
Keyword(s):  
Differential Equation ◽  
Iteration Method ◽  
Analytic Solution ◽  
Variational Iteration Method ◽  
Parabolic Differential Equation ◽  
Parabolic Differential Equations ◽  
Approximation Solution ◽  
Variational Iteration ◽  
Nonlinear Parabolic ◽  
Intial Condition

Variational iteration method is a semi analytic solution used to solve   the parabolic differential  equation both of homogen or nonhomogen. In the process of determining an approximation solution, this method did not use a linearization and a small pertubation. In this paper, the variational iteration method is implemented in the parabolic differential equation  in the form of  ut = uxx + f(u) + g(x, t) with appropriate intial condition. Furthermore, some examples of special parabolic differential equations are given to test the reliability and convergence of the method. Based on the result of study shows that the variational iteration method is  able to solve the parabolic differential equation with a good accuration.

scholarly journals Numerical Solution of Nonlinear Schrödinger Equation with Neumann Boundary Conditions Using Quintic B-Spline Galerkin Method

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 469 ◽  
Author(s):  
Azhar Iqbal ◽  
Nur Nadiah Abd Hamid ◽  
Ahmad Izani Md. Ismail
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Spline Method ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
B Spline

This paper is concerned with the numerical solution of the nonlinear Schrödinger (NLS) equation with Neumann boundary conditions by quintic B-spline Galerkin finite element method as the shape and weight functions over the finite domain. The Galerkin B-spline method is more efficient and simpler than the general Galerkin finite element method. For the Galerkin B-spline method, the Crank Nicolson and finite difference schemes are applied for nodal parameters and for time integration. Two numerical problems are discussed to demonstrate the accuracy and feasibility of the proposed method. The error norms L 2 , L ∞ and conservation laws I 1 ,   I 2 are calculated to check the accuracy and feasibility of the method. The results of the scheme are compared with previously obtained approximate solutions and are found to be in good agreement.

scholarly journals Application of He’s Variational Iteration Method to Nonlinear Integro-Differential Equations

2010 ◽  
Vol 65 (5) ◽  
pp. 418-430 ◽  
Author(s):  
Ahmet Yildirim
Keyword(s):  
Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Mathematical Tool ◽  
Variational Iteration ◽  
He’S Variational Iteration Method ◽  
Powerful Mathematical Tool

In this paper, an application of He’s variational iteration method is applied to solve nonlinear integro-differential equations. Some examples are given to illustrate the effectiveness of the method. The results show that the method provides a straightforward and powerful mathematical tool for solving various nonlinear integro-differential equations

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