Variational iteration method with He's polynomials for MHD Falkner-Skan flow over permeable wall based on Lie symmetry method

2014 ◽  
Vol 24 (6) ◽  
pp. 1348-1362 ◽  
Author(s):  
Buhe Eerdun ◽  
Qiqige Eerdun ◽  
Bala Huhe ◽  
Chaolu Temuer ◽  
Jing-Yu Wang
Keyword(s):  
Boundary Layer ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Lie Symmetry ◽  
Content Type ◽  
Variational Iteration ◽  
Lie Symmetry Method ◽  
He’S Polynomials ◽  
Layer Flow ◽  
Symmetry Method

Purpose – The purpose of this paper is to consider a steady two-dimensional magneto-hydrodynamic (MHD) Falkner-Skan boundary layer flow of an incompressible viscous electrically fluid over a permeable wall in the presence of a magnetic field. Design/methodology/approach – The governing equations of MHD Falkner-Skan flow are transformed into an initial values problem of an ordinary differential equation using the Lie symmetry method which are then solved by He's variational iteration method with He's polynomials. Findings – The approximate solution is compared with the known solution using the diagonal Pad’e approximants and the geometrical behavior for the values of various parameters. The results reveal the reliability and validity of the present work, and this combinational method can be applied to other nonlinear boundary layer flow problems. Originality/value – In this paper, an approximate analytical solution of the MHD Falkner-Skan flow problem is obtained by combining the Lie symmetry method with the variational iteration method and He's polynomials.

scholarly journals Solitary and compacton solutions of fractional KdV-like equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 328-336 ◽  
Author(s):  
Bo Tang ◽  
Yingzhe Fan ◽  
Jianping Zhao ◽  
Xuemin Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Reliable Tool ◽  
Variational Iteration ◽  
He’S Polynomials ◽  
Liouville Derivative

AbstractIn this paper, based on Jumarie’s modified Riemann-Liouville derivative, we apply the fractional variational iteration method using He’s polynomials to obtain solitary and compacton solutions of fractional KdV-like equations. The results show that the proposed method provides a very effective and reliable tool for solving fractional KdV-like equations, and the method can also be extended to many other fractional partial differential equations.

The estimation of the length constant of a long cooling fin by variational iteration method

2015 ◽  
Vol 25 (4) ◽  
pp. 887-891 ◽  
Author(s):  
Yan Zhang ◽  
Qiaoling Chen ◽  
Fujuan Liu ◽  
Ping Wang
Keyword(s):  
Nonlinear Equation ◽  
Nonlinear Equations ◽  
Iteration Method ◽  
Design Methodology ◽  
Variational Iteration Method ◽  
Engineering Problem ◽  
Content Type ◽  
Variational Iteration ◽  
Length Constant ◽  
Good Agreement

Purpose – The purpose of this paper is to validate the variational iteration method (VIM) is suitable for various nonlinear equations. Design/methodology/approach – The He’s VIM is applied to solve nonlinear equation which is derived from actual engineering problem. The result was compared with other method. Findings – The result obtained from VIM shows good agreement with Xu’s result which provide a solid evidence that VIM is convenient and effective for solving nonlinear equation in the engineering. Originality/value – The VIM can be extended to many academic and engineering fields for nonlinear equations solving.

Convergent optimal variational iteration method and applications to heat and fluid flow problems

2016 ◽  
Vol 26 (3/4) ◽  
pp. 790-804 ◽  
Author(s):  
Mustafa Turkyilmazoglu
Keyword(s):  
Fluid Flow ◽  
Iteration Method ◽  
Sufficient Conditions ◽  
Variational Iteration Method ◽  
Content Type ◽  
Flow Problems ◽  
Variational Iteration ◽  
Physical Problems ◽  
Novel Approach ◽  
Heat And Fluid Flow

Purpose – In an earlier paper (Turkyilmazoglu, 2011a), the author introduced a new optimal variational iteration method. The idea was to insert a parameter into the classical variational iteration formula in an aim to prevent divergence or to accelerate the slow convergence property of the classical approach. The purpose of this paper is to approve the superiority of the proposed method over the traditional one on several physical problems treated before by the classical variational iteration method. Design/methodology/approach – A sufficient condition theorem with an upper bound for the error is also presented to further justify the convergence of the new variational iteration method. Findings – The optimal variational iteration method is found to be useful for heat and fluid flow problems. Originality/value – The optimal variational iteration method is shown to be convergent under sufficient conditions. A novel approach to obtain the optimal convergence parameter is introduced.

Variational Iteration Method for the Hirota-Satsuma Model Using He’s Polynomials

2010 ◽  
Vol 65 (6-7) ◽  
pp. 525-528 ◽  
Author(s):  
Syed Tauseef Mohyud-Din ◽  
Ahmet Yildirim
Keyword(s):  
Applied Sciences ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Numerical Results ◽  
Variational Iteration ◽  
He’S Polynomials

This paper out lines the implementation of the variational iteration method using He’s polynomials (VMHP) for solving the Hirota-Satsuma model which occurs quite often in applied sciences. Numerical results show that the proposed VIMHP is quite efficient.

Solving imprecisely defined vibration equation of large membranes

2017 ◽  
Vol 34 (8) ◽  
pp. 2528-2546 ◽  
Author(s):  
Smita Tapaswini ◽  
Chunlai Mu ◽  
Diptiranjan Behera ◽  
Snehashish Chakraverty
Keyword(s):  
Fuzzy Number ◽  
Iteration Method ◽  
Initial Conditions ◽  
Computational Cost ◽  
Variational Iteration Method ◽  
Interval Uncertainty ◽  
Parametric Form ◽  
Content Type ◽  
Vibration Equation ◽  
Variational Iteration

Purpose Vibration of large membranes has great utility in engineering application such as in important parts of drums, pumps, microphones, telephones and other devices. So, to obtain a numerical solution of this type of problems is necessary and important. In general, in existing approaches, involved parameters and variables are defined exactly. Whereas in actual practice, it may contain uncertainty owing to error in observations, maintenance-induced error, etc. So, the main purpose of this paper is to solve this important problem numerically under fuzzy and interval uncertainty to have an uncertain solution and to study its behaviour. Design/methodology/approach In this study, the authors have considered a new approach is known as double parametric form of fuzzy number to model uncertain parameters. Along with this a semianalytical approach, i.e. variational iteration method, has been used to obtain uncertain bounds of the solution. Findings The variational iteration method has been successfully implemented along with the double parametric form of fuzzy number to find the uncertain solution of the vibration equation of a large membrane. The advantage of this approach is that the solution can be written in a power series or a compact form. Also, this method converges rapidly to obtain an accurate solution. Various cases depending on the functional value involved in the initial conditions have been studied and the behaviour has been analysed. Applying the double parametric form reduces the computational cost without separating the fuzzy equation into coupled differential equations as done in traditional approaches. Originality/value The vibration equation of large membranes has been solved under fuzzy and interval uncertainty. Uncertainties have been considered in the initial conditions. New approaches, i.e. variational iteration method along with the double parametric form, have been applied to solve the vibration equation of large membranes.

scholarly journals Variational Iteration Method for Initial and Boundary Value Problems Using He's Polynomials

2010 ◽  
Vol 2010 ◽  
pp. 1-28 ◽  
Author(s):  
Syed Tauseef Mohyud-Din ◽  
Ahmet Yildirim ◽  
M. M. Hosseini
Keyword(s):  
Boundary Value Problems ◽  
Iteration Method ◽  
Boundary Value ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
He’S Polynomials ◽  
User Friendly

This paper outlines a detailed study of the coupling of He's polynomials with correction functional of variational iteration method (VIM) for solving various initial and boundary value problems. The elegant coupling gives rise to the modified versions of VIM which is very efficient in solving nonlinear problems of diversified nature. It is observed that the variational iteration method using He's polynomials (VIMHP) is very efficient, easier to implements, and more user friendly. Several examples are given to reconfirm the efficiency of the proposed VIMHP.

A numerical approach for a class of astrophysics equations using piecewise spectral-variational iteration method

2017 ◽  
Vol 27 (2) ◽  
pp. 358-378 ◽  
Author(s):  
Mohammad Heydari ◽  
Ghasem Barid Loghmani ◽  
Abdul-Majid Wazwaz
Keyword(s):  
Iteration Method ◽  
Design Methodology ◽  
Variational Iteration Method ◽  
Mathematical Physics ◽  
Numerical Approach ◽  
Nonlinear System Of Equations ◽  
Content Type ◽  
Variational Iteration ◽  
Analytical Integration ◽  
Chebyshev Interpolation

Purpose The main purpose of this paper is to implement the piecewise spectral-variational iteration method (PSVIM) to solve the nonlinear Lane-Emden equations arising in mathematical physics and astrophysics. Design/methodology/approach This method is based on a combination of Chebyshev interpolation and standard variational iteration method (VIM) and matching it to a sequence of subintervals. Unlike the spectral method and the VIM, the proposed PSVIM does not require the solution of any linear or nonlinear system of equations and analytical integration. Findings Some well-known classes of Lane-Emden type equations are solved as examples to demonstrate the accuracy and easy implementation of this technique. Originality/value In this paper, a new and efficient technique is proposed to solve the nonlinear Lane-Emden equations. The proposed method overcomes the difficulties arising in calculating complicated and time-consuming integrals and terms that are not needed in the standard VIM.

Symmetry reduction and numerical solution of a nonlinear boundary value problem in fluid mechanics

2018 ◽  
Vol 28 (3) ◽  
pp. 518-531 ◽  
Author(s):  
Sudao Bilige ◽  
Yanqing Han
Keyword(s):  
Fluid Mechanics ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Lie Symmetry ◽  
Nonlinear Pdes ◽  
Initial Value ◽  
Content Type ◽  
Lie Symmetry Method ◽  
Homotopy Perturbation ◽  
Symmetry Method

Purpose The purpose of this paper is to study the applications of Lie symmetry method on the boundary value problem (BVP) for nonlinear partial differential equations (PDEs) in fluid mechanics. Design/methodology/approach The authors solved a BVP for nonlinear PDEs in fluid mechanics based on the effective combination of the symmetry, homotopy perturbation and Runge–Kutta methods. Findings First, the multi-parameter symmetry of the given BVP for nonlinear PDEs is determined based on differential characteristic set algorithm. Second, BVP for nonlinear PDEs is reduced to an initial value problem of the original differential equation by using the symmetry method. Finally, the approximate and numerical solutions of the initial value problem of the original differential equations are obtained using the homotopy perturbation and Runge–Kutta methods, respectively. By comparing the numerical solutions with the approximate solutions, the study verified that the approximate solutions converge to the numerical solutions. Originality/value The application of the Lie symmetry method in the BVP for nonlinear PDEs in fluid mechanics is an excellent and new topic for further research. In this paper, the authors solved BVP for nonlinear PDEs by using the Lie symmetry method. The study considered that the boundary conditions are the arbitrary functions Bi(x)(i = 1,2,3,4), which are determined according to the invariance of the boundary conditions under a multi-parameter Lie group of transformations. It is different from others’ research. In addition, this investigation will also effectively popularize the range of application and advance the efficiency of the Lie symmetry method.

Sign in / Sign up

Export Citation Format

Share Document