Variational principle and its fractal approximate solution for fractal Lane-Emden equation

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
KangLe Wang
Keyword(s):  
Differential Equation ◽  
Variational Principle ◽  
Iteration Method ◽  
Approximate Analytical Solution ◽  
Variational Iteration Method ◽  
Content Type ◽  
Emden Equation ◽  
Variational Iteration ◽  
Fractal Models ◽  
Fractal Differential Equation

Purpose The purpose of this paper is to describe the Lane–Emden equation by the fractal derivative and establish its variational principle by using the semi-inverse method. The variational principle is helpful to research the structure of the solution. The approximate analytical solution of the fractal Lane–Emden equation is obtained by the variational iteration method. The example illustrates that the suggested scheme is efficient and accurate for fractal models. Design/methodology/approach The author establishes the variational principle for fractal Lane–Emden equation, and its approximate analytical solution is obtained by the variational iteration method. Findings The variational iteration method is very fascinating in solving fractal differential equation. Originality/value The author first proposes the variational iteration method for solving fractal differential equation. The example shows the efficiency and accuracy of the proposed method. The variational iteration method is valid for other nonlinear fractal models as well.

scholarly journals Time- or Space-Dependent Coefficient Recovery in Parabolic Partial Differential Equation for Sensor Array in the Biological Computing

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Guanglu Zhou ◽  
Boying Wu ◽  
Wen Ji ◽  
Seungmin Rho
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Sensor Array ◽  
Variational Iteration Method ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Variational Iteration

This study presents numerical schemes for solving a parabolic partial differential equation with a time- or space-dependent coefficient subject to an extra measurement. Through the extra measurement, the inverse problem is transformed into an equivalent nonlinear equation which is much simpler to handle. By the variational iteration method, we obtain the exact solution and the unknown coefficients. The results of numerical experiments and stable experiments imply that the variational iteration method is very suitable to solve these inverse problems.

A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Hale Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Convergence Region ◽  
Riccati Differential Equation ◽  
Adomian Polynomials ◽  
Riccati Differential Equations ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

AbstractIn this paper, we introduce a modified variational iteration method (MVIM) for solving Riccati differential equations. Also the fractional Riccati differential equation is solved by variational iteration method with considering Adomians polynomials for nonlinear terms. The main advantage of the MVIM is that it can enlarge the convergence region of iterative approximate solutions. Hence, the solutions obtained using the MVIM give good approximations for a larger interval. The numerical results show that the method is simple and effective.

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.

The Variational-Iteration Method to Solve the Nonlinear Boltzmann Equation

2008 ◽  
Vol 63 (3-4) ◽  
pp. 131-139 ◽  
Author(s):  
Essam M. Abulwafa ◽  
Mohammed A. Abdou ◽  
Aber H. Mahmoud
Keyword(s):  
Differential Equation ◽  
Distribution Function ◽  
Boltzmann Equation ◽  
Generating Function ◽  
Iteration Method ◽  
Nonlinear Partial Differential Equation ◽  
Variational Iteration Method ◽  
Inhomogeneous Media ◽  
Nonlinear Boltzmann Equation ◽  
Variational Iteration

The time-dependent nonlinear Boltzmann equation, which describes the time evolution of a single-particle distribution in a dilute gas of particles interacting only through binary collisions, is considered for spatially homogeneous and inhomogeneous media without external force and energy source. The nonlinear Boltzmann equation is converted to a nonlinear partial differential equation for the generating function of the moments of the distribution function. The variational-iteration method derived by He is used to solve the nonlinear differential equation of the generating function. The moments for both homogeneous and inhomogeneous media are calculated and represented graphically as functions of space and time. The distribution function is calculated from its moments using the cosine Fourier transformation. The distribution functions for the homogeneous and inhomogeneous media are represented graphically as functions of position and time.

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.

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