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.

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.

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.

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.

New numerical technique for solving the fractional Huxley equation

2014 ◽  
Vol 24 (8) ◽  
pp. 1736-1754
Author(s):  
Talaat El-Sayed El-Danaf ◽  
Mfida Ali Zaki ◽  
Wedad Moenaaem
Keyword(s):  
Differential Operator ◽  
Iteration Method ◽  
Decomposition Method ◽  
Variational Iteration Method ◽  
Decomposition Methods ◽  
Content Type ◽  
Variational Iteration ◽  
Adomian Decomposition ◽  
Fractional Differential ◽  
Huxley Equation

Purpose – The purpose of this paper is to investigate the possibility of extension to the variational iteration and the Adomian decomposition methods for solving nonlinear Huxley equation with time-fractional derivative. Design/methodology/approach – Objectives achieved the main methods: the fractional derivative of f (x) in the Caputo sense is first stated. Second, the time-fractional Huxley equation is written in a differential operator form where the differential operator is in Caputo sense. After acting on both sides by the inverse operator of the fractional differential operator in Caputo sense, the Adomian's decomposition is then used to get the power series solution of the resulted time-fractional Huxley equation. Also, a second objective is achieved by applying the variational iteration method to get approximate solutions for the time-fractional Huxley equation. Findings – There are some important findings to state and summarize here. First, the variational iteration method and the decomposition method provide the solutions in terms of convergent series with easily computable components for this considered problem. Second, it seems that the approximate solution of time-fractional Huxley equation using the decomposition method converges faster than the approximate solution using the variational iteration method. Third, the variational iteration method handles nonlinear equations without any need for the so-called Adomian polynomials. However, Adomian decomposition method provides the components of the exact solution, where these components should follow the summation given in Equation (21). Originality/value – This paper presents new materials in terms of employing the variational iteration and the Adomian decomposition methods to solve the problem under consideration. It is expected that the results will give some insightful conclusions for the used techniques to handle similar fractional differential equations. This emphasizes the fact that the two methods are applicable to a broad class of nonlinear problems in fractional differential equations.

A general numerical algorithm for nonlinear differential equations by the variational iteration method

2020 ◽  
Vol 30 (11) ◽  
pp. 4797-4810 ◽  
Author(s):  
Ji-Huan He ◽  
Habibolla Latifizadeh
Keyword(s):  
Differential Equations ◽  
Numerical Algorithm ◽  
Iteration Method ◽  
Solution Process ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Iteration Algorithm ◽  
Content Type ◽  
Variational Iteration ◽  
Laplace Transform Technique

Purpose The purpose of this paper is to suggest a general numerical algorithm for nonlinear problems by the variational iteration method (VIM). Design/methodology/approach Firstly, the Laplace transform technique is used to reconstruct the variational iteration algorithm-II. Secondly, its convergence is strictly proved. Thirdly, the numerical steps for the algorithm is given. Finally, some examples are given to show the solution process and the effectiveness of the method. Findings No variational theory is needed to construct the numerical algorithm, and the incorporation of the Laplace method into the VIM makes the solution process much simpler. Originality/value A universal iteration formulation is suggested for nonlinear problems. The VIM cleans up the numerical road to differential equations.

Solving shallow water equations with crisp and uncertain initial conditions

2018 ◽  
Vol 28 (12) ◽  
pp. 2801-2815 ◽  
Author(s):  
Perumandla Karunakar ◽  
Snehashish Chakraverty
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Iteration Method ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Variational Iteration Method ◽  
Triangular Fuzzy Number ◽  
Governing Equations ◽  
Content Type ◽  
Variational Iteration

Purpose This paper aims to deal with the application of variational iteration method and homotopy perturbation method (HPM) for solving one dimensional shallow water equations with crisp and fuzzy uncertain initial conditions. Design/methodology/approach Firstly, the study solved shallow water equations using variational iteration method and HPM with constant basin depth and crisp initial conditions. Further, the study considered uncertain initial conditions in terms of fuzzy numbers, which leads the governing equations to fuzzy shallow water equations. Then using cut and parametric concepts the study converts fuzzy shallow water equations to crisp form. Then, HPM has been used to solve the fuzzy shallow water equations. Findings Results obtained by both methods HPM and variational iteration method are compared graphically in crisp case. Solution of fuzzy shallow water equations by HPM are presented in the form triangular fuzzy number plots. Originality/value Shallow water equations with crisp and fuzzy initial conditions have been solved.

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.

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.

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.

Fractional calculus for nanoscale flow and heat transfer

2014 ◽  
Vol 24 (6) ◽  
pp. 1227-1250 ◽  
Author(s):  
Hong-Yan Liu ◽  
Ji-Huan He ◽  
Zheng-Biao Li
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Content Type ◽  
Variational Iteration ◽  
Fractal Media ◽  
Fractional Differential ◽  
Analytical Approaches

Purpose – Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable water or air permeation and remarkable thermal conductivity. The purpose of this paper is to reveal the phenomena by the fractional calculus. Design/methodology/approach – This paper begins with the continuum assumption in conventional theories, and then the fractional Gauss’ divergence theorems are used to derive fractional differential equations in fractal media. Fractional derivatives are introduced heuristically by the variational iteration method, and fractal derivatives are explained geometrically. Some effective analytical approaches to fractional differential equations, e.g. the variational iteration method, the homotopy perturbation method and the fractional complex transform, are outlined and the main solution processes are given. Findings – Heat conduction in silk cocoon and ground water flow are modeled by the local fractional calculus, the solutions can explain well experimental observations. Originality/value – Particular attention is paid throughout the paper to giving an intuitive grasp for fractional calculus. Most cited references are within last five years, catching the most frontier of the research. Some ideas on this review paper are first appeared.

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