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This paper aims to review some effective methods for fully fourth-order nonlinear integral boundary value problems with fractal derivatives.

Boundary value problems arise everywhere in engineering, hence two-scale thermodynamics and fractal calculus have been introduced. Some analytical methods are reviewed, mainly including the variational iteration method, the Ritz method, the homotopy perturbation method, the variational principle and the Taylor series method. An example is given to show the simple solution process and the high accuracy of the solution.

An elemental and heuristic explanation of fractal calculus is given, and the main solution process and merits of each reviewed method are elucidated. The fractal boundary value problem in a fractal space can be approximately converted into a classical one by the two-scale transform.

This paper can be served as a paradigm for various practical applications.

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.

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.

Heat conduction in silk cocoon and ground water flow are modeled by the local fractional calculus, the solutions can explain well experimental observations.

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.

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.

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.

Some well-known classes of Lane-Emden type equations are solved as examples to demonstrate the accuracy and easy implementation of this technique.

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.

The purpose of this paper is to find approximate solutions for a general class of fractional order boundary value problems that arise in engineering applications.

A newly developed semi-analytical scheme will be applied to find approximate solutions for fractional order boundary value problems. The technique is regarded as an extension of the well-established variation iteration method, which was originally proposed for initial value problems, to cover a class of boundary value problems.

It has been demonstrated that the method yields approximations that are extremely accurate and have uniform distributions of error throughout their domain. The numerical examples confirm the method’s validity and relatively fast convergence.

The generalized variational iteration method that is presented in this study is a novel strategy that can handle fractional boundary value problem more effectively than the classical variational iteration method, which was designed for initial value problems.

A three-dimensional (3D) unsteady potential flow might admit a variational principle. The purpose of this paper is to adopt a semi-inverse method to search for the variational formulation from the governing equations.

A suitable trial functional with a possible unknown function is constructed, and the identification of the unknown function is given in detail. The Lagrange multiplier method is used to establish a generalized variational principle, but in vain.

Some new variational principles are obtained, and the semi-inverse method can easily overcome the Lagrange crisis.

The semi-inverse method sheds a promising light on variational theory, and it can replace the Lagrange multiplier method for the establishment of a generalized variational principle. It can be used for the establishment of a variational principle for fractal and fractional calculus.

This paper establishes some new variational principles for the 3D unsteady flow and suggests an effective method to eliminate the Lagrange crisis.

The purpose of this paper is to implement variational iteration method (VIM) and homotopy perturbation method (HPM) to solve modified Camassa‐Holm (mCH) and modified Degasperis‐Procesi (mDP) equations.

Perturbation method is a traditional method depending on a small parameter which is difficult to be found for real‐life nonlinear problems. To overcome the difficulties and limitations of the above method, two new ones have recently been introduced by He, i.e. VIM and HPM. In this paper, mCH and mDP equations are solved through these methods.

To assess the accuracy of the solutions, the comparison of the obtained results with the exact solutions reveals that both methods are tremendously effective.

The paper shows that VIM and HPM can be implemented to solve mCH and mDP equations.

The purpose of this paper is to develop an effective numerical algorithm for a gas-melt two-phase flow and use it to simulate a polymer melt filling process. Moreover, the suggested algorithm can deal with the moving interface and discontinuities of unknowns across the interface.

The algebraic sub-grid scales-variational multi-scale (ASGS-VMS) finite element method is used to solve the polymer melt filling process. Meanwhile, the time is discretized using the Crank–Nicolson-based split fractional step algorithm to reduce the computational time. The improved level set method is used to capture the melt front interface, and the related equations are discretized by the second-order Taylor–Galerkin scheme in space and the third-order total variation diminishing Runge–Kutta scheme in time.

The numerical method is validated by the benchmark problem. Moreover, the viscoelastic polymer melt filling process is investigated in a rectangular cavity. The front interface, pressure field and flow-induced stresses of polymer melt during the filling process are predicted. Overall, this paper presents a VMS method for polymer injection molding. The present numerical method is extremely suitable for two free surface problems.

For the first time ever, the ASGS-VMS finite element method is performed for the two-phase flow of polymer melt filling process, and an effective numerical method is designed to catch the moving surface.

The purpose of this paper is to solve an unsteady nonlinear convective‐radiative equation and a nonlinear convective‐radiative‐conduction equation containing two small parameters of ε₁ and ε₂ by variational homotopy perturbation method.

The heat transfer equations are described. The variational homotopy perturbation method as a powerful method for solving linear and nonlinear equations is applied to find the solutions of our model equations.

The solutions of heat transfer equations are calculated in the form of convergent series with easily computable components. Two examples are solved as illustrations, using symbolic computation.

The results show that the suggested method is easy to implement and has high level of accuracy. The method introduces a reliable tool for solving many linear and nonlinear differential equations.

The purpose of this paper is to validate the variational iteration method (VIM) is suitable for various nonlinear equations.

The He’s VIM is applied to solve nonlinear equation which is derived from actual engineering problem. The result was compared with other method.

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.

The VIM can be extended to many academic and engineering fields for nonlinear equations solving.

The purpose of this paper is to apply He's variational iteration method (VIM) coupled with an auxiliary parameter, which proves very effective to control the convergence region of approximate solution.

The proposed algorithm is tested on generalized Hirota‐Satsuma coupled KdV equation.

Numerical results explicitly reveal the complete reliability, efficiency and accuracy of the suggested technique.

It is observed that the approach may be implemented on other nonlinear models of physical nature.