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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.

To optimize the shape of a cascade where velocity (or pressure) distribution is optimally given.

The semi‐inverse method suggested by Ji‐Huan He is applied to establishment of a variational theory for the discussed inverse problem. The boundary conditions on unknown shape are converted into the natural boundary conditions of the obtained variational functional.

Ji‐Huan He's semi‐inverse method is a powerful tool to the search for the variational formulation for the discussed problem. The derivation procedure is very simple and convenient; the finite element method based on the variational theory with moving boundary provides a very effective and robust numerical approach to the inverse problem.

The design method is limited to frictionless flow.

The numerical method based on variational principle with moving boundary can be readily extended to other cases with moving surfaces or free boundaries.

The suggested numerical method can satisfy the demand of various cascade designs, where the velocity (or pressure) distribution can be optimally given from different aspects of engineering requirement: aerodynamics, strength, manufacture, etc.

It is extremely difficult to establish a variational principle for plasma. Kalaawy obtained a variational principle by using the semi-inverse method in 2016, and Li and He suggested a modification in 2017. This paper aims to search for a generalized variational formulation with a free parameter.

The semi-inverse method is used by suitable construction of a trial functional with some free parameters.

A modification of Li-He’s variational principle with a free parameter is obtained.

This paper suggests a new approach to construction of a trial-functional with some free parameters.

A generalized variational principle of 2D unsteady compressible flow around oscillating airfoils is established directly from the governing equations and boundary/initial conditions via the semi‐inverse method proposed by He. In this method, an energy integral with an unknown F is used as a trial‐functional. The identification of the unknown F is similar to the identification of the Lagrange multiplier. Based on the variational theory with variable domain, a variational principle for the inverse problem (given as the time‐averaged pressure over the airfoil contour, while the corresponding airfoil shape is unknown) is constructed, and all the boundary/initial conditions are converted into natural ones, leading to well‐posedness and the unique solution of the inverse problems.

The variational principle views a complex problem in an energy way, it gives good physical understanding of an iteration method, and the variational-based numerical methods always have a conservation scheme with a fast convergent rate. The purpose of this paper is to establish a variational principle for a fractal nano/microelectromechanical (N/MEMS) system.

This paper begins with an approximate variational principle in literature for the studied problem, and a genuine variational principle is obtained by the semi-inverse method.

The semi-inverse method is a good mathematical tool to the search for a genuine fractal variational formulation for the N/MEMS system.

The established variational principle can be used for both analytical and numerical analyses of the N/MEMS systems, and it can be extended to some more complex cases.

The variational principle can be used for variational-based finite element methods and energy-based analytical methods.

The new and genuine variational principle is obtained. This paper discovers the missing piece of the puzzle for the establishment of a variational principle from governing equations for a complex problem by the semi-inverse method. The new variational theory opens a new direction in fractal MEMS systems.

The purpose of this paper is the coupled nonlinear fractal Schrödinger system is defined by using fractal derivative, and its variational principle is constructed by the fractal semi-inverse method. The approximate analytical solution of the coupled nonlinear fractal Schrödinger system is obtained by the fractal variational iteration transform method based on the proposed variational theory and fractal two-scales transform method. Finally, an example illustrates the proposed method is efficient to deal with complex nonlinear fractal systems.

The coupled nonlinear fractal Schrödinger system is described by using the fractal derivative, and its fractal variational principle is obtained by the fractal semi-inverse method. A novel approach is proposed to solve the fractal model based on the variational theory.

The fractal variational iteration transform method is an excellent method to solve the fractal differential equation system.

The author first presents the fractal variational iteration transform method to find the approximate analytical solution for fractal differential equation system. The example illustrates the accuracy and efficiency of the proposed approach.

This paper aims to show how to establish a variational formulation directly from the governing equations. A modified Burger equation arising in dusty plasma is used as an example to show the derivation process.

Ji-Huan He’s semi-inverse method is adopted in the derivation process. To effectively use the semi-inverse method, a potential function is introduced, and the trial functional is constructed with some nonzero parameters, which are such identified that the stationary conditions satisfy the governing equations.

The derivation is simple, and the obtained variational principle is conciser than that obtained by Kalaawy, who introduced two special functions.

This paper suggests an effective approach to the inverse problem of the calculus of variations.

On a microgravity condition, a motion of soliton might be subject to a microgravity-induced motion. There is no theory so far to study the effect of air density and gravity on the motion property. Here, the author considers the air as discrete molecules and a motion of a soliton is modeled based on He’s fractal derivative in a microgravity space. The variational principle of the alternative model is constructed by semi-inverse method. The variational principle can be used to establish the conservation laws and reveal the structure of the solution. Finally, its approximate analytical solution is found by using two-scale method and homotopy perturbation method (HPM).

The author establishes a new fractal model based on He’s fractal derivative in a microgravity space and its variational principle is obtained via the semi-inverse method. The approximate analytical solution of the fractal model is obtained by using two-scale method and HPM.

He’s fractal derivative is a powerful tool to establish a mathematical model in microgravity space. The variational principle of the fractal model can be used to establish the conservation laws and reveal the structure of the solution.

The author proposes the first fractal model for the soliton motion in a microgravtity space and obtains its variational principle and approximate solution.

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.

The author establishes the variational principle for fractal Lane–Emden equation, and its approximate analytical solution is obtained by the variational iteration method.

The variational iteration method is very fascinating in solving fractal differential equation.

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.

Using the semi‐inverse method proposed by the present author, a family of variational principle for direct problem of S2‐flow in mixed‐flow turbomachinery is obtained; then, applying the functional variation with variable domain, two families of variational principles are established for the hybrid problems of determining the unknown shape of bladings, where pressure or velocity is over‐specified. The present variational models are well posed for redundant data at boundaries. The theory provides both rational ways for best contouring the hub/casing walls to meet various practical design requirements and a theoretical basis for introducing the finite element method into computational aerodynamics of turbomachinery.