The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics

An extension of the so-called new iterative method (NIM) has been used to handle linear and nonlinear fractional partial differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. Therefore, a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation, and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the NIM with those obtained by both Adomian decomposition method (ADM) and the variational iteration method (VIM) reveals that the NIM is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.

PurposeThis paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense.Design/methodology/approachNumerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation and fractional Klein‐Gordon equation are investigated to show the pertinent features of the technique.FindingsHPM is a powerful and efficient technique in finding exact and approximate solutions for fractional partial differential equations in fluid mechanics. The implementation of the noise terms, if they exist, is a powerful tool to accelerate the convergence of the solution. The results so obtained reinforce the conclusions made by many researchers that the efficiency of the HPM and related phenomena gives it much wider applicability.Originality/valueThe essential idea of this method is to introduce a homotopy parameter, say p, which takes values from 0 to 1. When p = 0, the system of equations usually reduces to a sufficiently simplied form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.

The variational iteration method is used to deal with linear and nonlinear differential equations. The main characteristics of the method lie in its flexibility and ability to accurately and easily solve nonlinear equations. In this work, a general framework is presented for a variational iteration method for the analytical treatment of partial differential equations in fluid mechanics. The Caputo sense is used to describe fractional derivatives. The time-fractional Kaup-Kupershmidt (KK) equation is investigated, as it is the solution of the system of partial differential equations via the Boussinesq-Burger equation. By comparing the results that are obtained by the variational iteration method with those obtained by the two-dimensional Legendre multiwavelet, the optimal homotopy asymptotic method (OHAM), the q-homotopy analysis transform method, the Laplace Adomian Decomposition Method, and the homotopy perturbation method, the first method proved to be very effective and convenient. The main methodology in this work is anticipated to be applied to various fractional calculus, linear, and nonlinear problems.

By handling the one dimensional partial differential equation with three methods i.e. Adomain decomposition method(ADM), Variation iteration method(VIM) and the New iterative method(NIM) and applied logarithmic and exponential functions as initial condition. A general framework of these methods is presented for analytical treatment of fractional partial differential equation arises in fluid mechanics. The fractional derivatives are described in the Caputo sense. The equation used in this paper is fractional wave equation, fractional burgers equation and fractional Klein-Gordon equation. After comparison of the results, the series of solution are found which is very helpful. The basic idea described in this paper is accepted to be further in use to solve other similar linear problems in fractional calculus.

This paper presents the results of applying a new iterative method to linear and nonlinear fractional partial differential equations in fluid mechanics. A numerical analysis was performed to find an exact solution of the fractional wave equation and fractional Burgers’ equation, as well as an approximate solution of fractional KdV equation and fractional Boussinesq equation. Fractional derivatives of the order α are described using Caputo's definition with <i>0</i> < α ≤ <i>1</i> or <i>1</i> < α ≤ <i>2</i>. A comparative analysis of the results obtained using a new iterative method with those obtained by the Adomian decomposition method showed the first method to be more efficient and simple, providing accurate results in fewer computational operations. Given its flexibility and ability to solve nonlinear equations, the iterative method can be used to solve more complex linear and nonlinear fractional partial differential equations.

In this paper, the generalized differential transform method is implemented for solving several linear fractional partial differential equations arising in fluid mechanics. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor's formula. Numerical illustrations of the time-fractional diffusion equation and the time-fractional wave equation are investigated to demonstrate the effectiveness of this method. Results obtained by using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere. The results reveal the method is feasible and convenient for handling approximate solutions of linear or nonlinear fractional partial differential equations.

We use the fractional variational iteration method (FVIM) with modified Riemann-Liouville derivative to solve some equations in fluid mechanics and in financial models. The fractional derivatives are described in Riemann-Liouville sense. To show the efficiency of the considered method, some examples that include the fractional Klein-Gordon equation, fractional Burgers equation, and fractional Black-Scholes equation are investigated.

This paper deals with the solutions of linear inhomogeneous time-fractional partial differential equations in applied mathematics and fluid mechanics. The fractional derivatives are described in the Caputo sense. The fractional Green function method is used to obtain solutions for time-fractional wave equation, linearized time-fractional Burgers equation, and linear time-fractional KdV equation. The new approach introduces a promising tool for solving fractional partial differential equations.

In this paper, a numerical approximation is shown to solve solutions of time fractional linear differential equations of variable order in fluid mechanics where the considered fractional derivatives of variable order are in the Caputo sense. Existence, uniqueness of solutions and Ulam–Hyers stability results are displayed. To solve the considered equations a numerical approximation based on the shifted Legendre polynomials are proposed. To perform the method, an operational matrix of fractional derivative with variable-order is derived for the shifted Legendre polynomials to be applied for developing the unknown function. By substituting the aforesaid operational matrix into the considered equations and using the properties of the shifted Legendre polynomials together with the collocation points, the main equations are reduced to a system of algebraic equations. The approximate solution is calculated by solving the obtained system which is technically easier for checking. We also study the error analysis for the approximate solution yielded by the introduced method. Finally, the accuracy and performance of the proposed method are checked by some illustrative examples. The illustrative examples results establish the applicability and usefulness of the proposed method.

The aim of this paper is to combined the variational iteration method with Aboodh transform method to solve linear and nonlinear fractional partial differential equations. Some illustrative examples are given as the linear and nonlinear fractional Klein-Gordon equations and the time fractional diffusion equation. The results reveal that this method is very effective, simple and can be applied to other physical differential equations with fractional order. The fractional derivative is taken in the Caputo sense.

Analytical approach to linear fractional partial differential equations arising in fluid mechanics

The generalized differential transform method is implemented for solving time-fractional partial differential equations in fluid mechanics. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor’s formula. Results obtained by using the scheme presented here agree well with the numerical results presented elsewhere. The results reveal the method is feasible and convenient for handling approximate solutions of time-fractional partial differential equations.

The improved fractional sub-equation method and its applications to the space–time fractional differential equations in fluid mechanics

An efficient numerical scheme to solve fractional diffusion-wave and fractional Klein–Gordon equations in fluid mechanics

Two examples on the property of the noise in the systems of equations of fluid mechanics