Time Fractional Burgers Research Articles

Lie symmetry analysis, conservation laws and exact solutions of the generalized time fractional Burgers equation

Under investigation in this work are the invariance properties of the generalized time fractional Burgers equation, which can be used to describe the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. The Lie group analysis method is applied to consider its vector fields and symmetry reductions. Furthermore, based on the sub-equation method, a new type of explicit solutions for the equation is well constructed with a detailed analysis. By means of the power series theory, exact power series solutions of the equation are also constructed. Finally, by using the new conservation theorem, conservation laws of the equation are well constructed with a detailed derivation.

A unified approach for the numerical solution of time fractional Burgers’ type equations

In this paper, a relatively new approach is devised for obtaining approximate solution of time fractional partial differential equations. Time fractional diffusion equation and time fractional Burgers-Fisher equation are solved with Haar wavelet method where fractional derivatives are Caputo derivative. Time discretization of the problems made by L1 discretization formula and space derivatives discretized by Haar series. $ L_{2}$ and $ L_{\infty}$ error norms are used for measuring accuracy of the proposed method. Numerical results obtained with proposed method compared with exact solutions as well as with available results from the literature. The numerical results verify the feasibility of Haar wavelet combined with L1 discretization formula for the considered problems.

A linear finite difference scheme for generalized time fractional Burgers equation

This paper is concerned with the numerical solutions of the generalized time fractional burgers equation. We propose a linear implicit finite difference scheme for solving the equation. Iterative methods become dispensable. As a result, the computational cost can be significantly reduced compare to the usual implicit finite difference schemes. Meanwhile, the finite difference method is proved to be unconditional globally stable and convergent. Numerical examples are shown to demonstrate the accuracy and stability of the method.

Consistent Riccati Expansion Method and Its Applications to Nonlinear Fractional Partial Differential Equations**Supported by the National Natural Science Foundation of China under Grant Nos. 11101332, 11201371, 11371293 and the Natural Science Foundation of Shaanxi Province under Grant No. 2015JM1037

In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie's modified Riemann–Liouville derivative. The efficiency and power of this approach are demonstrated by applying it successfully to some important fractional differential equations, namely, the time fractional Burgers, fractional Sawada–Kotera, and fractional coupled mKdV equation. A variety of new exact solutions to these equations under study are constructed.

The New Iterative Method for Approximate Solutions of Time Fractional Kdv, K(2,2), Burgers, and Cubic Boussinesq Equations

The New Iterative Method for Approximate Solutions of Time Fractional Kdv, K(2,2), Burgers, and Cubic Boussinesq Equations

Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order

AbstractThe fractional Burgers equation describes the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. The Laplace homotopy perturbation method is discussed to obtain the approximate analytical solution of space-fractional and time-fractional Burgers equations. The method used combines the Laplace transform and the homotopy perturbation method. Numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional orders.

Numerical solution of time fractional Burgers equation

Abstract In this article, the time fractional order Burgers equation has been solved by quadratic B-spline Galerkin method. This method has been applied to three model problems. The obtained numerical solutions and error norms L2 and L∞ have been presented in tables. Absolute error graphics as well as those of exact and numerical solutions have been given.

Application of the Subequation Method to Some Differential Equations of Time-Fractional Order

The main goal of this paper is to develop subequation method for solving nonlinear evolution equations of time-fractional order. We use the subequation method to calculate the exact solutions of the time-fractional Burgers, Sharma–Tasso–Olver, and Fisher's equations. Consequently, we establish some new exact solutions for these equations.

Multi-wave solutions of the space–time fractional Burgers and Sharma–Tasso–Olver equations

Abstract Based on the improved generalized exp-function method, the space–time fractional Burgers and Sharma–Tasso–Olver equations were studied. The single-wave, double-wave, three-wave and four-wave solution discussed. With the best of our knowledge, some of the results are obtained for the first time. The improved generalized exp-function method can be applied to other fractional differential equations.

An algorithm for the approximate solution of nonlinear total time fractional-order Burger equation

In this paper we have established the simplest equation method to find approximate solutions for total Burgurs equations with time fractional derivative. This method is effective in finding approximate traveling wave solutions of nonlinear, fractional evolution equations (NLEEs) in mathematical physics. The effectiveness of this manageable method has been shown by applying it to several examples in time fractional Burgers equations. The present approach has the potential to apply to other nonlinear fractional differential equations. All numerical calculations in the present study have been performed on a PC, applying programs written in Mathematica and Maple.

Numerical Solution of Time Fractional Burgers Equation by Cubic B-spline Finite Elements

We present some numerical examples which support numerical results for the time fractional Burgers equation with various boundary and initial conditions obtained by collocation method using cubic B-spline base functions. The aim of this paper is to show that the finite element method based on the cubic B-spline collocation method approach is also suitable for the treatment of the fractional differential equations. The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme.

On the Solution of Burgers’ Equation with the New Fractional Derivative

Abstract Firstly in this article, the exact solution of a time fractional Burgers’ equation, where the derivative is conformable fractional derivative, with dirichlet and initial conditions is found byHopf-Cole transform. Thereafter the approximate analytical solution of the time conformable fractional Burger’s equation is determined by using a Homotopy Analysis Method(HAM). This solution involves an auxiliary parameter ~ which we also determine. The numerical solution of Burgers’ equation with the analytical solution obtained by using the Hopf-Cole transform is compared.

Exact Solutions of Fractional Burgers and Cahn-Hilliard Equations Using Extended Fractional Riccati Expansion Method

Based on a general fractional Riccati equation and with Jumarie’s modified Riemann-Liouville derivative to an extended fractional Riccati expansion method for solving the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation, the exact solutions expressed by the hyperbolic functions and trigonometric functions are obtained. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.

Generalized Kudryashov Method for Time-Fractional Differential Equations

In this study, the generalized Kudryashov method (GKM) is handled to find exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. These time-fractional equations can be turned into another nonlinear ordinary differantial equation by travelling wave transformation. Then, GKM has been implemented to attain exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. Also, some new hyperbolic function solutions have been obtained by using this method. It can be said that this method is a generalized form of the classical Kudryashov method.

Time-fractional Burgers equation for dust acoustic waves in a two different temperatures dusty plasma

The reductive perturbation method has been used to derive the Burgers equation for dust acoustic shock waves in unmagnetized plasma having electrons, singly charged ions, hot and cold dust species with Boltzmann distributions for electrons and ions in the presence of the cold (hot) dust viscosity coefficients. The time-fractional Burgers equation is formulated using Euler-Lagrange variational technique and is solved using the variational-iteration method. The effect of time fractional parameter on the behavior of the shock waves in the dusty plasma has been investigated.

Parametric spline functions for the solution of the one time fractional Burgers’ equation

This paper aims to present a general framework of the cubic parametric spline functions to develop a numerical method for obtaining an approximate solution for the time fractional Burgers’ equation. The truncation error of the method is theoretically analyzed. Using Von Neumann method, the proposed method is also shown to be conditionally stable. Two numerical examples are then included to illustrate the practical implementation of the proposed method. The obtained results reveal that the proposed technique is very effective, convenient and quite accurate to such considered problems.

Lie Group Classifications and Non-differentiable Solutions for Time-Fractional Burgers Equation

Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained.

Analytical Approach to Space- and Time-Fractional Burgers Equations

A scheme is developed to study numerical solution of the space- and time-fractional Burgers equations under initial conditions by the homotopy analysis method. The fractional derivatives are considered in the Caputo sense. The solutions are given in the form of series with easily computable terms. Numerical solutions are calculated for the fractional Burgers equation to show the nature of solution as the fractional derivative parameter is changed.

Numerical algorithm based on Adomian decomposition for fractional differential equations

In this paper, a novel algorithm based on Adomian decomposition for fractional differential equations is proposed. Comparing the present method with the fractional Adams method, we use this derived computational method to find a smaller “efficient dimension” such that the fractional Lorenz equation is chaotic. We also apply this new method to the time-fractional Burgers equation with initial and boundary value conditions. Numerical results and computer graphics show that the constructed numerical is efficient.

Approximate analytical solution of the space-and time-fractional Burgers equations

This paper presents approximate analytical solutions for the generalized fractional time-space Burgers equations. The time-and space-fractional derivatives are considered in the Caputo sense. The application of variational iteration method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of the space-and time-fractional Burgers equation. The solutions of our model equation are calculated in the form of convergent series with easily computable components. Two examples are presented to show the application of the present technique. The effect of varying the order of the time-and space-fractional derivatives on the behaviour of solutions has been investigated.