Time Fractional Burgers Equation Research Articles

Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation

This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs.

Solving Inverse System of Time-Fractional Burgers Equation

Solving Inverse System of Time-Fractional Burgers Equation

An Efficient Cubic B-Spline Technique for Solving the Time Fractional Coupled Viscous Burgers Equation

The second order Burger’s equation model is used to study the turbulent fluids, suspensions, shock waves, and the propagation of shallow water waves. In the present research, we investigate a numerical solution to the time fractional coupled-Burgers equation (TFCBE) using Crank–Nicolson and the cubic B-spline (CBS) approaches. The time derivative is addressed using Caputo’s formula, while the CBS technique with the help of a θ-weighted scheme is utilized to discretize the first- and second-order spatial derivatives. The quasi-linearization technique is used to linearize the non-linear terms. The suggested scheme demonstrates unconditionally stable. Some numerical tests are utilized to evaluate the accuracy and feasibility of the current technique.

A Spectral Galerkin Method Based on the Modified Bases for Solving Time-fractional Burgers Equation

A Spectral Galerkin Method Based on the Modified Bases for Solving Time-fractional Burgers Equation

Two linear finite difference schemes based on exponential basis for two-dimensional time fractional Burgers equation

In present paper, we propose tailored finite point method (TFPM) based on exponential basis for solving two-dimensional time fractional Burgers equation. We use the L1 and L1-2 discretization formulas for the Caputo fractional derivative in time direction and use the TFPM to approximate the spatial derivatives. This method with the help of exponential functions fit the properties of the local solution in time and space simultaneously which act as basis functions in the frame of TFPM. We focus on constructing two linear implicit finite difference schemes, i.e. a five-point node centered scheme and a four-point cell-centered scheme based on exponential functions for deriving the numerical solution of given problem. The stability and convergence of fully discrete schemes based on five-point node centered are discussed. At last, we use some numerical examples to show the accuracy and efficiency of the presented schemes.

A Potent Collocation Approach Based on Shifted Gegenbauer Polynomials for Nonlinear Time Fractional Burgers’ Equations

This paper presents a numerical strategy for solving the nonlinear time fractional Burgers's equation (TFBE) to obtain approximate solutions of TFBE. During this procedure, the collocation approach is used. The proposed numerical approximations are supposed to be a double sum of the products of two sets of basis functions. The two sets of polynomials are presented here: a modified set of shifted Gegenbauer polynomials and a shifted Gegenbauer polynomial set. Some specific integers and fractional derivatives are explicitly given as a combination of basis functions to apply the proposed collocation procedure. This method transforms the governing boundary-value problem into a set of nonlinear algebraic equations. Newton's approach can be used to solve the resulting nonlinear system. An analysis of the precision of the proposed method is provided. Various examples are presented and compared to some existing methods in the literature to prove the reliability of the suggested approach.

Analytical solutions and mathematical simulation of traveling wave solutions to fractional order nonlinear equations

We investigate the tanh–coth method’s usefulness in understanding space–time fractional nonlinear evolution problems. To use this method and then examine their plots, we take into consideration three significant equations: the space–time fractional Burgers equation, the space–time fractional regularized long wave equation, and the space–time fractional Boussinesq equation. The modified Riemann–Liouville derivatives are used to define the fractional derivative. The approach provides a linearization- or small perturbation-free analytical solution in the form of a convergent series with easily calculable components. For the aforementioned nonlinear fractional equations, we have discovered their precise solutions. We utilize a generalized fractional complex transform to translate these to ordinary differential equations which subsequently resulted into number of exact solutions. The method provides us abundant analytical traveling wave solutions containing fewer number parameters with the aid of Mathematica and MATLAB.

Solution of the Modified Time Fractional Coupled Burgers Equations Using Laplace Adomian Decompostion Method

Abstract In this work, a coupled system of time-fractional modified Burgers’ equations is considered. Three different fractional operators: Caputo, Caputo-Fabrizio and Atangana-Baleanu operators are implemented for the equations. Also, two different scenarios are examined for each fractional operator: when the initial conditions are u(x, y, 0) = sin(xy), v(x, y, 0) = sin(xy), and when they are u(x, y, 0) = e{−kxy}, v(x, y, 0) = e{−kxy}, where k, α are some positive constants. With the aid of computable Adomian polynomials, the solutions are obtained using Laplace Adomian decomposition method (LADM). The method does not need linearization, weak nonlinearity assumptions or perturbation theory. Simulations are also presented to support theoretical results, and the behaviour of the solutions under the three different fractional operators compared.

A new approximate method to the time fractional damped Burger equation

<abstract><p>In this article, we study a Caputo fractional model, namely, the time fractional damped Burger equation. As the main mathematical tool of this article, we apply a new approximate method which is called the approximate-analytical method (AAM) to deal with the time fractional damped Burger equation. Then, a new approximate solution of this considered equation was obtained. It may be used to characterize nonlinear phenomena of the shallow water wave phenomena. Thereby, it provides a new window for us to find the time fractional damped Burger equation new evolutionary mechanism.</p></abstract>

Variational Iteration Method for Solving Time Fractional Burgers Equation Using Maple

The Time Fractional Burger equation was solved in this study using the Mabel software and the Variational Iteration approach. where a number of instances of the Time Fractional Burger Equation were handled using this technique. Tables and images were used to present the collected numerical results. The difference between the exact and numerical solutions demonstrates the effectiveness of the Mabel program’s solution, as well as the accuracy and closeness of the results this method produced. It also demonstrates the Mabel program’s ability to quickly and effectively produce the numerical solution.

Application of the B-spline Galerkin approach for approximating the time-fractional Burger's equation

<abstract> <p>This paper presents a numerical scheme based on the Galerkin finite element method and cubic B-spline base function with quadratic weight function to approximate the numerical solution of the time-fractional Burger's equation, where the fractional derivative is considered in the Caputo sense. The proposed method is applied to two examples by using the $L_2$ and $ {L_\infty } $ error norms. The obtained results are compared with a previous existing method to test the accuracy of the proposed method.</p> </abstract>

A local projection stabilization virtual element method for the time-fractional Burgers equation with high Reynolds numbers

We propose and analyze a local projection stabilization virtual element method for time-fractional Burgers equation on polygonal meshes, whose solutions display a weak singularity at the initial time. Based on the L1 scheme on a graded mesh in time and virtual element method in space, the stability and existence and uniqueness of the fully discrete scheme are proved; and the priori error estimates in L2 norm is derived with local projection stabilization term. Numerical experiments show that our method is effective for time fractional Burgers equation with high Reynolds number.

Lie symmetry analysis of time fractional Burgers equation, Korteweg-de Vries equation and generalized reaction-diffusion equation with delays

In this paper, Lie symmetry analysis method is applied to time fractional Burgers equation, Korteweg-de Vries equation and generalized reaction-diffusion equation with delays, respectively. The Lie symmetries for fractional partial differential equations with delays (DFPDEs) are obtained, and the group classifications of the equations are established. The obtained group generators are used to reduce the DFPDEs to fractional ordinary differential equations with delays (DFODEs). Some exact solutions constructed for the DFODEs generate group-invariant solutions of the discussed DFPDEs.

On fractional symmetry group scheme to the higher-dimensional space and time fractional dissipative Burgers equation

In this paper, we studied a higher-dimensional space and time fractional model, namely, the (3+1)-dimensional dissipative Burgers equation which can be used to describe the shallow water waves phenomena. Here, the analyzed tool is the Lie symmetry scheme in the sense of the Riemann–Liouville fractional derivative. First of all, the symmetry of this considered equation was yielded. Then, based on the above obtained symmetry, the one-parameter Lie group was obtained. Subsequently, this model can be changed into the lower-dimensional equation with the Erdélyi–Kober fractional operators. Lastly, conservation laws of this studied equation via a new conservation theorem were also received. After such a series of processing, these new results play an important role in our understanding of this higher-dimensional space and time differential equations.

L1/LDG Method for the Generalized Time-Fractional Burgers Equation in Two Spatial Dimensions

L1/LDG Method for the Generalized Time-Fractional Burgers Equation in Two Spatial Dimensions

Solution of Non-linear Fractional Burger's Type Equations Using The Laplace Transform Decomposition Method

Our goal in this paper is to use combined Laplace transform (CLT) and Adomian decomposition method(ADM) (that will be explained in section 3), to study approximate solutions for non-linear time-fractionalBurger's equation, fractional Burger's Kdv equation and the fractional modi?ed Burger's equation for theCaputo and Conformable derivatives. Comparison between the two solutions and the exact solution is made.Here we report that the Laplace transform decomposition method (LTDM) proved to be e?cient and beused to obtain new analytical solutions of nonlinear fractional di?erential equations (FDEs).

Truncation and convergence dynamics: KdV Burgers model in the sense of Caputo derivative

This study examines the time fractional KdV Burgers equation with the initial conditions by using the extended result on Caputo formula, finite difference method (FDM). For this reason, various fractional differential operators are defined and analyzed. In order to check the stability of the numerical scheme, the Fourier-von Neumann technique is used. By presenting an example of KdV Burgers equation above mentioned issues are discussed and numerical solutions of the error estimates have been found for the FDM. For the errors in $L_2$ and $L_\infty$ the method accuracy has been controlled. Moreover, the obtained results have been compared with the exact solution for different cases of non-integer order and the behavior of the potentials u is presented as a graph. The numerical results have been shown in tables.

Fourier spectral approximation for generalized time fractional Burgers equation

Fourier spectral approximation for generalized time fractional Burgers equation

Application of Pell collocation method for solving the general form of time-fractional Burgers equations

Application of Pell collocation method for solving the general form of time-fractional Burgers equations

An accurate, robust, and efficient finite difference scheme with graded meshes for the time-fractional Burgers’ equation

This work presents the L1 implicit difference scheme based on non-uniform meshes for the time-fractional Burgers equation. We find that the difficulty arising from the singularity of the exact solution at t=0 can be overcome by non-uniform meshes. We yield the unconditional stability and optimal convergence rate of h2+τmin{rα,2−α} by means of the energy method, when selecting r=(2−α)/α. Convergence rates are higher than some recently studied schemes. Numerical experiments confirm the predicted theoretical estimate.