Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace method

In this paper we study the analytic solutions of Burgers-type nonlinear fractional equations by means of the Invariant Subspace Method. We first study a class of nonlinear equations directly related to the time-fractional Burgers equation. Some generalizations linked to the forced time-fractional Burgers equations and variable-coefficient diffusion are also considered. Finally we study a Burgers-type equation involving both space and time-fractional derivatives

Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics

A fractional time derivative is introduced into the Burger's equation to model losses of nonlinear waves. This term amounts to a time convolution product, which greatly penalizes the numerical modeling. A diffusive representation of the fractional derivative is adopted here, replacing this nonlocal operator by a continuum of memory variables that satisfy local-in-time ordinary differential equations. Then a quadrature formula yields a system of local partial differential equations, well-suited to numerical integration. The determination of the quadrature coefficients is crucial to ensure both the well-posedness of the system and the computational efficiency of the diffusive approximation. For this purpose, optimization with constraint is shown to be a very efficient strategy. Strang splitting is used to solve successively the hyperbolic part by a shock-capturing scheme, and the diffusive part exactly. Numerical experiments are proposed to assess the efficiency of the numerical modeling, and to illustrate the effect of the fractional attenuation on the wave propagation.Comment: submitted to Siam SIA

Local Fractional Operator for a One-Dimensional Coupled Burger Equation of Non-Integer Time Order Parameter

In this study, approximate solutions of a system of time-fractional coupled Burger equations were obtained by means of a local fractional operator (LFO) in the sense of the Caputo derivative. The LFO technique was built on the basis of the standard differential transform method (DTM). Illustrative examples used in demonstrating the effectiveness and robustness of the proposed method show that the solution method is very efficient and reliable as "“ unlike the variational iteration method "“ it does not depend on any process of identifying Lagrange multipliers, even while still maintaining accuracy