Exact solutions and numerical study of time fractional Burgers' equations

Abstract In this paper, we investigate the exact and numerical solutions of the time fractional Burgers’ equation. By using Cole–Hopf transformation and the method of variables separation, the exact solution is presented in terms of the Mittag-Leffler functions. Moreover, by using the L1-scheme on graded meshes and the Legendre–Galerkin spectral method, the numerical approximation is obtained. A numerical example is presented to confirm all the results.

[1]  Chein-Shan Liu,et al.  Recovering a source term in the time-fractional Burgers equation by an energy boundary functional equation , 2018, Appl. Math. Lett..

[2]  Dongfang Li,et al.  Linearized compact ADI schemes for nonlinear time-fractional Schrödinger equations , 2018, Appl. Math. Lett..

[3]  Talaat S. El-Danaf,et al.  Parametric spline functions for the solution of the one time fractional Burgers’ equation , 2012 .

[4]  Seakweng Vong,et al.  Unconditional Convergence in Maximum-Norm of a Second-Order Linearized Scheme for a Time-Fractional Burgers-Type Equation , 2018, J. Sci. Comput..

[5]  E. Hopf.  The partial differential equation ut + uux = μxx , 1950 .

[6]  Chengjian Zhang,et al.  A linear finite difference scheme for generalized time fractional Burgers equation , 2016 .

[7]  Bangti Jin,et al.  Correction of High-Order BDF Convolution Quadrature for Fractional Evolution Equations , 2017, SIAM J. Sci. Comput..

[8]  Zohreh Asgari,et al.  Efficient numerical schemes for the solution of generalized time fractional Burgers type equations , 2018, Numerical Algorithms.

[9]  Jie Shen,et al.  Spectral Methods: Algorithms, Analysis and Applications , 2011 .

[10]  Roberto Garra,et al.  Fractional-calculus model for temperature and pressure waves in fluid-saturated porous rocks. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Jose L. Gracia,et al.  Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation , 2017, SIAM J. Numer. Anal..

[12]  D. Kaya,et al.  Numerical and exact solutions for time fractional Burgers' equation , 2017 .

[13]  Eskandar Naraghirad,et al.  A new computational method based on optimization scheme for solving variable-order time fractional Burgers' equation , 2019, Math. Comput. Simul..

[14]  Zhimin Zhang,et al.  Linearized Galerkin FEMs for Nonlinear Time Fractional Parabolic Problems with Non-smooth Solutions in Time Direction , 2019, J. Sci. Comput..

[15]  A. Esen,et al.  A unified approach for the numerical solution of time fractional Burgers’ type equations , 2016 .