Application of Bessel functions and Jacobian free Newton method to solve time-fractional Burger equation

Abstract

In this paper, a novel method based on Bessel functions (BF), generalized Bessel functions (GBF), the collocation method and the Jacobian free Newton-Krylov sub-space (JFNK) will be introduced to solve the nonlinear time-fractional Burger equation. In this paper, an implicit formula is introduced to calculate Riemann–Liouville fractional derivative of GBFs, that can be very useful in spectral methods. In this work, the nonlinear time-fractional Burger equation is converted to a nonlinear system of algebraic equations via collocation algorithm based on BFs and GBFs without any linearization and descretization methods. Finally, by using JFNK, the solution of this nonlinear system will be achieved. To show the reliability and applicability of the proposed method, we solve some examples of time-fractional Burger equation and compare our results with others.

Keywords

• nonlinear time-fractional burger equation
• generalized bessel function
• collocation method
• jacobian free newton method
• riemann–liouville fractional operations