The arbitrary-order fractional hyperbolic nonlinear scalar conservation law.

In this paper, we use a new powerful technique of arbitrary-order fractional (AOF) characteristic method (CM) to solve the AOF hyperbolic nonlinear scalar conservation law (HNSCL) of time and space. We present the existence and uniqueness of this class of equations in time and one-dimensional space of fractional arbitrary order. We extend Jumarie's modification of Riemann–Liouville and Caputo's definition of the fractional arbitrary order to introduce some formulae (Appl. Math. Lett. 22:378–385, 2009; Appl. Math. Lett. 18:739–748, 2005). Then, we use these formulae to prove the main theorem. In the application section, we use the analytical technique that is presented in the theorem to solve examples that are given. [ABSTRACT FROM AUTHOR]