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.
In this paper, the existence and uniqueness of the interface coupling (IC) of time and spatial (TS) arbitrary-order fractional (AOF) nonlinear hyperbolic scalar conservation laws (NHSCL) are investigated. The technique of arbitrary fractional characteristic method (AFCM) is used to accomplish this task. We apply Jumarie’s modification of Riemann–Liouville and Liouville–Caputo’s definition to extend some formulae to the arbitrary-order fractional calculus. Then these formulae are utilized to prove the main theorem. In this process, we develop an analytic method, which gives us the ability to find the solution of IC AOF NHSCL. The feature of this method is that it enables us to verify that the obtained solution satisfies the fractional partial differential equation (FPDE), and the solution is unique. Furthermore, a few examples illustrate the implementation of this technique in the application section.
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