Abstract
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.
Highlights
The notion of hyperbolic conservation laws (HCLs) was raised about five decades ago [1, 2]
Several phenomena occur in mathematical physics, and their mathematical models are expressed in the form of the interface coupling (IC) HCLs
We have used the generalization of the classical characteristic method that is extended to the fractional characteristic method
Summary
The notion of hyperbolic conservation laws (HCLs) was raised about five decades ago [1, 2]. The properties for a partial differential equations’ (PDEs’) system of this type were distinguished. The interface coupling (IC) of HCLs has important applications. Several phenomena occur in mathematical physics, and their mathematical models are expressed in the form of the IC HCLs. many researchers have tried to develop new techniques to find analytical and numerical solutions for IC HCLs. Many of them have been successful in introducing methods to find numerical solutions
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