The exact solutions for the fractional Riemann wave equation in quantum mechanics and optics

In this paper, the fractional Riemann wave equation with M-truncated derivative (FRWE-MTD)is considered. The Jacobi elliptic function method and the modified extended tanh function method are applied to acquire new elliptic, rational, hyperbolic, and trigonometric functions solutions. Moreover, we expand some earlier studies. The obtained solutions are important in explaining some exciting physical phenomena, since the Riemann wave equation is used in various fields, including quantum mechanics, optics, signal processing, and general relativity. Also, this equation is used to describe the propagation of waves in various dispersive systems, where wave motion is affected by the medium through which it travels. Several 3D and 2D graphs are shown to demonstrate how the M-truncated derivative affects the exact solutions of the FRWE-MTD.