The analytical solutions of long waves over geometries with linear and nonlinear variations in the form of power-law nonlinearities with solid inclined wall

In this paper, we derive the exact analytical solutions for the long-wave equation in both linear and nonlinear power-law form depth and breadth geometries containing a solid inclined wall. Firstly, we give general information about the concept of partial reflection and its components, and formulate the solid inclined wall boundary condition. For these specific power-law forms of depth and breadth geometries, we show that in the presence of the solid inclined wall, the long-wave equation admits solutions in terms of Bessel-Z functions and the Cauchy–Euler series. Since the presence of solid vertical wall removes the singular point from the domain, the solution admits both the first and the second kind of the Bessel functions and Cauchy–Euler series terms. We derive results for the general case and also discuss their significance using six different geometries with solid inclined wall.