The new wave solution of mathematical equations used in physics, engineering, and many applied sciences was found in this research using an alternative technique. For nonlinear partial differential equations without the term integer, our goal is to arrive at the analytical solution without the need for a new transformation to make the balancing term integer. To find the exact solutions to the Eckhaus equation and the cubic nonlinear Klein Gordon equation, as well as new type of complex hyperbolic trigonometric travelling wave solutions. In order to display the graphs showing the stationary wave, the parameters in these solutions are given specified values. Furthermore, few discussions about new complex solutions are presented. It is described by supplying the constants in traveling wave solutions, which are important both physically and mathematically, Finally, three-dimensional simulation is used to support these discussions

In this paper, we give a reconstruction formula for the potential q for a second order differential equation with boundary condition which contains spectral parameter. For this as methodology, we use Prufer substitution ¨ that has an advantage different from other methods. Because in this method, we do not need any information of eigenfunctions

In this paper, Conformable derivative order version of the Sturm-Liouville problem having modified Coulomb potential is obtained. The studied system proves the shape of the conformable derivative general statement of the Sturm-Liouville operator with boundary conditions. Furthermore, real of eigenvalues and 𝜶-orthogonal of eigenfunctions have been proved for the problem considered. Additionally, the representation of the solution of the Sturm-Liouville problem having modified Coulomb potential is found. The results are shown comparatively by figures.