## 1 Introduction
Meshless full-wave methods, such as methods of analytical regularization or integral equations, have guaranteed convergence and offer controlled accuracy, limited, in principle, only by the precision of the corresponding programming language, where the home-made code is created. Employing the functional approach of the Riemann-Hilbert problem, in [18] the authors effectively inverse the static part of the scattering operator, which arises in the \(H\)-polarized electromagnetic wave diffraction by graphene grating with the dielectric substrate. In [19], after regularization, the set of the integral equations with additional conditions is obtained for the pre-fractal grating of impedance strips. The approach involving the Helmholtz decomposition and the Galerkin method is applied in [20; 21] to both single graphene disks and several layers of graphene disks. As a result, second kind Fredholm equations are obtained. The method of singular integral equations (SIEs) with Nystrom-type discretization schemes is proposed in [22; 23; 24; 25; 26] to study the scattering of waves emitted by the magnetic line source by the dielectric cylinder of circular cross section with graphene strips, and of the \(H\)-polarized waves by the graphene grating inside the dielectric waveguide. The Nystrom-type algorithms of discretization can be considered as a numerical regularization. The \(H\)-polarization case is of interest (vector \(\overset{\rightarrow}{H}\) is parallel to the edges of the strips), because here, unlike in the \(E\)-polarization case, plasmon resonances can arise. In this situation, where natural waves are radiated from the dielectric waveguide due to the scattering by graphene strips, as in the case of PEC (perfectly electric conducting) analogs, the elevation angle of the main lobe strongly depends on the excitation frequency and the period. By varying the chemical potential, it is possible to control a number of characteristics, including the radiated power and the side-lobes level at a fixed frequency, as well as to change the plasmon resonance frequency. In this paper, we consider the radiation of the natural waves of the planar dielectric waveguide with the infinite graphene plane and with the graphene strip grating. The propagation constant characterizing the natural wave directly depends on the conductivity of the graphene plane. Thus, manipulating the chemical potential of this plane will allow us to regulate the main lobe elevation angle of the radiation pattern. Furthermore, by manipulating the chemical potential of graphene strips forming the grating, the radiated power and maximum of the main lobe can be controlled. We should mention that THz generators are piecemeal and not available on an industrial scale. Our purpose is the theoretical study of the influence of the chemical potential on the radiation characteristics with the use of the developed full-wave meshless method. Here we have two sub-problems. One is determining the propagation constants of the natural waves (eigenwaves) of the dielectric waveguide backed by the infinite graphene plane, which gives a transcendent complex-valued equation. Another is the scattering problem of these natural waves by the system of finite number of graphene strips placed on the waveguide. The solution gives a singular integral equation [27]. For graphene, the following boundary conditions can be used [10]:  
\[E_{\tau}^{+}=\frac{1}{\sigma}(H_{\tau}^{+}-H_{\tau}^{-}), \tag{1}\]  
\[E_{\tau}^{+}=E_{\tau}^{-}, \tag{2}\]  
where symbols \"\(\pm\)\" denote the limit values of the field components above (for \"\(+\)\") or below (for \"\(-\)\") the interfaces. The conductivity of graphene \(\sigma\) can be determined using the Kubo formalism [28; 29]. Spatial dispersion can be neglected for surface waves propagating along the graphene plane for the considered frequency range and graphene parameters. When spatial dispersion is taken into account, one more type of mode appears.