An approximation procedure for time optimal control problems for the linear wave equation is analyzed. Its asymptotic behavior is investigated and an optimality system including a transversality conditions for the regularized and unregularized problems are derived. The regularized problem also serves as a starting point for numerical techniques. For this purpose a family of parameterized optima...

Soliton equations are infinite-dimensional integrable systems described by nonlinear evolution equations. As one of the soliton equations, long wave equation takes on profound significance of theory and reality. By using the method of nonlinearization, the relation between long wave equation and second-order eigenvalue problem is generated. Based on the nonlinearized Lax pairs, Euler-Lagrange f...

Abstract The significance of different types periodic solutions in nonlinear equations is vital across various practical applications. Our objective this study was to uncover novel forms for the modified regularized long wave equation. This particular model holds great importance realm physics as it characterizes propagation weak nonlinearity and space-time dispersion waves, encompassing phenom...

In this article, we develop a high-order efficient numerical scheme to solve the initial-boundary problem of the MRLW equation. The method is based on a combination between the requirement to have a discrete counterpart of the conservation of the physical "energy" of the system and finite difference method. The scheme consists of a fourth-order compact finite difference approximation in space a...

Recent studies of the evolution of weakly nonlinear long waves in shear flows have revealed that when the wave field contains a critical layer, a new nonlinear wave equation is needed to describe the wave evolution. This equation is of the same type as the well-known Korteweg-de Vries equation but has a more complicated nonlinear structure. Our main interest is in the steady solitary wave solut...

<p style='text-indent:20px;'>We study a porous medium equation with dissipative and hyperdiffusive effects no sign restrictions on the diffusion coefficient.</p>

This paper is concerned with an inverse random source scattering problem in an inhomogeneous background medium. The wave propagation is modeled by the stochastic Helmholtz equation with the source driven by an additive white noise. The goal is to reconstruct the statistical properties of the random source such as the mean and variance from the boundary measurement of the radiated random wave fi...

In this paper, high accurate numerical solutions of the regularised long-wave (RLW) equation is going to be obtained by using effective algorithm including finite difference method, differential quadrature and Rubin-Graves type linearization technique. Solitary wave Maxwellian initial condition based generation are successfully. To observe development present algorithm, results compared with ma...