No abstract

This paper investigates the well-posedness of a boundary value problem on the semiaxis for a class of third-order operator-differential equations whose principal part has multiple real characteristics. We obtain sufficient conditions for the existence and uniqueness of the solution of a boundary value problem in the Sobolev-type space W 3 2 (R + ; H). These conditions are expressed in terms of the operator coefficients of the investigated equation. We find relations between the estimates of the norms of intermediate derivatives operators in the subspace W 3 2 (R + ; H) and the solvability conditions. Furthermore, we calculate the exact values of these norms. The results are illustrated with an example of the initial-boundary value problems for partial differential equations. MSC: 34G10; 47A50; 47D03; 47N20 Keywords: well-posed and unique solvability; operator-differential equation; multiple characteristic; self-adjoint operator; the Sobolev-type space; inter-mediate derivatives operators; factorization of pencilswhere E σ is the resolution of the identity for A.

In this paper, we use the improved modified extended tanh-function method to obtain exact solutions for the nonlinear longitudinal wave equation in magneto-electro-elastic circular rod. With the aid of this method, we get many exact solutions like bright and singular solitons, rational, singular periodic, hyperbolic, Jacobi elliptic function and exponential solutions. Moreover, the two-dimensional and the three-dimensional graphs of some solutions are plotted for knowing the physical interpretation.

In this paper, a class of operator-differential equation of the first order with multiple characteristics is considered, for which the initial boundary value problem on the semi-axis is well-posed and uniquely solvable in the Sobolev space.