Cauchy problem of the generalized double dispersion equation

Convergence of Finite Difference Schemes for a Multidimensional Boussinesq Equation

“…denotes the space of continuous functions with continuous derivatives up to order m with respect to x and order n with respect to t. The existence of a classical (local or global) solution with the smoothness prescribed above is proved in the 1D case in [14], while for the multi-dimensional case similar results for local solutions are established in [15].…”

“…d = 1). The existence (both local and global in time) and uniqueness of weak and strong solutions in Sobolev spaces for the 1D problem are treated in [8,13,14]. Sufficient conditions for blow-up of the solution are given in [6,13].…”

“…d > 1) is less studied. The dependence of existence, smoothness and blow-up of the solution on the nonlinear function f (u) is investigated in [14,15] for isotropic Sobolev spaces and in [12] for specially designed anisotropic Sobolev spaces. The numerical investigation of the 2D BE is also in its initial stage (see e.g.…”

“…The discrete balance law (12) valid for the solution to the scheme (11) fully corresponds to the energy equation [14] valid for the solution to the initial problem (2)-(4). The scheme (10) does not have a strict conservation of the discretized energy functional (E h v) (k) , but it satisfies similar balance identities given below.…”

Явные Решения Обобщенного Уравнения Буссинеска

“…Это уравнение исследовалось в работах [3], [4]. В работах [5]- [8] рассматривалась начально-краевая задача и за-дача Коши для обобщенного уравнения с двумя дисперсиями, которое включает уравнение (3) как частный случай. Симметрийные редукции и точные решения имеют немало различных важных приложений в контексте дифференциальных уравнений.…”