A Stable Difference Scheme for a Third-Order Partial Differential Equation

Abstract

The nonlocal boundary-value problem for a third-order partial differential equation$$ \left\{\begin{array}{l}\frac{d^3u(t)}{dt^3}+A\frac{du(t)}{dt}=f(t),\kern1em 0<t<1,\\ {}u(0)=\gamma u\left(\lambda \right)+\varphi, \kern1em {u}^{\prime }(0)=\alpha {u}^{\prime}\left(\lambda \right)+\psi, \left|\gamma \right|<1,\\ {}{u}^{{\prime\prime} }(0)=\beta {u}^{{\prime\prime}}\left(\lambda \right)+\xi, \kern1em \left|1+\beta \alpha \right|>\left|\alpha +\beta \right|,0<\lambda \le 1\end{array}\right. $$in a Hilbert space H with a self-adjoint positive definite operator A is considered. A stable three-step difference scheme for the approximate solution of the problem is presented. The main theorem on stability of this difference scheme is established. As applications, the stability estimates for the solution of difference schemes of the approximate solution of three nonlocal boundary-value problems for third-order partial differential equations are obtained. Numerical results for one- and two-dimensional third-order partial differential equations are provided.