The initial boundary-value problem for systems of fourth-order partial differential equations with two independent variables is considered. By using a new unknown eigenfunction, the problem under consideration is reduced to an equivalent nonlocal problem for a system of second-order hyperbolic-type integro-differential equations with integral conditions. An algorithm for finding an approximate solution of the resulting equivalent problem is proposed, and its convergence is proved. Conditions for the existence of a unique classical solution of the initial boundary-value problem for systems of fourth-order differential equations are established in terms of the coefficients of the system and the boundary matrices.

中文翻译：

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四阶双曲方程组初边值问题的一种求解方法

考虑了具有两个自变量的四阶偏微分方程组的初始边值问题。通过使用新的未知本征函数，所考虑的问题被简化为具有积分条件的二阶双曲型积分微分方程系统的等效非局部问题。提出了一种寻找所得等效问题近似解的算法，并证明了其收敛性。根据系统的系数和边界矩阵，建立了四阶微分方程组初始边值问题唯一经典解的存在条件。