On a partial differential equation in 4-dimensional Euclidean space

The objective of this article is to construct in a hyper-rectangular region of the 4-dimensional Euclidean space a solution of the Goursat problem for the following equation:
$$L(u)=\sum_{i_1=0}^{m_1}\sum_{i_2=0}^{m_2}\sum_{i_3=0}^{m_3}\sum_{i_4=0}^{m_4}a_{i_1i_2i_3i_4}(x_1,x_2,x_3,x_4)\frac{\partial^{i_1+i_2+i_3+i_4}u}{\partial x_1^{i_1}\partial x_2^{i_2}\partial x_3^{i_3}\partial x_4^{i_4}}=F(x_1,x_2,x_3,x_4).$$