Goursat boundary value problem (Q2730566)

The author proposes an algorithm of bilateral method for the approximate integration of the system of partial differential equations NEWLINE\[NEWLINED^{(1,1)} u_{i}(x,y)=f_{i}(x,y,u,D^{(1,0)}u, D^{(0,1)}u), \quad i=1,2;NEWLINE\]NEWLINE NEWLINE\[NEWLINEu=(u_1(x,y),u_2(x,y)), \quad D^{(k,j)}u_{i}(x,y)=\partial^{k+j} u_{i}(x,y)/\partial x^{k}\partial y^{j}NEWLINE\]NEWLINE with the boundary conditions NEWLINE\[NEWLINEu_1(x_0,y)=\phi_1(y),\;u_2(x_1,y)=\phi_2(y),\;u_1(x,y_1)=\psi_1(x),\;u_2(x,y_0)=\psi_2(x)NEWLINE\]NEWLINE and the consistency conditions \(\phi_1(y_1)=\psi_1(x_0), \phi_2(y_0)=\psi_2(x_1)\).