Approximate solution of a boundary problem for a linear complex differential equation of the second order

Abstract

The paper considers the problem $$ D^{(2)}w+a(z,\bar z)Dw+b(z,\bar z)w=f(z,\bar z), $$ with boundary conditions $$ \aligned c_1(z)\a_{g(z)}w+c_2\a_{g(z)}Dw&=c_3(z), d_1(z)\a_{h(z)}w+d_2\a_{h(z)}Dw&=d_3(z). \endaligned $$ The problem is solved approximately, by using the formulas $$ \align 2\df{z^2}{h^2}(w_{i+1}-2w_i+w_{i-1})+a_i\df zh(w_{i+1}-w_{i- 1})+b_iw_i&=f_i,\quad i=1,\dots,n-1, c_1(z)w_0+c_2(z)\df zh(-w_2+4w_1-3w_0)&=c_3(z), d_1(z)w_n+d_2(z)\df zh(3w_n-4w_{n-1}+w_{n-2})&=d_3(z). \endalign $$