On integral equations for the Riemann function
The paper is devoted to the determination of the Riemann function of some equations. For the equation \[ u_{xy}+ au_x+ bu_y+ cu= 0,\tag{1} \] the Riemann function is a solution of the conjugated equation \(v_{xy}- (av)_x- (bv)_y+ cv= 0\), satisfying the conditions \(v(x,\tau, t,\tau)= \exp \int^x_t b(\xi,\tau) d\xi\), \(v(t,y,t,\tau)= \exp\int^y_\tau a(\tau,\eta) d\eta\), where \(a,b,c\in \mathbb{C}^2\) in the domain under consideration, and there exist \(a_x,b_x\in \mathbb{C}^2\). It is proved that in the case when \(a= a_1(y)+ \lambda x\), \(b= b_1(x)+ \lambda y\), \(\lambda= \text{const}\), \(c-ab-\lambda= \varphi(x)\psi(y)\), the Riemann function of the equation (1) is given by the formula \[ v= J_0\Biggl\{ 2\Biggl[ \int^x_t \varphi(\xi) d\xi \int^y_\tau \psi(\eta) d\eta\Biggr]^{1/2}\Biggr\} \exp\Biggl [\int^x_t b_1(\xi) d\xi+ \int^y_\tau a_1(\eta) d \eta+\lambda(xy- t\tau)\Biggr]. \] For the equation \[ u_{xyz}+ au_{xy}+ bu_{yz}+ cu_{xz}+ du_x+ eu_y+ fu_z+ gu= 0,\tag{2} \] the Riemann function is a solution of the equation \(v_{xyz}- (av)_{xy}- (bv)_{yz}- (cv)_{xz}+ (dv)_x+ (ev)_y+ (fv)_z- gv= 0\), satisfying the conditions \[ \begin{aligned} v(t,\tau,z,t,\tau,\vartheta) & = \exp \int^z_\vartheta a(t,\tau,\zeta) d\zeta,\;v(t,y,\vartheta, t,\tau,\vartheta)= \exp \int^y_\tau c(t,\eta,\vartheta) d\eta,\\ v(x,\tau,\vartheta, t,\tau,\vartheta) & = \exp \int^x_t b(\xi,\tau,\vartheta) d\xi.\end{aligned} \] The following expressions are considered: \[ \begin{aligned} h_1 & = a_x+ ab- e,\;h_2= a_y+ ac- d,\;h_3= b_y+ bc- f,\;h_4= b_z+ ab- e,\\ h_5 & = c_x+ bc- f, h_6= c_z+ ac- d, h_7= d_x+ bd- g, h_8= e_y+ ce- g, h_9= f_z+ af- g.\end{aligned} \] It is proved that the Riemann function of the equation (2) is given by \[ v(x,y,z,t,\tau, \vartheta)= J_0(2\sqrt\omega) \exp\sigma(x,y,z,t,\tau, \vartheta) \] if \(h_4\equiv h_6\equiv h_9\equiv 0\), \(c= c_1(y, z)+ \alpha(z)x\), \(b= b_1(x, z)+ \alpha(y)y\), \(f- bc- \alpha= \varphi(x, z)\psi(y,z)\), where \(\omega= \int^x_t \varphi(\xi,\vartheta) d\xi \int^y_\tau \psi(\eta,\vartheta) d\eta\), \[ \sigma= \int^x_t b_1(\xi,\vartheta) d\xi+ \int^y_\tau c_1(\eta, \vartheta) d\eta+ \int^x_\vartheta a(x,y,\zeta) d\zeta+ (xy- t\tau) \alpha(z). \] Some similar cases are considered as well.
integral equations for the Riemann function
scientific article; zbMATH DE number 1309950