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On integral equations for the Riemann function. (English. Russian original) Zbl 0932.35043
Russ. Math. 42, No. 1, 24-28 (1998); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1998, No. 1, 26-30 (1998).
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.

MSC:
 35G05 Linear higher-order PDEs 35C15 Integral representations of solutions to PDEs