The Riemann method of the Goursat problem for a class of third order differential equations (Q1405698)

In [\textit{V. I. Zhegalov}, Three-dimensional analog of the Goursat problem (Russian), in: Nonclassical equations and equations of mixed type, Novosibirsk, 94--98 (1990; Zbl 0799.35044), \textit{V. F. Volkodavov} and \textit{G. N. Kadirov}, Goursat and Darboux problems for some differential equations in three-dimensional spaces, VINITI, Dep. No. 3857-B91 (1991), \textit{V. F. Volkodavov} and \textit{V. F. Zakharov}, Riemann functions for one class of differential equations in three-dimensional Euclidean space and their applications, Samara Pedagogical Inst. Press (1996)] the solution of the Goursat problem for third-order equations in the three-dimensional space was expressed via a large number of addents. There arise difficulties to use this solution in the study of other boundary value problems. In this paper the equation \[ {\mathfrak I}(U)\equiv U_{xyz}+ a_1 U_{xy}+ a_2 U_{yz}+ b_1 U_x+ b_2U_y+ b_3 U_z+ cU= 0 \] is considered in the first octant \(G= \{(x,y,z): 0< x,y,z<+\infty\}\) as \(a_{1xy}\), \(a_{2xz}\), \(a_{3yz}\), \(b_{1x}\), \(b_{2y}\), \(b_{3z}\), \(c\in C(G)\). For the Goursat problem: Find the solution of the equation in the domain \(G\), which is continuous in \(\overline G\) and satisfies the boundary value conditions \[ U(0,y,z)= \varphi_1(y,z),\quad 0\leq y,\;z<+\infty, \] \[ U(x,0,z)= 0,\quad 0\leq x,\;z<+\infty,\quad U(x,y,0)= 0,\quad 0\leq x,\;y<+\infty \] it is proved the following Theorem: If \(\varphi_{1ts}(t, s)\in [0,+\infty;0,+\infty]\), then the unique solution of the Goursat problem is defined by the formula \[ \begin{multlined} U(x,y,z)= \int^y_0 dt \int^z_0 [\varphi_{1ts}(t,s)+ a_1(0,t,s) \varphi_{1t}(t+ s)+ a_2(0,t,s) \varphi_{1t}(t,s)+\\ b_1(0,t,s) \varphi_1(t,s)] R(0,t,s; x,y,z)\,ds.\end{multlined} \] In such way a compact form of the solution is obtained which is an essential contribution.