The Tricomi problem for the Lavrent'ev--Bitsadze equation with retarded argument
For the Lavrent'ev-Bitsadze equation with retarded argument \[ u_{xx}(x,y)+(\operatorname {sgn}x)u_{yy}(x,y)-H(x-\tau)u(x-\tau ,y)=0, \] where \(0<\tau\) and \(H(\xi)\) is the Heaviside function, the authors consider the following BVP: Find a solution of the above equation in the domain \(D=\{0<x<\infty\), \(0<y<\pi \}\cup \{-\frac{\pi }{2}<x<0\), \(-x<y<\pi +x\}\), which satiesfies the boundary conditions \(u(x,0)=u(x,\pi)=0,0<x<\infty\); \(u(-y,y)=\psi (y)\), \(0<y<\pi /2\); \(\lim u(x,y)=0\), \(0<y<\pi /2\), \(x\rightarrow \infty\). Using some appropriate integral equations, the uniqueness of the solution for \(\tau \leq \sqrt{2}\) is proved. Construction of a solution is given by its appropriate representation with some biorthogonal series.
scientific article; zbMATH DE number 1915931