Abstract:
We consider the equation $y^mu_{xx}-u_{yy}-b^2y^mu=0$ in the rectangular area $\{(x,y)\mid0<x<1,\ 0<y<T\}$, where $m>0$, $b\ge0$, $T>0$ are given real numbers. For this equation we study problems with initial conditions $u(x,0)=\tau(x)$, $u_y(x,0)=\nu(x)$, $0\le x\leq1$, and nonlocal boundary conditions $u(0,y)=u(1,y)$, $u_x(0,y)=0$ or $u_x(0,y)=u_x(1,y)$, $u(1,y)=0$ with $0\le y\le T$. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems.
Keywords:nonlocal problem, spectral method, completeness, sum of biorthogonal series.
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