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Solution of boundary value problems on a graph by an operational method. (Russian) Zbl 0727.35032
The boundary value problem for a partial differential equation on a graph is considered. The equation in every branch with number \(k=1,...,n\) is \[ \frac{\partial u_ k}{\partial x}(x,t)=A_ k(x)u_ k(x,t)+\sum^{d}_{i=1}B_{k,r}(x)\frac{\partial^ ru_ k}{\partial t^ r}(x,t)+f_ k(x,t). \] Here \(x\in [0,\ell_ k]\), \(\ell_ k\) is the length of the branch, \(t\geq 0\), \(u_ k(\cdot): [0,\ell_ k]\times [0,+\infty)\to {\mathbb{R}}^ m\). There are boundary conditions at the nodes of the graph. The main assumption is that J. Hadamard’s examples of special type do not exist (this proposal is a simpler variant of a dissipativity condition). An explicit formula is obtained for the Laplace transform \(\tilde v(x,\lambda\)) of the function \(v(x,t)=[u_ 1(\ell_ 1x,t),...,u_ n(\ell_ nx,t)]\), where \(x\in [0,1]\). The function \(\tilde v(x,\lambda\)) is proved to be the solution of a boundary value problem for an ordinary differential equation on the interval [0,1] with boundary conditions in \(x=0\) and \(x=1\).
Reviewer: N.G.Dokuchaev
35G15 Boundary value problems for linear higher-order PDEs
44A10 Laplace transform