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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
##### MSC:
 35G15 Boundary value problems for linear higher-order PDEs 44A10 Laplace transform