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Consistent models for electrical networks with distributed parameters. (English) Zbl 0762.35044
The authors consider the mathematical model of an electric circuit with distributed parameter lines and lumped capacitors. This model is reduced to the following initial-boundary value problem \[ {\mathcal M}u_ t(t,x)={\mathcal N}u_{xx}(t,x)-{\mathcal P}u(t,x), \]
\[ S(u_ x)_ b(t)=-{\mathcal G}u_ b(t)+\int^ t_ 0{\mathcal K}(t-s)u_ b(s)ds+B(t),\quad u(0,x)=u_ 0(x), \] where \(u(t,x)=(u_ 1,\dots,u_ n)\), \(u_ b(s)=(u_ 1(s,0),u_ 1(s,d_ 1),\dots,u_ n(s,0),u_ n(s,d_ n)),\) \({\mathcal M},{\mathcal N},{\mathcal P}\) are given diagonal \(n\times n\) matrices, \({\mathcal S}\) is a given \(2n\times 2n\) diagonal matrix, \({\mathcal G}\) and \({\mathcal K}\) are given \(2n\times 2n\) matrices, \(B\) is a given \(2n\)-dimensional vector, \(u_ 0\) is given \(n\)-dimensional vector.
It is proved that under certain conditions on the data this problem has a unique variational (weak) solution. A concrete circuit is considered which is investigated by the above mathematical model.
MSC:
35K45 Initial value problems for second-order parabolic systems
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35A15 Variational methods applied to PDEs
47B44 Linear accretive operators, dissipative operators, etc.
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