Calculation of Eigenvalues of Initial-Boundary Value Problems Defined on Finite Connected Quantum Graphs with Time-Varying Edges

The development of new computationally efficient methods for solving spectral problems for discrete semi-bounded differential operators defined on graphs with time-varying parameters is associated with the advancement of new technologies in science and engineering. In this article, algorithms for calculating the eigenvalues of initial-boundary value problems for partial differential operators given on finite connected graphs with time-varying edges are developed using parabolic equations as an example. Analytical formulas are presented for finding approximate values of the eigenvalues of these operators at the required time instants. The developed method allows one to extend a previously obtained technique for solving inverse spectral problems defined on quantum graphs with constant geometry to quantum graphs with time-varying geometry. Numerical experiments to calculate the eigenvalues of model problems were conducted in the Maple mathematical environment. The experimental results suggest the good computational efficiency of the developed technique.

Keywords

initial-boundary value problems; connected graphs; eigenvalues and eigenfunctions of operators; discrete and semi-bounded operators; Galerkin method; regularized trace method.

References

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