. The impedance change of an induction coil surrounding a metal tube adapter is investigated by the truncated region eigenfunction expansion (TREE) method. Conventional TREE method is inapplicable to this problem as a consequence of the numerical overflow of eigenfunctions of the air-metal multi-subdomain regions. The difficulty is surmounted by a normalization procedure for the numerical eigenfunctions obtained from the finite element method (FEM). Efficient algorithm is devised by the Clenshaw-Curtis quadrature rule for integrals involving the numerical eigenfunctions. Numerical results of the TREE and FEM simulation coincide very well in all cases, and the efficiency of the proposed method is also confirmed.

In this study, we consider delta fractional Sturm--Liouville (DFSL) initial value problems in the sense of delta Caputo and delta Riemann-Liouville (R--L) operators. One of the properties of delta fractional difference operators which makes it different from nabla counterpart is to shift its domain. This feature makes it more complex than the nabla fractional operator. We obtain sum representation of solutions for DFSL initial value problems with the help of $\mathcal{Z}-$ transformation. Moreover, we get analytical solutions of homogeneous DFSL problem within Riemann-Liouville (R--L) and Caputo sense, discrete Sturm--Liouville (DSL) problem, continuous fractional Sturm--Liouville (FSL) problem in the sense of R--L and Caputo operators, and continuous Sturm--Liouville (SL) differential problem. From this point of view, we compare all the solutions with each other. Consequently, we show that all results for these four eigenvalue problems are compatible with each other and approach to each other while the orders tends to one, i.e. $\Delta^{\mu }\left( \Delta x\left( t-\mu \right) \right)\cong D_{0^{+}}^{\mu }\left( x^{\prime }\left( t\right) \right)\cong \Delta^2x(n-1) \cong x^{\prime \prime }\left( t\right) =\lambda x\left( t\right),\ \mu\rightarrow1 $ . We support our results comparatively by tables and simulations in detail.

Fourier expansions by the polyharmonic-Neumann eigenfunctions showed improved convergence compared to the Fourier expansions by the classical trigonometric system due to the rapid decay of the corresponding Fourier coefficients. Based on this evidence, we investigate interpolations on a finite interval that are exact for the polyharmonic-Neumann eigenfunctions and show the same benefits. Further, we improve the convergence of the interpolations by applying the idea of quasi-periodicity, where the basis functions are periodic on a slightly extended interval. We show that those interpolations have much better convergence rates away from the endpoints of the approximation interval. Moreover, the interpolations are more accurate on the entire interval. We prove those properties for a specific case of the polyharmonic-Neumann eigenfunctions known as the modified Fourier system. For other basis functions, we provide evidence based on the results of numerical experiments.

The article considers an internal boundary value problem of the distribution of an electrostatic field in a lens formed by two identical semi-infinite circular cylinders coaxially located inside an infinite external cylinder. The problem is reduced to solving a system of singular Wiener-Hopf integral equations, which is further solved by the Wiener-Hopf method using factorized Bessel functions. Solutions to the problem for each region inside the infinite outer cylinder are presented as exponentially converging series in terms of eigenfunctions and eigenvalues.