This paper considers the problem of viscous dissipation in the flow of Newtonian fluid through a tube of annular cross section, with Dirichlet boundary conditions. The solution of the problem is obtained by a series expansion about the complete eigenfunctions system of a Sturm-Liouville problem. Eigenfunctions and eigenvalues of this Sturm-Liouville problem are obtained by Galerkin’s method.

In this work, we prove the existence of a spectral function for singular q-Sturm-Liouville operator. Further, we establish a Parseval equality and expansion formula in eigenfunctions by terms of the spectral function.

We present a computational method for solving a class of boundary-value problems in Sturm-Liouville form. The algorithms are based on global polynomial collocation methods and produce discrete representations of the eigenfunctions. Error control is performed by evaluating the eigenvalue problem residuals generated when the eigenfunctions are interpolated to a finer discretization grid; eigenfun...

We consider the use of eigenfunctions of polyharmonic operators, equipped with homogeneous Neumann boundary conditions, to approximate nonperiodic functions in compact intervals. Such expansions feature a number of advantages in comparison with classical Fourier series, including uniform convergence and more rapid decay of expansion coefficients. Having derived an asymptotic formula for expansi...

A numerical method for solving the time-independent Schrödinger equation of a particle moving freely in a three-dimensional axisymmetric region is developed. The boundary of the region is defined by an arbitrary analytic function. The method uses a coordinate transformation and an expansion in eigenfunctions. The effectiveness is checked and confirmed by applying the method to a particular exam...

The linearized Vlasov equation for a plasma system in an external constant magnetic field and the corresponding linear Vlasov operator are studied. The solution of the Vlasov equation is found by the resolvent method. The spectrum and eigenfunctions of the Vlasov operator are also found. The spectrum of this operator consists of two parts: one is continuous and real; the other is discrete and c...

We present a novel surface smoothing framework using the Laplace-Beltrami eigenfunctions. The Green’s function of an isotropic diffusion equation on a manifold is analytically represented using the eigenfunctions of the Laplace-Beltraimi operator. The Green’s function is then used in explicitly constructing heat kernel smoothing as a series expansion of the eigenfunctions. Unlike many previous ...

This paper presents a stochastic meshless method for probabilistic analysis of linear-elastic structures with spatially varying random material properties. Using Karhunen-Loève (K-L) expansion, the homogeneous random field representing material properties was discretized by a set of orthonormal eigenfunctions and uncorrelated random variables. Two numerical methods were developed for solving th...

We present a novel surface smoothing framework using the Laplace-Beltrami eigenfunctions. The Green’s function of an isotropic diffusion equation on a manifold is analytically represented using the eigenfunctions of the Laplace-Beltraimi operator. The Green’s function is then used in explicitly constructing heat kernel smoothing as a series expansion of the eigenfunctions. Unlike many previous ...

A fully relativistic time-dependent close-coupling method is developed based on Dirac’s covariant formulation of quantum mechanics. The expansion of a one-electron wave function in spin-orbit eigenfunctions yields the well-known coupled Dirac equations in two radial wave functions, while the expansion of a twoelectron wave function in coupled spin-orbit eigenfunctions yields close-coupled Dirac...