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A new computational procedure is offered to provide simple, accurate and flexible methods for using modern computers to give numerical evaluations of the various Bessel functions. The Trapezoidal Rule, applied to suitable integral…
Often the easiest way to discretize an ordinary or partial differential equation is by a rectangular numerical method, in which n basis functions are sampled at m>>n collocation points. We show how eigenvalue problems can be solved in this…
The focus of this article is the approximation of functions which are analytic on a compact interval except at the endpoints. Typical numerical methods for approximating such functions depend upon the use of particular conformal maps from…
The method of computing eigenvectors from eigenvalues of submatrices can be shown as equivalent to a method of computing the constraint which achieves specified stationary values of a quadratic optimization. Similarly, we show computation…
Conventionally, piecewise polynomials have been used in the boundary elements method (BEM) to approximate unknown boundary values. Since infinitely smooth radial basis functions (RBFs) are more stable and accurate than the polynomials for…
Estimating the number of eigenvalues located in a given interval of a large sparse Hermitian matrix is an important problem in certain applications and it is a prerequisite of eigensolvers based on a divide-and-conquer paradigm. Often an…
Numerical solutions of differential equations are usually not smooth functions. However, they should resemble the smoothness of the corresponding real solutions in one way or another. In two of our recent papers, a kind of spacial…
Despite its numerical challenges, finite element method is used to compute viscous fluid flow. A consensus on the cause of numerical problems has been reached; however, general algorithms---allowing a robust and accurate simulation for any…
This paper presents a new approach for the computation of eigenvalues of the generalized spheroidal wave equations. The novelty of the present method is in the use of the analytical derivatives of the eigenvalues to minimize losses in…
We present a comparative numerical study for three functionals used for variational mesh adaptation. One of them is a generalisation of Winslow's variable diffusion functional while the others are based on equidistribution and alignment.…
Many real life problems can be reduced to the solution of a complex exponentials approximation problem which is usually ill posed. Recently a new transform for solving this problem, formulated as a specific moments problem in the plane, has…
This work deals with the problem of choosing a time step for the numerical solution of boundary value problems for parabolic equations. The problem solution is derived using the fully implicit scheme, whereas a time step is selected via…
In [2] we introduced a method combining together an observability inequality and a spectral decomposition to get a logarithmic stability estimate for the inverse problem of determining both the potential and the damping coefficient in a…
We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility…
A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full…
Many computer vision and human-computer interaction applications developed in recent years need evaluating complex and continuous mathematical functions as an essential step toward proper operation. However, rigorous evaluation of this kind…
This paper proposes a methodology to estimate characteristic functions of stochastic differential equations that are defined over polynomials and driven by L\'evy noise. For such systems, the time evolution of the characteristic function is…
Variable-exponent fractional models attract increasing attentions in various applications, while the rigorous analysis is far from well developed. This work provides general tools to address these models. Specifically, we first develop a…
We calculate eigenvalues of one-dimensional quantum-systems by the exact numerical solution of the Lippmann-Schwinger equation, analogous to the scattering problem. To illustrate our method, we treat elementary problems: the harmonic and…
When the eigenvalues of the coefficient matrix for a linear scalar ordinary differential equation are of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The…