We apply the modified variational iteration method MVIM for solving the singular and nonsingular initial and boundary value problems in this paper. The proposed modification is made by introducing Adomian’s polynomials in the correct functional. The suggested algorithm is quite efficient and is practically well suited for use in such problems. The proposed iterative scheme finds the solution wi...

A theory for explicit Runge-Kutta schemes applied to the initial value problem for a first-order system of differential equations with a singularity of the first kind is developed. It is shown that, in general, the order of convergence is at most two but that the usual order up to a logarithmic term can be obtained for level three and four schemes applied to specific problems.

The algorithm of Mathias and Stewart [A block QR algorithm and the singular value decomposition, Linear Algebra and Its Applications, 182:91-100, 1993] is examined as a tool for constructing regularized solutions to rank-deficient and ill-posed linear equations. The algorithm is based on a sequence of QR factorizations. If it is stopped after the first step it produces that same solution as the...

In this work a new method is developed to obtain C1-approximate solutions of initial and boundary-value problems generated from a one parameter second order singular ordinary differential equation. Information about the order of approximation is also given by introducing the so called growth index of a function. Conditions are given for the existence of such approximations for initial and bound...

A problem of paramount importance in both pure (Restricted Invertibility problem) and applied mathematics (Feature extraction) is the one of selecting a submatrix of a given matrix, such that this submatrix has its smallest singular value above a specified level. Such problems can be addressed using perturbation analysis. In this paper, we propose a perturbation bound for the smallest singular ...

Using the methods of mathematical analysis and computer algebra, we developed a new reliable algorithm of initial-value method for solving nonlinear singular perturbation problems. To mechanize the algorithm a MatLab procedure called IVPHZ was established. The IVPHZ can handle problems with one or two boundary layers as well as problems with an internal shock layer. Testing of the IVPHZ was mad...

In this paper, we consider some inverse singular value problems for Toeplitz-related matrices. We construct a Toeplitz-plus-Hankel matrix from prescribed singular values including a zero singular value. Then we find a solution to the inverse singular value problem for Toeplitz matrices which have double singular values including a double zero singular value.