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This paper is concerned with computations of a few smaller eigenvalues (in absolute value) of a large extremely ill-conditioned matrix. It is shown that smaller eigenvalues can be accurately computed for a diagonally dominant matrix or a…
We introduce basic aspects of new operator method, which is very suitable for practical solving differential equations of various types. The main advantage of the method is revealed in opportunity to find compact exact operator solutions of…
Nowadays, fractional differential equations are a well established tool to model phenomena from the real world. Since the analytical solution is rarely available, there is a great effort in constructing efficient numerical methods for their…
Integral transformations are used to estimate high order derivatives of various special functions. Applications are given to numerical integration, where estimates of high order derivatives of the integrand are needed to achieve bounds on…
This paper provides a new algorithm for solving inverse problems, based on the minimization of the $L^2$ norm and on the control of the Total Variation. It consists in relaxing the role of the Total Variation in the classical Total…
We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to…
We introduce a new system of split variational inequality problems which is a natural extension of split variational inequality problem in semi-inner product spaces. We use the retraction technique to propose an iterative algorithm for…
We present a class of nonconforming virtual element methods for general fourth order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element…
A typical system of k difference (or differential) equations can be compressed, or folded into a difference (or ordinary differential) equation of order k. Such foldings appear in control theory as the canonical forms of the controllability…
Much recent work has addressed the solution of a family of partial differential equations by computing the inverse operator map between the input and solution space. Toward this end, we incorporate function-valued reproducing kernel Hilbert…
This paper proposes a new and efficient numerical algorithm for recovering the damping coefficient from the spectrum of a damped wave operator, which is a classical Borg-Levinson inverse spectral problem. The algorithm is based on inverting…
For partial differential equations (PDEs) that have $n\geq2$ independent variables and a symmetry algebra of dimension at least $n-1$, an explicit algorithmic method is presented for finding all symmetry-invariant conservation laws that…
We present a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is…
We give a new method for the reduction of tensor integrals to finite integral representations and UV divergent analytic expressions. This includes a new method for the handling of the gamma-algebra. TYPO IN EQUATION (5) CORRECTED, MACROS…
We consider structure-preserving methods for conservative systems, which rigorously replicate the conservation property yielding better numerical solutions. There, corresponding to the skew-symmetry of the differential operator, that of…
In the present work, by taking advantage of a so-called practical limitation of fractional derivatives, namely, the absence of a simple chain and Leibniz's rules, we proposed a generalized fractional calculus of variation where the…
The first part of this paper introduces sufficient conditions to determine conservation laws of diffusion equations of arbitrary fractional order in time. Numerical methods that satisfy a discrete analogue of these conditions have…
We study a scalar integro-differential conservation law. The equation was first derived in [2] as the slow erosion limit of granular flow. Considering a set of more general erosion functions, we study the initial boundary value problem for…
This paper presents the Euler-Lagrange equations for fractional variational problems with multiple integrals. The fractional Noether-type theorem for conservative and nonconservative generalized physical systems is proved. Our approach uses…
Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far…