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We study the problem of placing effective upper bounds for the number of zeros of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of…
The work deals with establishing the solvability of a system of integro-differential equations in the situation of the double scale anomalous diffusion. Each equation of such system involves the sum of the two negative Laplace operators…
We propose a new method for solution of the integrability problem for evolutionary differential-difference equations of arbitrary order. It enables us to produce necessary integrability conditions, to determine whether a given equation is…
We derive an explicit system of Picard-Fuchs differential equations satisfied by Abelian integrals of monomial forms and majorize its coefficients. A peculiar feature of this construction is that the system admitting such explicit…
A numerical procedure providing guaranteed two-sided bounds on the effective coefficients of elliptic partial differential operators is presented. The upper bounds are obtained in a standard manner through the variational formulation of the…
The Euler scheme is up to date the most important numerical method for ordinary differential inclusions, because the use of the available higher-order methods is prohibited by their enormous complexity after spatial discretization.…
We established a new eighth-order iterative method, consisting of three steps, for solving nonlinear equations. Per iteration the method requires four evaluations (three function evaluations and one evaluation of the first derivative).…
In this paper, we study existence results for initial value problems for hybrid fractional integro-differential equations. Our investigation is based on the Dhage hybrid fixed point theorem. Some fundamental fractional differential…
We present a new upper bound for the orders of derivatives in the Rosenfeld-Groebner algorithm. This algorithm computes a regular decomposition of a radical differential ideal in the ring of differential polynomials over a differential…
In this work we shall present a survey on problems and results on singular holomorphic foliations and Pfaff systems on complex manifolds assuming that these objects possess invariant analytic varieties. We will focus on recent results which…
This paper proves a local higher integrability result for the spatial gradient of weak solutions to doubly nonlinear parabolic systems. The new feature of the argument is that the intrinsic geometry involves the solution as well as its…
In this paper, a high-order exponential scheme is developed to solve the 1D unsteady convection-diffusion equation with Neumann boundary conditions. The present method applies fourth-order compact exponential difference scheme in spatial…
We introduce a novel monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. This problem is prevalent in…
The main contribution of this work is to construct and analyze stable and high order schemes to efficiently solve the two-dimensional time Caputo-Fabrizio fractional diffusion equation. Based on a third-order finite difference method in…
We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over $\Z$. The result improves previous work of Philippon, Berenstein-Yger and Krick-Pardo. We also present degree and height estimates of…
By using variational methods, the existence of infinitely many solutions for a nonlinear algebraic system with a parameter is established in presence of a perturbed Lipschitz term. Our goal was achieved requiring an appropriate behavior of…
We present bounds for the sparseness and for the degrees of the polynomials in the Nullstellensatz. Our bounds depend mainly on the unmixed volume of the input polynomial system. The degree bounds can substantially improve the known ones…
A unified approach to derive optimal finite differences is presented which combines three critical elements for numerical performance especially for multi-scale physical problems, namely, order of accuracy, spectral resolution and…
Weighted averaged finite difference methods for solving fractional diffusion equations are discussed and different formulae of the discretization of the Riemann-Liouville derivative are considered. The stability analysis of the different…
The notion of differentiation index for DAE systems of arbitrary order with generic second members is discussed by means of the study of the behavior of the ranks of certain Jacobian associated sub-matrices. As a by-product, we obtain upper…