in this paper, we studied the numerical solution of nonlinear weakly singular volterra-fredholm integral equations by using the product integration method. also, we shall study the convergence behavior of a fully discrete version of a product integration method for numerical solution of the nonlinear volterra-fredholm integral equations. the reliability and efficiency of the proposed scheme are...

2 Ep-Groupoids and Generalized Maps 17 2.1 Ep-Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Functors and Equivalences . . . . . . . . . . . . . . . . . . . . 21 2.3 Inversion of Equivalences and Generalized Maps . . . . . . . . 28 2.4 Strong Bundles over Ep-Groupoids . . . . . . . . . . . . . . . 33 2.5 Auxiliary Norms . . . . . . . . . . . . . . . . . . . . . . . . . 4...

Recently, a beautiful identity due to Borodin and Okounkov was proved for Toeplitz determinants which shows how one can write a Toeplitz determinant as a Fredholm determinant. In this note we generalize this to the Wiener-Hopf case. The proof in the Wiener-Hopf case follows identically with the second one given in [1]. We include it here for completeness sake and because the nature of the ident...

Let A be a bounded operator on a separable, infinite dimensional Hilbert space H. Moreover, assume that A is not in the set C + K(H) of operators expressible as a sum of a scalar multiple of identity and a compact operator. What is the smallest similarity invariant semigroup containing operator A? Equivalently, which operators can be expressed as products of operators, similar to A? A partial a...

We prove that the operator G, the closure of the first-order differential operator −d/dt + D(t) on L2(R, X), is Fredholm if and only if the not well-posed equation u′(t) = D(t)u(t), t ∈ R, has exponential dichotomies on R+ and R− and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, D(t)...

We study interfaces in an Allen-Cahn equation, separating two metastable states. Our focus is on a directional quenching scenario, where a parameter renders the system bistable in a half plane and monostable in its complement, with the region of bistability expanding at a fixed speed. We show that the growth mechanism selects a contact angle between the boundary of the region of bistability and...

We investigate the relationship between the decay at infinity of the right-hand side f and solutions u of an equation Lu = f when L is a second order elliptic operator on RN . It is shown that when L is Fredholm, u inherits the type of decay of f (for instance, exponential, or power-like). In particular, the generalized eigenfunctions associated with all the Fredholm eigenvalues of L, isolated ...

I n recent years, many numerical methods have been proposed for solving fuzzy linear integral equations. For example, in [10], the authors used the divided differences and finite differences methods for solving a parametric of the fuzzy Fredholm integral equations of the second kind. Also, in [9], a numerical method is proposed for the approximate solution of fuzzy linear Fredholm functional in...

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrödinger operators in L2(Ω; dnx), where Ω ⊂ Rn, n = 2, 3, are open sets with a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of modified Fredholm perturbation determinants associated with operators in L2(Ω; dnx) to modifie...