In present paper, a numerical approach for solving Cauchy type singular integral equations is discussed. Lagrange interpolation with Gauss Legendre quadrature nodes and Taylor series expansion are utilized to reduce the computation of integral equations into some algebraic equations. Finally, five examples with exact solution are given to show efficiency and applicability of the method. Also, w...

In this paper, we are interested in solving the Fermat-type equations x5 + y5 = dzp, where d is a positive integer and p a prime number ≥ 7. We describe a new method based on modularity theorems which allows us to improve all earlier results for this equation. We finally discuss the present limits of the method by looking at the case d = 3.

Convergence results are presented for rank-type difference equations, whose evolution rule is defined at each step as the kth largest of p univariate difference equations. If the univariate equations are individually contractive, then the equation converges to a fixed point equal to the kth largest of the individual fixed points of the univariate equations. Examples are max-type equations for k...

be linear and quasilinear evolution equations of parabolic type in a Banach space X respectively. By "parabolic type" we mean that A(t) and A(t,u) are all the infinitesimal generators of analytic linear semigroups on X we do not necessarily assume that the domains of the operators A(t) and A("t,u) are dense subspaces of X, so the semigroups generated by them may not be of class c0 J. The domain...

As important examples in the superstring theory and non-Kähler complex geometry, the complex manifolds #k(S 3×S3) for any k ≥ 2 [4, 11] also admit a non-vanishing holomorphic three-form [4] and a balanced metric [5]. Moreover, we know that #k(S 3 × S3) satisfies the ∂∂̄–lemma [4]. A natural question to ask is, whether #k(S 3 ×S3) admit a balanced metric ω0 such that (1.2) holds. Such a metric ω0...

A supersymmetric version is proposed for the well known Harry Dym system. A general class super Lax operator which leads to consistent equations is considered.