We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painlevé equation (or higher-order analogues), and admitting a large family of monodromy-preserving deformations. The solutions are certain semiclassical biorthogonal functions (and their Cauchy transforms), biorthogonal with respect to higher-order analogues of Spiridon...

We give a new characterisation of Borel summability of formal power series solutions to the n-dimensional heat equation in terms of holomorphic properties of the integral means of the Cauchy data. We also derive the Borel sum for the summable formal solutions. Mathematics Subject Classification (2010). 35K05, 35C10.

This paper contains a theoretical formulation of multiple interface cracks in two bonded dissimilar half planes subjected to in-plane traction. The distributed dislocation technique is used to construct integral equations for a dissimilar mediums weakened by several interface cracks. These equations are of Cauchy singular type at the location of dislocation, which are solved numerically to obta...

In this paper, we consider an acoustic wave transmission problem with mixed boundary conditions of Dirichlet, Neumann, and impedance type. We will derive a formulation as direct, space-time retarded integral equation, where both Cauchy data are kept unknowns on the part boundary. This requires definition single-trace spaces which incorporate homogeneous Dirichlet Neumann corresponding parts pro...

We consider a general class of convolution-type nonlocal wave equations modeling bidirectional propagation nonlinear waves in continuous medium. In the limit vanishing nonlocality we study behavior solutions to Cauchy problem. prove that, as kernel functions convolution integral approach Dirac delta function, converge strongly corresponding classical elasticity equation. An energy estimate with...

A variety of problems in differential equations ((abstract) functional differential equations, age-dependent population models (with and without delay), evolution equations with boundary conditions e.g.) can be written as semilinear Cauchy problems with a Lipschitz perturbation of a closed linear operator which is not non-densely defined but satisfies the estimates of the Hille&Yosida theorem. ...

won{gil park [won{gil park, j. math. anal. appl., 376 (1) (2011) 193{202] proved the hyers{ulam stability of the cauchy functional equation, the jensen functional equation and the quadraticfunctional equation in 2{banach spaces. one can easily see that all results of this paper are incorrect.hence the control functions in all theorems of this paper are not correct. in this paper, we correctthes...

The purpose of this paper is to approximate the solution of a Cauchy integral equation of the second kind, using projection, finite rank approximations and a regularization procedure. We us the Kantorovich projection, the Sloan projection, the Galerkin projection, respectively. We give a general framework and we prove the existence of the solution for a projection schemes. 2012 Elsevier Inc. Al...

Based on the theory of elasticity, previous analytical solutions concerning a penny-shaped interface crack employ the derivative of the crack surface opening displacements as the primary unknowns, thus leading to singular integral equations with Cauchy-type singularity. The solutions to the resulting integral equations permit only the determination of stress intensity factors and energy release...

In this brief note,  using the technique of measures of noncompactness, we give some extensions of Darbo fixed point theorem. Also we prove  an existence result for a quadratic  integral equation of Hammerstein type on an unbounded interval in two variables  which includes several classes of nonlinear integral equations of Hammerstein type. Furthermore, an example is presented to show the effic...