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An integral equation method for solving the Yukawa-Beltrami equation on a multiply-connected sub-manifold of the unit sphere is presented. A fundamental solution for the Yukawa-Beltrami operator is constructed. This fundamental solution can…
We study L\'evy flights {{with arbitrary index $0< \mu \leq 2$}} inside a potential well of infinite depth. Such problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in…
The Legendre-based ultraspherical spectral method for ordinary differential equations is combined with a formula for the convolution of two Legendre series to produce a new technique for solving linear Fredholm and Volterra…
We study linear systems of ordinary differential equations of an arbitrary order on a finite interval with the most general (generic) inhomogeneous boundary conditions in Sobolev spaces. We investigate the character of solvability of…
In the first part of the article we establish the existence in the sense of sequences of solutions in $H^{2}(R)$ for some nonhomogeneous linear differential equation in which one of the terms has the argument translated by a constant. It is…
In this paper, we employ the fractional $q$-calculus in solving a triple system of $q$-Integral equations, where the kernel is the third Jackson $q$-Bessel functions. The solution is reduced to two simultaneous Fredholm $q$-integral…
This paper addresses the problem of finding an asymptotic solution for first and second order integro-differential equations containing an arbitrary kernel, by evaluating the corresponding inverse Laplace and Fourier transforms. The aim of…
This work presents a collocation method for solving linear Fredholm integral equations of the second kind defined on a closed contour in the complex plane. The right-hand side of the equation is a piecewise continuous function that may have…
We analyze a discretization method for solving nonlinear integral equations that contain multiple integrals. These equations include integral equations with a Volterra series, instead of a single integral term, on one side of the equation.…
We investigate the problem of pointwise convergence of the family of non-linear integral operators: \begin{equation} L_\lambda(f,x) = \int_a^b \sum_{m=1}^N f^m(t) K_{\lambda ,m}(x,t) dt, \end{equation} where $\lambda $ is a real parameters,…
A global approximation method of Nystr\"om type is explored for the numerical solution of a class of nonlinear integral equations of the second kind. The cases of smooth and weakly singular kernels are both considered. In the first…
This paper is motivated by Phong and Stein's paper on non-standard singular integrals with mixed homogeneities. Our purpose is to study these new non-standard convolution singular integrals and establish the boundedness of these singular…
We consider elliptic equations of order $2m$ in a bounded domain $Q\subset\mathbb R^n$ with nonlocal boundary-value conditions connecting the values of a solution and its derivatives on $(n-1)$-dimensional smooth manifolds $\Gamma_i$ with…
This paper explores the solution of Fredholm-like equations with infinite dimensional solution spaces. We set out to find a method for determining a particular solution to a Fredholm-like equation subject to a given constraint. The…
A semilinear parabolic problem of second order with an unknown time-convolution kernel is considered. The missing kernel is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is…
In this paper, we obtain some new results about weakly singular integral inequalities. These inequalities are used to discuss the global existence and uniqueness results for fractional differential equations of Riemann-Liouville type. Some…
We study multivariate linear tensor product problems with some special properties in the worst case setting. We consider algorithms that use finitely many continuous linear functionals. We use a unified method to investigate tractability of…
The Fredholm integral equations of the first kind are a classical example of ill-posed problem in the sense of Hadamard. If the integral operator is self-adjoint and admits a set of eigenfunctions, then a formal solution can be written in…
Some corrections are made in our article, which was published in Appl. Anal. Optim. Vol. 3 (2019), No. 1, 103--127. These corrections are intended to transform the equation \eqref{eq:1.1} \begin{equation}\label{eq:1.1} x(t) +…
The purpose of this paper is to give a convergence analysis of the iterative scheme: \bee u_n^\dl=qu_{n-1}^\dl+(1-q)T_{a_n}^{-1}K^*f_\dl,\quad u_0^\dl=0,\eee where $T:=K^*K,\quad T_a:=T+aI,\quad q\in(0,1),\quad a_n:=\alpha_0q^n,…