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Sonine kernel is characterized by the Sonine condition (denoted by SC) and is an important class of kernels in nonlocal differential equations and integral equations. This work proposes a SC with a more general form (denoted by gSC), which is more convenient than SC to accommodate complex kernels and equations. A typical kernel is given, and the first-kind Volterra integral equation under gSC is accordingly transformed and then analyzed. Based on these results, it is finally proved that the gSC is indeed equivalent to the original SC, which indicates that the Sonine kernel may be essentially characterized by the behavior of its convolution with the associated kernel at the starting point. [ABSTRACT FROM AUTHOR]

The interest of the majorizing (scalar) sequences lies in that, from their convergence, we can deduce the convergence of an iterative method in Banach spaces. We propose a new technique to construct majorizing sequences that generalizes that given by Kantorovich for Newton's method. [ABSTRACT FROM AUTHOR]

This paper mainly studies numerical solution to the Volterra integral equation of the second kind. By using the roots of Chebyshev polynomial as collocation points, a new collocation method is proposed to solve the Volterra integral equation of the second kind. The proposed method firstly interpolates the product of the kernel function and the unknown solution at the roots of Chebyshev polynomial. Then, the Volterra integral equation is transformed into a system of linear algebra equations by properties of Chebyshev polynomials. Finally, the numerical solution of the Volterra integral equation is obtained by the Chebyshev polynomial interpolation. In addition, the error estimates of the proposed method are provided in a semi-posteriori sense; and numerical examples are given to show effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We show that, by correctly selecting the probability distribution function p ( s ) for a particle’s distance-to-collision, the nonclassical diffusion equation can be represented exactly by the nonclassical linear Boltzmann equation for an infinite homogeneous medium. This choice of p ( s ) preserves the true mean-squared free path of the system, which sheds new light on the results obtained in previous work. [ABSTRACT FROM AUTHOR]

Abstract: In the anti-plane problem about a high-frequency diffraction by an interface crack located between two different elastic materials we propose a new asymptotic approach, which reduces the problem to the Wiener–Hopf integral equations. The key point of the method is a factorization of the symbolic function which is performed in an efficient way. As a result, the leading asymptotic term is written out in an explicit analytical form. [Copyright &y& Elsevier]

Abstract: By making use of asymptotic properties of nonoscillatory solutions, the oscillation behavior of solutions for the integro-dynamic equation and the integral equation on time scales is investigated. Easily verifiable sufficient conditions are established for the oscillation of all solutions. The results are new for both continuous and discrete cases. The paper is concluded by an open problem. [Copyright &y& Elsevier]

Abstract: We consider an American put option under the CEV process. This corresponds to a free boundary problem for a PDE. We show that this free boundary satisfies a nonlinear integral equation, and analyze it in the limit of small , where is the interest rate and is the volatility. We use perturbation methods to find that the free boundary behaves differently for five ranges of time to expiry. [Copyright &y& Elsevier]

Abstract: We establish the existence of analytic solutions to integral equations in the complex domain when the nonlinearity satisfies either a growth condition or a monotonic type condition. [Copyright &y& Elsevier]

Abstract: In this work, a functional approximation method for calculating the linear functional of the solution of second-kind Fredholm integral equations is developed. When the method is applied to the collocation method or to the multi-projection method, it generates approximations which exhibit superconvergence. [Copyright &y& Elsevier]

Abstract: A direct function theoretic method is applied to solve a weakly singular integral equation whose kernel involves logarithmic singularity. This method avoids the occurrence of strong singularity. The solution of this integral equation is then applied to re-investigate the well known problem of water wave scattering by a partially immersed vertical barrier. [Copyright &y& Elsevier]

In this paper, we are concerned with the existence and non-existence of positive solutions to the following integral equation related to the sharp Hardy–Littlewood–Sobolev inequality f q − 1 (x) = ∫ Ω K (x) f (y) K (y) | x − y | n − α d y + λ ∫ Ω f (y) | x − y | n − α − β d y , f ≥ 0 , x ∈ Ω ¯ , where q > 1 , 1 < α < n , 0 < β < n − α , Ω is a smooth bounded domain, K (x) is positive continuous in Ω ¯. For K ≡ c o n s t , β = 1 , the existence and non-existence of positive solutions to the equation has been studied by Dou and Zhu (2019). This paper is devoted to the case K (x) ⁄ ≡ c o n s t. We want to point out that the weight function will bring difficulty when estimating the energy of the energy functional according to the integral equation. [ABSTRACT FROM AUTHOR]