Recent zbMATH articles in MSC 45Ghttps://zbmath.org/atom/cc/45G2021-05-28T16:06:00+00:00WerkzeugOn the solution of quadratic nonlinear integral equation with different singular kernels.https://zbmath.org/1459.652402021-05-28T16:06:00+00:00"Basseem, M."https://zbmath.org/authors/?q=ai:basseem.m"Alalyani, Ahmad"https://zbmath.org/authors/?q=ai:alalyani.ahmadSummary: All the previous authors discussed the quadratic equation only with continuous kernels by different methods. In this paper, we introduce a mixed nonlinear quadratic integral equation (MQNLIE) with singular kernel in a logarithmic form and Carleman type. An existence and uniqueness of MQNLIE are discussed. A quadrature method is applied to obtain a system of nonlinear integral equation (NLIE), and then the Toeplitz matrix method (TMM) and Nystrom method are used to have a nonlinear algebraic system (NLAS). The Newton-Raphson method is applied to solve the obtained NLAS. Some numerical examples are considered, and its estimated errors are computed, in each method, by using Maple 18 software.On solution of a system of Hammerstein-Nemytskiĭ type nonlinear integral equations on the whole axis.https://zbmath.org/1459.450042021-05-28T16:06:00+00:00"Khachatryan, Kh. A."https://zbmath.org/authors/?q=ai:khachatryan.khachatur-aSummary: We study a system of Hammerstein-Nemytskii type nonlinear integral equations on whole axis. The system has direct application in the theory of Ricker's nonlinear system for travelling waves. The combination of special iterative methods with the theory of primitive matrices and properties of convolution operations allow us to prove existence of a positive solution (by components) in the space of integrable functions possessing zero limit at infinity.The coupled method for singularly perturbed Volterra integro-differential equations.https://zbmath.org/1459.652482021-05-28T16:06:00+00:00"Tao, Xia"https://zbmath.org/authors/?q=ai:tao.xia"Zhang, Yinghui"https://zbmath.org/authors/?q=ai:zhang.yinghuiSummary: In this work a coupled (LDG-CFEM) method for singularly perturbed Volterra integro-differential equations with a smooth kernel is implemented. The existence and uniqueness of the coupled solution is given, provided that the source function and the kernel function are sufficiently smooth. Furthermore, the coupled solution achieves the optimal convergence rate \(p+1\) in the \(L^{2}\) norm and a superconvergence rate \(2p\) at nodes for the numerical solution \(\hat{U}_{N}\) with the one-sided flux inside the boundary layer region under layer-adapted meshes uniformly with respect to the singular perturbation parameter \(\epsilon\).On the solutions of a delay functional integral equation of Volterra-Stieltjes type.https://zbmath.org/1459.450032021-05-28T16:06:00+00:00"El-Sayed, A. M. A."https://zbmath.org/authors/?q=ai:el-sayed.ahmed-mohamed-ahmed"Omar, Y. M. Y."https://zbmath.org/authors/?q=ai:omar.yasmin-m-ySummary: In this work, a delay functional integral equation of Volterra-Stieltjes type and an initial value problem of a delay integro-differential equation of Volterra-Stieltjes type will be considered. We study the existence of at least one or exact one solution. The continuous dependent of the unique solution will be proved.Stein-Weiss inequalities with the fractional Poisson kernel.https://zbmath.org/1459.350062021-05-28T16:06:00+00:00"Chen, Lu"https://zbmath.org/authors/?q=ai:chen.lu"Liu, Zhao"https://zbmath.org/authors/?q=ai:liu.zhao|liu.zhao.1"Lu, Guozhen"https://zbmath.org/authors/?q=ai:lu.guozhen"Tao, Chunxia"https://zbmath.org/authors/?q=ai:tao.chunxiaSummary: In this paper, we establish the following Stein-Weiss inequality with the fractional Poisson kernel:
\begin{align*}\qquad \int_{\mathbb{R}^n_+} \int_{\partial\mathbb{R}^n_+} |\xi|^{-\alpha} f(\xi) &\,P(x,\xi,\gamma)\, g(x)\, |x|^{-\beta}\, d\xi\, dx \\ &\leq C_{n,\alpha,\beta,p,q'}\, \|g\|_{L^{q'}(\mathbb{R}^n_+)}\, \|f\|_{L^p(\partial \mathbb{R}^n_+)}, \tag{\(\star\)}\end{align*}
where
\[P(x,\xi,\gamma)=\frac{x_n}{(|x'-\xi|^2+x_n^2)^{(n+2-\gamma)/2}},\]
\(2\le \gamma < n\), \(f\in L^p(\partial\mathbb{R}^n_+)\), \(g\in L^{q'}(\mathbb{R}^n_+)\), and \(p, q'\in (1,\infty)\) and satisfy \((n-1)/(np)+1/q'+(\alpha+\beta+2-\gamma)/{n}=1\). Then we prove that there exist extremals for the Stein-Weiss inequality \((\star)\), and that the extremals must be radially decreasing about the origin. We also provide the regularity and asymptotic estimates of positive solutions to the integral systems which are the Euler-Lagrange equations of the extremals to the Stein-Weiss inequality \((\star)\) with the fractional Poisson kernel. Our result is inspired by the work of Hang, Wang and Yan [\textit{F. Hang} et al., Commun. Pure Appl. Math. 61, No. 1, 54--95 (2008; Zbl 1173.26321)], where the Hardy-Littlewood-Sobolev type inequality was first established when \(\gamma=2\) and \(\alpha=\beta=0\). The proof of the Stein-Weiss inequality \((\star)\) with the fractional Poisson kernel in this paper uses recent work on the Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel by Chen, Lu and Tao, and the present paper is a further study in this direction.