Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2501.04701

Existence and Uniqueness of the Solution of Two-dimensional Fuzzy Volterra Integral Equation with Piecewise Kernel

Samad Noeiaghdam
View original source ↗Read source document ↗Export metadata ↓

Abstract

This study investigates the existence and uniqueness of solutions to Volterra integral equations with discontinuous kernels in both linear and nonlinear cases. The problem is two-dimensional, and the collocation method is employed to analyze the equations. The research aims to provide a comprehensive understanding of the solution properties of these integral equations, which are crucial in various mathematical and physical applications. By examining the existence and uniqueness of solutions, this study contributes to the development of numerical methods for solving Volterra integral equations with discontinuous kernels. The findings of this research have the potential to impact various fields, including physics, engineering, and economics, where integral equations play a significant role in modeling complex phenomena.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Samad Noeiaghdam. 2024-12-11. Existence and Uniqueness of the Solution of Two-dimensional Fuzzy Volterra Integral Equation with Piecewise Kernel. https://arxiv.org/abs/2501.04701

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗