Solvability of Nth Order Linear Boundary Value Problems

Copyright © 2015 P. Almenar and L. Jódar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.This paper presents a method that provides necessary and sufficient conditions for the existence of solutions of nth order linear boundary value problems. The method is based on the recursive application of a linear integral operator to some functions and the comparison of the result with these same functions. The recursive comparison yields sequences of bounds of extremes that converge to the exact values of the extremes of the BVP for which a solution exists.This work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2013-41765-P.Almenar, P.; Jódar Sánchez, LA. (2015). Solvability of Nth Order Linear Boundary Value Problems. 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Решение линейной краевой задачи без начальных условий для гиперболического уравнения второго порядка

Розглянуто крайову задачу без початкових умов для лінійного неоднорідного гіперболічного рівняння другого порядку вигляду utt – a²uxx = f(x,t), 0 ≤ x ≤ π, 0 ≤ t ≤ T, u(0,t) = u(π,t) = 0, 0 ≤ t ≤ T. Використовуючи методи теорії диференціальних рівнянь у частинних похідних і теорії інтегральних рівнянь, для довільної функції μ(z) ∈ C¹(R) побудовано точний розв’язок вказаної задачі у вигляді u(x,t) = u⁰(x,t) + ũ(x,t), де u⁰(x,t) = 1/(2a) at–x∫at+x μ(α)dα – розв’язок однорідного рівняння, а ũ(x,t) = 1/(2a) 0∫t dτ x–a(t–τ)∫ x+a(t–τ)f(ξ,τ)dξ – частинний розв’язок неоднорідного рівняння. Встановлено нові умови існування розв’язків вказаної задачі. Виділено класи функцій B₀⁻ = {μ : μ(z) = –μ(–z) = μ(π – z)}, B⁻ = {f : f(x,t) = f(π – x,t) = –f(–x,t)}, у яких існує класичний розв’язок лінійної крайової задачі без початкових умов для гіперболічного рівняння другого порядку. На основі встановлених результатів побудовано оператор A, який переводить клас функцій B⁻ ={f : f(x,t) = f(π – x,t) = –f(–x,t)}, у самого себе. Це дає змогу використовувати його при побудові наближених обчислень розв’язку крайових задач для квазілінійних гіперболічних рівнянь. Отримані результати є початком вивчення крайових задач без початкових умов для гіперболічних рівнянь другого порядку вигляду utt–a²uxx = f(x,t,ut,ux). Запропонований метод побудови розв’язку можна застосувати також для розв’язування напівлінійних крайових задач.This paper studies boundary-value problem without initial conditions for the linear non-homogeneous second-order hyperbolic equation appearance utt – a²uxx = f(x,t), 0 ≤ x ≤ π, 0 ≤ t ≤ T, u(0,t) = u(π,t) = 0, 0 ≤ t ≤ T. Using the methods of the theory of differential equations in partial derivatives and methods of the theory of integral equations, for arbitrary functions μ(z) ∈ C¹(R) the exact solution of the indicated problem is constructed as u(x,t) = u⁰(x,t) + ũ(x,t), where u⁰(x,t) = 1/(2a) at–x∫at+x μ(α)dα – the solution of the homogeneous equation and ũ(x,t) = 1/(2a) 0∫t dτ x–a(t–τ)∫x+a(t–τ) f(ξ,τ)dξ – particular solution of the non-homogeneous equation. New existence conditions of the indicated problem are established. The classes of functions B₀⁻ = {μ : μ(z) = –μ(–z) = μ(π – z)}, B⁻ = {f : f(x,t) = f(π – x,t) = –f(–x,t)}, in which there is a classical solution of the linear boundary-value problem without initial conditions for the second order hyperbolic equations are discriminated. Based on the results operator A, which translates the class of functions B⁻ = {f : f(x,t) = f(π – x,t) = –f(–x,t)}in itself was built. This allows using it in the construction of approximate computations of the solution of boundary-value problems for the quasilinear hyperbolic equations. The results are beginning of the boundary-value problems study without initial conditions for the second order hyperbolic equations in form utt – a²uxx = f(x,t,ut,ux).The proposed method of construction of the solution can be applied also to solve the semi-linear boundary-value problems.Рассмотрена краевая задача без начальных условий для линейного неоднородного гиперболического уравнения второго порядка вида utt – a²uxx = f(x,t), 0 ≤ x ≤ π, 0 ≤ t ≤ T, u(0,t) = u(π,t) = 0, 0 ≤ t ≤ T. Используя методы теории дифференциальных уравнений в частных производных и теории интегральных уравнений, для произвольной функции μ(z) ∈ C¹(R) построено точное решение указанной задачи в виде u(x,t) = u⁰(x,t) + ũ(x,t), где u⁰(x,t) = 1/(2a) at–x∫at+x μ(α)dα – решение однородного уравнения, а ũ(x,t) = 1/(2a) 0∫tdτ x–a(t–τ)∫x+a(t–τ) f(ξ,τ)dξ – частное решение неоднородного уравнения. Установлены новые условия существования решений указанной задачи. Выделены классы функций B₀⁻ = {μ : μ(z) = –μ(–z) = μ(π – z)}, B⁻ = {f : f(x,t) = f(π – x,t) = –f(–x,t)}, в которых существует классическое решение линейной краевой задачи без начальных условий для гиперболического уравнения второго порядка. На основе установленных результатов построен оператор A, переводящий класс функций B⁻ = {f : f(x,t) = f(π – x,t) = –f(–x,t)}, в самого себя. Это позволяет использовать его при построении приближенных вычислений решения краевых задач для квазилинейных гиперболических уравнений. Полученные результаты являются началом изучения краевых задач без начальных условий для гиперболических уравнений второго порядка вида utt – a²uxx = f(x,t,ut,ux). Предложенный метод построения решения можно применить также для решения полулинейных краевых задач

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Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients

This is the post-print version of the Article. The official publised version can be accessed from the links below. Copyright @ 2013 Springer BaselEmploying the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original boundary value problems is proved. It is established that the corresponding localized boundary-domain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik-Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.This research was supported by the grant EP/H020497/1: "Mathematical Analysis of Localized Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems" from the EPSRC, UK