Corrigendum to “Admissible boundary conditions and solutions to internally pressurized thin arbitrarily laminated cylindrical shell boundary-value problems” [Compos. Struct. 86 (2008) 385–400]

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

Download Full-text

Solving nonlinear third-order three-point boundary value problems by boundary shape functions methods

Advances in Difference Equations ◽

10.1186/s13662-021-03288-x ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Ji Lin ◽

Yuhui Zhang ◽

Chein-Shan Liu

Keyword(s):

Boundary Conditions ◽

Boundary Value Problems ◽

Initial Conditions ◽

Boundary Value ◽

Accurate Solution ◽

Point Boundary ◽

Third Order ◽

Boundary Shape ◽

Free Function ◽

New Algorithms

AbstractFor nonlinear third-order three-point boundary value problems (BVPs), we develop two algorithms to find solutions, which automatically satisfy the specified three-point boundary conditions. We construct a boundary shape function (BSF), which is designed to automatically satisfy the boundary conditions and can be employed to develop new algorithms by assigning two different roles of free function in the BSF. In the first algorithm, we let the free functions be complete functions and the BSFs be the new bases of the solution, which not only satisfy the boundary conditions automatically, but also can be used to find solution by a collocation technique. In the second algorithm, we let the BSF be the solution of the BVP and the free function be another new variable, such that we can transform the BVP to a corresponding initial value problem for the new variable, whose initial conditions are given arbitrarily and terminal values are determined by iterations; hence, we can quickly find very accurate solution of nonlinear third-order three-point BVP through a few iterations. Numerical examples confirm the performance of the new algorithms.

Download Full-text

Recovering differential operators with two constant delays under Dirichlet/Neumann boundary conditions

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2019-0074 ◽

2020 ◽

Vol 28 (2) ◽

pp. 237-241

Author(s):

Biljana M. Vojvodic ◽

Vladimir M. Vladicic

Keyword(s):

Boundary Conditions ◽

Boundary Value Problems ◽

Differential Operators ◽

Boundary Value ◽

Neumann Boundary ◽

Neumann Boundary Conditions

AbstractThis paper deals with non-self-adjoint differential operators with two constant delays generated by {-y^{\prime\prime}+q_{1}(x)y(x-\tau_{1})+(-1)^{i}q_{2}(x)y(x-\tau_{2})}, where {\frac{\pi}{3}\leq\tau_{2}<\frac{\pi}{2}<2\tau_{2}\leq\tau_{1}<\pi} and potentials {q_{j}} are real-valued functions, {q_{j}\in L^{2}[0,\pi]}. We will prove that the delays and the potentials are uniquely determined from the spectra of four boundary value problems: two of them under boundary conditions {y(0)=y(\pi)=0} and the remaining two under boundary conditions {y(0)=y^{\prime}(\pi)=0}.

Download Full-text

Non-Instantaneous Impulsive Boundary Value Problems Containing Caputo Fractional Derivative of a Function with Respect to Another Function and Riemann–Stieltjes Fractional Integral Boundary Conditions

Axioms ◽

10.3390/axioms10030130 ◽

2021 ◽

Vol 10 (3) ◽

pp. 130

Author(s):

Suphawat Asawasamrit ◽

Yasintorn Thadang ◽

Sotiris K. Ntouyas ◽

Jessada Tariboon

Keyword(s):

Boundary Conditions ◽

Fractional Derivative ◽

Boundary Value Problems ◽

Caputo Fractional Derivative ◽

Fractional Integral ◽

Boundary Value ◽

Integral Boundary Conditions ◽

Integral Boundary ◽

Uniqueness Results ◽

Nonlinear Alternative

In the present article we study existence and uniqueness results for a new class of boundary value problems consisting by non-instantaneous impulses and Caputo fractional derivative of a function with respect to another function, supplemented with Riemann–Stieltjes fractional integral boundary conditions. The existence of a unique solution is obtained via Banach’s contraction mapping principle, while an existence result is established by using Leray–Schauder nonlinear alternative. Examples illustrating the main results are also constructed.

Download Full-text