On Symmetric Hyperbolic Boundary Problems with Nonhomogeneous Conservative Boundary Conditions

2012 ◽  
Vol 44 (3) ◽  
pp. 1925-1949 ◽  
Author(s):  
Matthias Eller
Keyword(s):  
Boundary Conditions ◽  
Boundary Problems ◽  
Hyperbolic Boundary

scholarly journals The Ritz Method for Boundary Problems with Essential Conditions as Constraints

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Vojin Jovanovic ◽  
Sergiy Koshkin
Keyword(s):  
Boundary Conditions ◽  
Saddle Point ◽  
Lagrange Multipliers ◽  
Ritz Method ◽  
Higher Dimensions ◽  
Boundary Problems ◽  
Elementary Derivation ◽  
Convergence Proof

We give an elementary derivation of an extension of the Ritz method to trial functions that do not satisfy essential boundary conditions. As in the Babuška-Brezzi approach boundary conditions are treated as variational constraints and Lagrange multipliers are used to remove them. However, we avoid the saddle point reformulation of the problem and therefore do not have to deal with the Babuška-Brezzi inf-sup condition. In higher dimensions boundary weights are used to approximate the boundary conditions, and the assumptions in our convergence proof are stated in terms of completeness of the trial functions and of the boundary weights. These assumptions are much more straightforward to verify than the Babuška-Brezzi condition. We also discuss limitations of the method and implementation issues that follow from our analysis and examine a number of examples, both analytic and numerical.

scholarly journals Exact Static Analysis of In-Plane Curved Timoshenko Beams with Strong Nonlinear Boundary Conditions

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Sen-Yung Lee ◽  
Qian-Zhi Yan
Keyword(s):  
Boundary Conditions ◽  
Function Method ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Strong Nonlinearity ◽  
Timoshenko Beams ◽  
Static Deflection ◽  
Boundary Problems ◽  
Beam System ◽  
Differential Characteristic

Analytical solutions have been developed for nonlinear boundary problems. In this paper, the shifting function method is applied to develop the static deflection of in-plane curved Timoshenko beams with nonlinear boundary conditions. Three coupled governing differential equations are derived via the Hamilton’s principle. The mathematical modeling of the curved beam system can be decomposed into a complete sixth-order ordinary differential characteristic equation and the associated boundary conditions. It is shown that the proposed method is valid and performs well for problems with strong nonlinearity.

scholarly journals The Iterative Positive Solution for a System of Fractional q-Difference Equations with Four-Point Boundary Conditions

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
Chuanzhi Bai ◽  
Dandan Yang
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Difference Equations ◽  
Iterative Approach ◽  
Point Boundary ◽  
Boundary Problems ◽  
Monotone Iterative

In this work, we investigate the following system of fractional q-difference equations with four-point boundary problems: Dqαut+ft,vt=0,0<t<1;Dqβvt+gt,ut=0,0<t<1;u0=0,u1=γ1uη1; and v0=0,v1=γ2uη2, where Dqα and Dqβ are the fractional Riemann–Liouville q-derivative of order α and β, respectively, 0<q<1, 1<β≤α≤2, 0<η1,η2<1, 0<γ1η1α−1<1, and 0<γ2η2β−1<1. By virtue of monotone iterative approach, the iterative positive solutions are obtained. An example to illustrate our result is given.

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