First-order elliptic systems degenerated at the boundary

1993 ◽  
Vol 123 (6) ◽  
pp. 1203-1212
Author(s):  
Abduhamid Dzhuraev
Keyword(s):  
Boundary Conditions ◽  
Explicit Formula ◽  
Elliptic Systems ◽  
Bounded Solutions ◽  
Order Linear ◽  
Multiply Connected ◽  
First Order

SynopsisIn this paper we state that the bounded solutions of general first order linear elliptic systems of two equations in bounded multiply-connected plane domains, degenerated at the boundary, are determined in domains without any boundary conditions, provided the boundary is not characteristic for this system. The explicit formula for calculating the index of the system is derived.

A numerical method of solving second-order linear differential equations with two-point boundary conditions

1957 ◽  
Vol 53 (2) ◽  
pp. 442-447 ◽  
Author(s):  
E. Cicely Ridley
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Riccati Equation ◽  
Direct Method ◽  
Linear Equations ◽  
Factorization Method ◽  
Order Linear ◽  
First Order ◽  
The Matrix ◽  
Linear Differential

ABSTRACTA direct method of integrating the equation y″ + g(x) y = h(x), with the two-point linear boundary conditions y′(a) + αy(a) = A, y′(b) + βy(b) = B, is based on the factorization of the equation into two first-order linear equations v′ − sv = h and y′ + sy = v, where s is a solution of the Riccati equation s′ − s2 = g. The first-order equations for v and y are integrated in succession, one in the direction of x increasing, and one in the direction of x decreasing, one boundary condition being used in each of these integrations. The appropriate solution of the Riccati equation is determined by the boundary condition at the end of the range from which the integration of the equation for v is started. The process is compared with the matrix factorization method of Thomas and Fox, and its stability discussed.

On BVPS for Strongly Elliptic Systems with Higher Order Boundary Conditions

2007 ◽  
Vol 14 (1) ◽  
pp. 145-167
Author(s):  
Flavia Lanzara
Keyword(s):  
Boundary Conditions ◽  
Explicit Formula ◽  
Singular Integral ◽  
Elliptic Systems ◽  
Higher Order ◽  
Boundary Operator ◽  
Compatibility Conditions ◽  
Layer Potential ◽  
Integral System ◽  
Regular Type

Abstract We consider BVPs for strongly elliptic systems of order 2𝑙 with the boundary conditions of order 𝑙 + 𝑛, 𝑛 ⩾ 0. By representing the solution by means of a simple layer potential of order 𝑛 and by imposing the boundary conditions, we get a singular integral system which is of regular type if and only if the boundary operator satisfies the Lopatinskiĭ condition and which can be solved if suitable compatibility conditions are satisfied. An explicit formula for computing the index of the BVP is given.

Chaotic Oscillations of Second Order Linear Hyperbolic Equations with Nonlinear Boundary Conditions: A Factorizable but Noncommutative Case

2015 ◽  
Vol 25 (11) ◽  
pp. 1530032 ◽  
Author(s):  
Liangliang Li ◽  
Yu Huang ◽  
Goong Chen ◽  
Tingwen Huang
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Hyperbolic Equations ◽  
Nonlinear Boundary Conditions ◽  
Second Order ◽  
Chaotic Oscillations ◽  
Van Der Pol ◽  
Order Linear ◽  
First Order

If a second order linear hyperbolic partial differential equation in one-space dimension can be factorized as a product of two first order operators and if the two first order operators commute, with one boundary condition being the van der Pol type and the other being linear, one can establish the occurrence of chaos when the parameters enter a certain regime [Chen et al., 2014]. However, if the commutativity of the two first order operators fails to hold, then the treatment in [Chen et al., 2014] no longer works and significant new challenges arise in determining nonlinear boundary conditions that engenders chaos. In this paper, we show that by incorporating a linear memory effect, a nonlinear van der Pol boundary condition can cause chaotic oscillations when the parameter enters a certain regime. Numerical simulations illustrating chaotic oscillations are also presented.

scholarly journals On the Composition of Simultaneous Differential Systems of the First Order

1932 ◽  
Vol 3 (2) ◽  
pp. 128-131
Author(s):  
M. Mursi-Ahmed
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Linear Differential Equations ◽  
Differential Systems ◽  
Order Linear ◽  
First Order ◽  
Linear Differential ◽  
Image Position

§ 1. Consider the system of n first order linear differential equations:together with the n boundary conditionswhere aij, bij are constants and where we assume for simplicity of notation that gii = 0.

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