scholarly journals Multiple Positive Solutions for the Dirichlet Boundary Value Problems by Phase Plane Analysis

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
A. Kirichuka ◽  
F. Sadyrbaev
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Phase Plane ◽  
Boundary Value ◽  
Phase Plane Analysis ◽  
Multiple Positive Solutions ◽  
Degree Polynomial ◽  
Plane Analysis ◽  
Phase Plane Method ◽  
Scalar Differential Equation

We consider boundary value problems for scalar differential equationx′′+λfx=0,x(0)=0,x(1)=0, wheref(x)is a seventh-degree polynomial andλis a parameter. We use the phase plane method combined with evaluations of time-map functions and make conclusions on the number of positive solutions. Bifurcation diagrams are constructed and examples are considered illustrating the bifurcation processes.

scholarly journals Existence of Multiple Positive Solutions for Even Order Multi-Point Boundary Value Problems

2007 ◽  
Vol 14 (4) ◽  
pp. 775-792
Author(s):  
Youyu Wang ◽  
Weigao Ge
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Even Order

Abstract In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem: where (0,1), 0 < ξ 1 < ξ 2 < ⋯ < ξ 𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.

Phase plane analysis of left ventricular hemodynamics

2001 ◽  
Vol 90 (6) ◽  
pp. 2238-2244 ◽  
Author(s):  
Stephanie A. Eucker ◽  
Jennifer B. Lisauskas ◽  
Jasvindar Singh ◽  
Sándor J. Kovács
Keyword(s):  
Ventricular Function ◽  
Phase Plane ◽  
Pressure Rise ◽  
Left Ventricular ◽  
Time Dependent ◽  
Phase Plane Analysis ◽  
Plane Method ◽  
Plane Analysis ◽  
Pressure Contour ◽  
Phase Plane Method

We sought to extract additional physiological information from the time-dependent left ventricular (LV) pressure contour and thereby gain new insights into ventricular function. We used phase plane analysis to characterize high-fidelity pressure data in selected subjects undergoing elective cardiac catheterization. The standard hemodynamic indexes of LV systolic and diastolic function derived from the time-dependent LV pressure contour could be easily obtained using the phase plane method. Additional novel attributes of the phase plane pressure loop, such as phase plane pressure loop area, graphical representation of the isovolumic relaxation time constant, and quantitative measures of beat-to-beat systolic-diastolic coupling were characterized. The asymmetry between the pressures at which maximum isovolumic pressure rise and pressure fall occur, as well as their load dependence, were also easily quantitated. These results indicate that the phase plane method provides a novel window for physiological discovery and has theoretical and applied advantages in quantitative ventricular function characterization.

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