scholarly journals Glacial lake drainage: a stability analysis

1997 ◽  
Vol 24 ◽  
pp. 175-180
Author(s):  
Krzysztof Szilder ◽  
Edward P. Lozowski ◽  
Martin J. Sharp
Keyword(s):  
Differential Equations ◽  
Water Flow ◽  
Frictional Heating ◽  
Initial Perturbation ◽  
Tunnel Wall ◽  
Cross Sectional ◽  
Linear Ordinary Differential Equations ◽  
Simplifying Assumptions ◽  
Lake Drainage ◽  
The Stability

A model has been formulated to determine the stability regimes for water flow in a Subglacial conduit draining from a reservoir. The physics of the water flow is described with a set of differential equations expressing conservation of mass, momentum and energy. Non-steady flow of water in the conduit is considered, the conduit being simultaneously enlarged by frictional heating and compressed by plastic deformation in response to the pressure difference across the tunnel wall. With the aid of simplifying assumptions, a mathematical model has been constructed from two time-dependent, non-linear, ordinary differential equations, which describe the time evolution of the conduit cross-sectional area and the water depth in the reservoir. The model has been used to study the influence of conduit area and reservoir levels on the stability of the water flow for various glacier and ice-sheet configurations. The region of the parameter space where the system can achieve equilibrium has been identified. However, in the majority of cases the equilibrium is unstable, and an initial perturbation from equilibrium may lead to a catastrophic outburst of water which empties the reservoir.

scholarly journals Glacial lake drainage: a stability analysis

1997 ◽  
Vol 24 ◽  
pp. 175-180 ◽  
Author(s):  
Krzysztof Szilder ◽  
Edward P. Lozowski ◽  
Martin J. Sharp
Keyword(s):  
Differential Equations ◽  
Water Flow ◽  
Frictional Heating ◽  
Initial Perturbation ◽  
Tunnel Wall ◽  
Cross Sectional ◽  
Linear Ordinary Differential Equations ◽  
Simplifying Assumptions ◽  
Lake Drainage ◽  
The Stability

A model has been formulated to determine the stability regimes for water flow in a Subglacial conduit draining from a reservoir. The physics of the water flow is described with a set of differential equations expressing conservation of mass, momentum and energy. Non-steady flow of water in the conduit is considered, the conduit being simultaneously enlarged by frictional heating and compressed by plastic deformation in response to the pressure difference across the tunnel wall. With the aid of simplifying assumptions, a mathematical model has been constructed from two time-dependent, non-linear, ordinary differential equations, which describe the time evolution of the conduit cross-sectional area and the water depth in the reservoir. The model has been used to study the influence of conduit area and reservoir levels on the stability of the water flow for various glacier and ice-sheet configurations. The region of the parameter space where the system can achieve equilibrium has been identified. However, in the majority of cases the equilibrium is unstable, and an initial perturbation from equilibrium may lead to a catastrophic outburst of water which empties the reservoir.

A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov

1962 ◽  
Vol 58 (4) ◽  
pp. 694-702 ◽  
Author(s):  
P. C. Parks
Keyword(s):  
Differential Equations ◽  
Direct Proof ◽  
Real Constant ◽  
Integral Calculus ◽  
Linear Ordinary Differential Equations ◽  
Hurwitz Stability ◽  
Constant Coefficients ◽  
Linear Differential ◽  
The Stability ◽  
Complex Integral

ABSTRACTThe second method of Liapunov is a useful technique for investigating the stability of linear and non-linear ordinary differential equations. It is well known that the second method of Liapunov, when applied to linear differential equations with real constant coefficients, gives rise to sets of necessary and sufficient stability conditions which are alternatives to the well-known Routh-Hurwitz conditions. In this paper a direct proof of the Routh-Hurwitz conditions themselves is given using Liapunov's second method. The new proof is ‘elementary’ in that it depends on the fundamental concept of stability associated with Liapunov's second method, and not on theorems in the complex integral calculus which are required in the usual proofs. A useful by-product of this new proof is a method of determining the coefficients of a linear differential equation with real constant coefficients in terms of its Hurwitz determinants.

scholarly journals Stability theory of nano-fluid over an exponentially stretching cylindrical surface containing microorganisms

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
M. Ferdows ◽  
Amran Hossan ◽  
M. Z. I. Bangalee ◽  
Shuyu Sun ◽  
Faris Alzahrani
Keyword(s):  
Differential Equations ◽  
Cylindrical Surface ◽  
Velocity Ratio ◽  
Critical Value ◽  
Governing Equations ◽  
Linear Ordinary Differential Equations ◽  
Nano Fluid ◽  
Exponential Stretching ◽  
The Stability ◽  
Exponentially Stretching

Abstract This research is emphasized to describe the stability analysis in the form of dual solution of the flow and heat analysis on nanofluid over an exponential stretching cylindrical surface containing microorganisms. The research is also implemented to manifest the dual profiles of velocity, temperature and nanoparticle concentration in the effect of velocity ratio parameter ($$s = \frac{{U_{w} }}{{U_{\infty } }}$$ s = U w U ∞ ). Living microorganisms’ cell are mixed into the nanofluid to neglect the unstable condition of nano type particles. The governing equations are transformed to non-linear ordinary differential equations with respect to pertinent boundary conditions by using similarity transformation. The significant differential equations are solved using build in function bvp4c in MATLAB. It is seen that the solution is not unique for vertical stretching sheet. This research is reached to excellent argument when found results are compared with available result. It is noticed that dual results are obtained demanding on critical value ($$s_{c}$$ s c ), the meanings are indicated at these critical values both solutions are connected and behind these critical value boundary layer separates thus the solution are not stable.

Precise Stability Analysis of Stepped Telescopic Booms and Practical Algorithm

2013 ◽  
Vol 395-396 ◽  
pp. 871-876
Author(s):  
Liang Du ◽  
Peng Lan ◽  
Nian Li Lu
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Characteristic Equation ◽  
Energy Method ◽  
Cross Sectional ◽  
Component Method ◽  
Practical Algorithm ◽  
The Stability ◽  
Precise Expression ◽  
Constant Section

To analyze the stability of stepped telescopic booms accurately, using vertical and horizontal bending theory, this paper established the deflection differential equations of stepped column model of arbitrary sectioned telescopic boom, the stability were analyzed, and obtained the precise expression of the buckling characteristic equation; Took certain seven-sectioned telescopic booms as example, by comparing the results with ANSYS, the accuracy of the equations deduced in this paper was verified. Presented the equivalent component method for the stability analysis of multi-stepped column, the equivalent cross-sectional moment of inertia was deduced by energy method, thus the stability of stepped column equivalent to that of constant section component. By comparing the results with exact value, the precision of equivalent component method was verified which was convenient for stability analysis of telescopic boom.

scholarly journals A New Method for Numerical Integration of Higher-Order Ordinary Differential Equations Without Losing the Periodic Responses

2021 ◽  
Vol 7 ◽  
Author(s):  
John T. Katsikadelis
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Variable Coefficients ◽  
Space Representation ◽  
Parabolic Differential Equations ◽  
Linear Ordinary Differential Equations ◽  
First Order ◽  
Symmetrical Matrices ◽  
The Stability

A new numerical method is presented for the solution of initial value problems described by systems of N linear ordinary differential equations (ODEs). Using the state-space representation, a differential equation of order n > 1 is transformed into a system of L = n×N first-order equations, thus the numerical method developed recently by Katsikadelis for first-order parabolic differential equations can be applied. The stability condition of the numerical scheme is derived and is investigated using several well-corroborated examples, which demonstrate also its convergence and accuracy. The method is simply implemented. It is accurate and has no numerical damping. The stability does not require symmetrical and positive definite coefficient matrices. This advantage is important because the scheme can find the solution of differential equations resulting from methods in which the space discretization does not result in symmetrical matrices, for example, the boundary element method. It captures the periodic behavior of the solution, where many of the standard numerical methods may fail or are highly inaccurate. The present method also solves equations having variable coefficients as well as non-linear ones. It performs well when motions of long duration are considered, and it can be employed for the integration of stiff differential equations as well as equations exhibiting softening where widely used methods may not be effective. The presented examples demonstrate the efficiency and accuracy of the method.

On the stability of a system of two linear hybrid functional differential systems with aftereffect

2020 ◽  
pp. 299-306
Author(s):  
Pyotr M. Simonov
Keyword(s):  
Differential Equations ◽  
Matrix Function ◽  
Sufficient Conditions ◽  
Hybrid Functional ◽  
Cauchy Matrix ◽  
Linear Ordinary Differential Equations ◽  
Functional Differential Systems ◽  
Functional Differential ◽  
Linear Differential ◽  
The Stability

We consider a system of two hybrid vector equations containing linear difference (defined on a discrete set) and functional differential (defined on a half-axis) parts. To study it, a model system of two vector equations is chosen, one of which is linear difference with aftereffect (LDEA), and the other is a linear functional differential with aftereffect (LFDEA). Two equivalent representations of this system are shown: the first representation in the form of LFDEA, the second — in the form of LDEA. This allows us to study the stability issues of the system under consideration using the well-known results on the stability of LFDEA and LDEA. Using the results of the article [Gusarenko S. A. On the stability of a system of two linear differential equations with delayed argument // Boundary value problems. Interuniversity collection of scientific papers. Perm: PPI, 1989. P. 3–9], two examples are shown when a joint system of four equations will be stable with respect to the right side. In the first example, we use the LFDEA for which sufficient conditions for the sign-definiteness of the elements of the 2 Ч 2 Cauchy matrix function are known (in terms of the LFDEA coefficients). In the second example, LFDEA is given such that LFDEA is a system of linear ordinary differential equations (LODE) of the second order. In both cases, estimates of the components of the Cauchy matrix function are known. An exponential estimate with a negative exponent is given for the components of the Cauchy matrix function of LDEA.

Explicit Solutions for the Stability of No-Tension Beam-Columns

2003 ◽  
Vol 03 (02) ◽  
pp. 195-213 ◽  
Author(s):  
A. de Falco ◽  
M. Lucchesi
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Axial Load ◽  
Explicit Solutions ◽  
Cross Sectional ◽  
Beam Columns ◽  
The Cross ◽  
The Stability ◽  
No Tension Material ◽  
Explicit Relation

This work concerns the stability of rectangular cross-sectional piles made of a no-tension material and subjected to an axial load acting at the extremities within the middle third of the cross section. The resulting differential equations are solved, and an explicit relation between the load and a suitable deformability parameter obtained.

Sign in / Sign up

Export Citation Format

Share Document