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Experiments show that if the volume of a vessel containing a fixed amount of liquid is heated and expands at constant temperature, at a certain pressure, (), vapor, (denoted by dots at points and in Fig. 1) bubbles nucleate so the fluid is no longer homogeneous, but rather it has become a heterogeneous mixture of boiling liquid and condensing ...
The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov . In simple terms, if the solutions that start out near an equilibrium point x e {\displaystyle x_{e}} stay near x e {\displaystyle x_{e}} forever, then x e {\displaystyle x_{e}} is ...
The stability of fixed points of a system of constant coefficient linear differential equations of first order can be analyzed using the eigenvalues of the corresponding matrix. An autonomous system ′ =, where x(t) ∈ R n and A is an n×n matrix with real entries, has a constant solution =
Stability generally increases to the left of the diagram. [1] Some sink, source or node are equilibrium points. In mathematics , specifically in differential equations , an equilibrium point is a constant solution to a differential equation.
An algorithm for solving a linear evolutionary partial differential equation is stable if the total variation of the numerical solution at a fixed time remains bounded as the step size goes to zero. The Lax equivalence theorem states that an algorithm converges if it is consistent and stable (in this sense).
The stability of numerical schemes can be investigated by performing von Neumann stability analysis. For time-dependent problems, stability guarantees that the numerical method produces a bounded solution whenever the solution of the exact differential equation is bounded.
Fixed-point computation refers to the process of computing an exact or approximate fixed point of a given function. [1] In its most common form, the given function f {\displaystyle f} satisfies the condition to the Brouwer fixed-point theorem : that is, f {\displaystyle f} is continuous and maps the unit d -cube to itself.
In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact C 1-small perturbations). Examples of such qualitative properties are numbers of fixed points and periodic orbits (but not their periods