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Separable First-Order Equations
Published in Kenneth B. Howell, Ordinary Differential Equations, 2019
Any curve that is at least part of the graph of an implicit solution for a differential equation is called an integral curve for the differential equation. Remember, this is the graph of an equation. If a function y(x) is a solution to that differential equation, then y = y(x) must also satisfy any equation serving as an implicit solution, and, thus, the graph of that y(x) (which we will call a solution curve) must be at least a portion of one of the integral curves for that differential equation. Sometimes an integral curve will be a solution curve. That is “clearly” the case in Figure 4.3, because that curve is “clearly” the graph of a function (more on that later).
Transfer function-based 2D/3D interactive spatiotemporal visualizations of mesoscale eddies
Published in International Journal of Digital Earth, 2020
Fenglin Tian, Lingqi Cheng, Ge Chen
In a 2D Euclidean space, a time-dependent vector field is a map u(x, t) that assigns a vector to each point x in space at time t. A particle at (x, t) has its own pathline хpath (Figure 1), which is an integral curve of the vector field governed by the ODE (ordinary differential equation) (Johnson and Hansen 2011):The initial condition хpath (t0; x0, t0) = x0 is given at time t0, and the curve is parameterized by t that contains the point x0 for t = t0. The solution to the ODE (Equation (1)) is obtained by formal integration, which gives the following pathline:
A dynamical system perspective on mathematical programming
Published in Dynamical Systems, 2018
Suppose, in addition to the required smoothness, that f: RN → R is coercive in the sense and ∇h(x)∇h(x)T is always non-singular. If there is a unique solution for (2.4), then every integral curve of the associated vector field Ψ converges to such a unique solution.
A New Approach to Analytical Modeling of Mars’s Magnetic Field
Published in Applied Mathematics in Science and Engineering, 2022
I.E. Stepanova, T.V. Gudkova, A.M. Salnikov, A.V. Batov
At the second step, we analyze the previous stage results and solve an ordinary differential equation system to find the integral curve of the gravity or magnetic field. The points of this curve are included in an extended data set, and we repeat the procedure described above for the first stage;