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Results tagged with differential-equations
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user 238
Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.
6
votes
3
answers
1k
views
Jacobi fields on a "bump surface"
Consider a "bump surface" which looks like the following:
Such a surface is rotationally symmetric, $C^2$-smooth, has positive curvature in the middle and negative curvature along the ring (the ora …
- 8,532
3
votes
1
answer
2k
views
A formula for the Jacobian of a flow
Let $U : \mathbb R^d \to \mathbb R^d$ be a smooth vector field, and let $F_t : \mathbb R \times \mathbb R^d \to \mathbb R^d$ be the corresponding smooth flow, defined by the differential equation $$\t …
- 8,532
4
votes
2
answers
725
views
Analyzing the solution to a second-order, non-linear ODE
Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - …
- 8,532
11
votes
1
answer
988
views
Prescribing Gaussian curvature
Let $K(r)$ be the piecewise function
I want to solve the PDE $$\Delta u + K(|x|) e^{2u} = 0$$ for radially symmetric $u$ with boundary condition $u = 0$ …
- 8,532