Meaning of Cauchy problem | Babel Free

Definitions

For a given m-order partial differential equation, the problem of finding a solution function u on ℝⁿ that satisfies the boundary conditions that, for a smooth manifold S⊂ℝⁿ, u(x)=f_0(x) and (∂ᵏu(x))/(∂nᵏ)=f_k(x), ∀x∈S, k=1…m-1, given specified functions f_k defined on, and vector n normal to, the manifold.

Equivalents

Examples

“In this chapter we formulate and in many cases prove⟳ results on uniqueness and stability of solutions of the Cauchy problem for general partial differential equations.”

“To emphasize the similarity between the characteristic method and the probabilistic approach⟳ we recall⟳ that the method of characteristics allows [one] to reduce⟳ the Cauchy problem for a first order⟳ PDE to a Cauchy problem for an ODE while the probabilistic approach⟳ allows [one] to reduce⟳ the Cauchy problem for a second order⟳ PDE to a Cauchy problem for an SDE (stochastic differential equation).”

“In this monograph we discuss⟳ the C#92;infty well-posedness of the Cauchy problem for hyperbolic systems.”

CEFR level