"Cauchy problem" meaning in All languages combined

{
  "etymology_text": "After French mathematician Augustin Louis Cauchy.",
  "forms": [
    {
      "form": "Cauchy problems",
      "tags": [
        "plural"
      ]
    }
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      "expansion": "Cauchy problem (plural Cauchy problems)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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        {
          "kind": "other",
          "langcode": "en",
          "name": "Differential equations",
          "orig": "en:Differential equations",
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        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Geometry",
          "orig": "en:Geometry",
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        {
          "kind": "topical",
          "langcode": "en",
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        {
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      ],
      "examples": [
        {
          "ref": "2006, Victor Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Springer, page 41",
          "text": "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.",
          "type": "quotation"
        },
        {
          "ref": "2006, S. Albeverio, Ya. Belopolskaya, “Probabilistic Interpretation of the VV-Method for PDE Systems”, in Olga S. Rozanova, editor, Analytical Approaches to Multidimensional Balance Laws, Nova Science Publishers, page 2",
          "text": "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).",
          "type": "quotation"
        },
        {
          "ref": "2014, Tatsuo Nishitani, Hyperbolic Systems with Analytic Coefficients: Well-posedness of the Cauchy Problem, Springer, page v",
          "text": "In this monograph we discuss the C#x5C;infty well-posedness of the Cauchy problem for hyperbolic systems.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "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."
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          "partial differential equation"
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      "raw_glosses": [
        "(mathematics, mathematical analysis) 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."
      ],
      "topics": [
        "mathematical-analysis",
        "mathematics",
        "sciences"
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      "translations": [
        {
          "code": "it",
          "lang": "Italian",
          "sense": "class of problem involving simultaneous PDEs",
          "tags": [
            "masculine"
          ],
          "word": "problema di Cauchy"
        }
      ],
      "wikipedia": [
        "Augustin Louis Cauchy",
        "Cauchy problem"
      ]
    }
  ],
  "word": "Cauchy problem"
}

{
  "etymology_text": "After French mathematician Augustin Louis Cauchy.",
  "forms": [
    {
      "form": "Cauchy problems",
      "tags": [
        "plural"
      ]
    }
  ],
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        {
          "ref": "2006, Victor Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Springer, page 41",
          "text": "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.",
          "type": "quotation"
        },
        {
          "ref": "2006, S. Albeverio, Ya. Belopolskaya, “Probabilistic Interpretation of the VV-Method for PDE Systems”, in Olga S. Rozanova, editor, Analytical Approaches to Multidimensional Balance Laws, Nova Science Publishers, page 2",
          "text": "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).",
          "type": "quotation"
        },
        {
          "ref": "2014, Tatsuo Nishitani, Hyperbolic Systems with Analytic Coefficients: Well-posedness of the Cauchy Problem, Springer, page v",
          "text": "In this monograph we discuss the C#x5C;infty well-posedness of the Cauchy problem for hyperbolic systems.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "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."
      ],
      "links": [
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          "mathematics",
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          "vector",
          "vector"
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          "normal",
          "normal"
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      "raw_glosses": [
        "(mathematics, mathematical analysis) 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."
      ],
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        "mathematics",
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      "wikipedia": [
        "Augustin Louis Cauchy",
        "Cauchy problem"
      ]
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  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "class of problem involving simultaneous PDEs",
      "tags": [
        "masculine"
      ],
      "word": "problema di Cauchy"
    }
  ],
  "word": "Cauchy problem"
}