{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-partial-differential-equation",
  "semantic_family_id": "sf-partial-differential-equation",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "偏微分方程"
  },
  "summary": {
    "zh-Hant": "偏微分方程關聯多變數未知函數與一個或多個偏導數；殘差與主部分類需固定算子convention。"
  },
  "statement": {
    "zh-Hant": "本輪固定二變數二次多項式候選u與線性常係數算子，精確計算一階或二階殘差。二階主部採a u_xx+b u_xy+c u_yy，判別式為b²-4ac；這個convention必須明示。"
  },
  "formula": {
    "tex": "L(u)=g",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>L</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>g</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "L",
          "glyph": "L"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "u",
            "glyph": "u"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "g",
        "glyph": "g"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "79b03d25d49eae9c7e20e7fa8a627b2a8b60c59461aa86daa565913a9d707168"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫只處理多項式符號導數、點殘差、主部分類與多項式恆等式。"
    },
    {
      "id": "a2",
      "zh-Hant": "點殘差為零只證該點；只有符號殘差多項式全零才證指定多項式在全域滿足該算子方程。"
    }
  ],
  "symbols": [
    {
      "token": "L",
      "role": "linear_pde_operator",
      "role_zh": "線性PDE算子",
      "scope": "representative_formula"
    },
    {
      "token": "u",
      "role": "candidate_unknown_function",
      "role_zh": "候選未知函數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-partial-derivative",
      "title_zh": "偏導數",
      "reason_zh": "PDE算子使用一階或高階偏導。"
    },
    {
      "id": "mko-equation",
      "title_zh": "方程",
      "reason_zh": "PDE是函數與偏導之間的等式。"
    },
    {
      "id": "mko-derivative-rules",
      "title_zh": "微分法則",
      "reason_zh": "多項式偏導由微分法則導出。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "算子係數與混合項convention決定分類公式。",
      "點殘差不等於全域解證明。",
      "候選解通過也不自動給一般解、存在唯一或適定性。"
    ],
    "common_misconception_zh": "從有限取樣殘差為零宣稱全域解，或混用2b u_xy與b u_xy的判別式。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核指定線性常係數多項式切片；不處理一般解理論。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入邊界值問題、弱解與適定性。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-partial-differential-equation.md 整理；2026-10-13 脈衝 R06（使用者手動接續 2026-10-24 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}