Spaces:
Running
Running
programming_framework / processes /partial_differential_equations /partial_differential_equations-classical.html
| <html lang="en"> | |
| <head> | |
| <meta charset="UTF-8"> | |
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |
| <title>Classical PDEs - Mathematics Process</title> | |
| <script src="https://cdn.jsdelivr.net/npm/mermaid@10.6.1/dist/mermaid.min.js"></script> | |
| <style> | |
| * { margin: 0; padding: 0; box-sizing: border-box; } | |
| body { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background: linear-gradient(135deg, #667eea 0%, #764ba2 100%); min-height: 100vh; padding: 20px; } | |
| .container { max-width: 1200px; margin: 0 auto; background: white; border-radius: 15px; box-shadow: 0 20px 40px rgba(0,0,0,0.1); overflow: hidden; } | |
| .header { background: #2980b9; color: white; padding: 30px; } | |
| .header h1 { margin: 0 0 10px 0; font-size: 2em; font-weight: 300; } | |
| .header-meta { display: flex; flex-wrap: wrap; gap: 15px; margin-top: 15px; font-size: 0.9em; opacity: 0.9; } | |
| .meta-item { background: rgba(255,255,255,0.2); padding: 5px 12px; border-radius: 20px; } | |
| .nav-links { padding: 15px 30px; background: #f8f9fa; border-bottom: 1px solid #ecf0f1; } | |
| .nav-links a { color: #2980b9; text-decoration: none; margin-right: 20px; font-weight: 500; } | |
| .nav-links a:hover { text-decoration: underline; } | |
| .content { padding: 30px; } | |
| .description { margin-bottom: 30px; } | |
| .description h2 { color: #2c3e50; margin-bottom: 15px; } | |
| .flowchart-container { margin: 30px 0; } | |
| .flowchart-container h2 { color: #2c3e50; margin-bottom: 15px; } | |
| .mermaid { background: white; padding: 20px; border-radius: 10px; border: 1px solid #ecf0f1; overflow-x: auto; } | |
| .color-legend { background: #f8f9fa; padding: 20px; border-radius: 10px; margin: 30px 0; } | |
| .color-legend h3 { color: #2c3e50; margin-bottom: 15px; } | |
| .color-grid { display: grid; grid-template-columns: repeat(auto-fit, minmax(180px, 1fr)); gap: 15px; } | |
| .color-item { display: flex; align-items: center; gap: 10px; padding: 10px; background: white; border-radius: 5px; } | |
| .color-box { width: 30px; height: 30px; border-radius: 4px; border: 1px solid #ddd; } | |
| .info-card { background: #f8f9fa; padding: 20px; border-radius: 10px; margin-top: 20px; } | |
| .info-card h3 { color: #2c3e50; margin-bottom: 15px; } | |
| .info-card ul { list-style: none; padding: 0; } | |
| .info-card li { padding: 8px 0; border-bottom: 1px solid #ecf0f1; } | |
| .info-card li:last-child { border-bottom: none; } | |
| </style> | |
| </head> | |
| <body> | |
| <div class="container"> | |
| <div class="header"> | |
| <h1>Classical Partial Differential Equations</h1> | |
| <div class="header-meta"> | |
| <span class="meta-item">Partial Differential Equations</span> | |
| <span class="meta-item">Mathematics</span> | |
| </div> | |
| </div> | |
| <div class="nav-links"> | |
| <a href="../../mathematics_index.html">← Back to Mathematics Index</a> | |
| <a href="partial_differential_equations-index.html">Partial Differential Equations Index</a> | |
| <a href="https://arxiv.org/list/math.AP/recent" target="_blank">arXiv: Analysis of PDEs (math.AP)</a> | |
| </div> | |
| <div class="content"> | |
| <div class="description"> | |
| <h2>Description</h2> | |
| <p>Dependency graph for classical PDEs: Laplace, heat, and wave equations. Shows how definitions of elliptic, parabolic, and hyperbolic operators lead to existence, uniqueness, and regularity theorems.</p> | |
| </div> | |
| <div class="flowchart-container"> | |
| <h2>Dependency Flowchart</h2> | |
| <div class="mermaid"> | |
| graph TD | |
| D1["D1 Laplace operator\nΔu = Σ ∂²u/∂xᵢ²"] | |
| D2["D2 Heat operator\n∂u/∂t − αΔu = 0"] | |
| D3["D3 Wave operator\n∂²u/∂t² − c²Δu = 0"] | |
| D4["D4 Classification\nElliptic, Parabolic, Hyperbolic"] | |
| T1["T1 Existence of Laplace solutions\nDirichlet problem has solution"] | |
| T2["T2 Uniqueness for heat equation\nMaximum principle"] | |
| T3["T3 Uniqueness for wave equation\nEnergy method"] | |
| T4["T4 Regularity\nSmooth data ⇒ smooth solution"] | |
| T5["T5 Harnack inequality\nFor nonnegative harmonic functions"] | |
| D1 --> D2 | |
| D1 --> D3 | |
| D1 --> D4 | |
| D2 --> T2 | |
| D3 --> T3 | |
| D1 --> T1 | |
| D1 --> T5 | |
| D2 --> T4 | |
| D3 --> T4 | |
| D4 --> T4 | |
| T1 --> T5 | |
| classDef definition fill:#3498db,color:#fff,stroke:#2980b9 | |
| classDef theorem fill:#1abc9c,color:#fff,stroke:#16a085 | |
| class D1,D2,D3,D4 definition | |
| class T1,T2,T3,T4,T5 theorem | |
| </div> | |
| </div> | |
| <div class="color-legend"> | |
| <h3>Color Scheme</h3> | |
| <div class="color-grid"> | |
| <div class="color-item"><div class="color-box" style="background:#3498db"></div><div><strong>Blue</strong><br><small>Definitions (D1–D4)</small></div></div> | |
| <div class="color-item"><div class="color-box" style="background:#1abc9c"></div><div><strong>Teal</strong><br><small>Theorems (T1–T5)</small></div></div> | |
| </div> | |
| </div> | |
| <div class="info-card"> | |
| <h3>Info</h3> | |
| <ul> | |
| <li><strong>Subcategory:</strong> partial_differential_equations</li> | |
| <li><strong>Keywords:</strong> Laplace equation, heat equation, wave equation, elliptic, parabolic, hyperbolic</li> | |
| <li><strong>Research frontier:</strong> <a href="https://arxiv.org/list/math.AP/recent" target="_blank">arXiv math.AP</a></li> | |
| </ul> | |
| </div> | |
| </div> | |
| </div> | |
| <script> | |
| mermaid.initialize({ startOnLoad: true, theme: 'default', flowchart: { useMaxWidth: true, htmlLabels: true, curve: 'step', nodeSpacing: 25, rankSpacing: 40 } }); | |
| </script> | |
| </body> | |
| </html> | |