garywelz/programming_framework
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1<!DOCTYPE html>2<html lang="en">3<head>4 <meta charset="UTF-8">5 <meta name="viewport" content="width=device-width, initial-scale=1.0">6 <title>Classical PDEs - Mathematics Process</title>7 <script src="https://cdn.jsdelivr.net/npm/mermaid@10.6.1/dist/mermaid.min.js"></script>8 <style>9 * { margin: 0; padding: 0; box-sizing: border-box; }10 body { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background: linear-gradient(135deg, #667eea 0%, #764ba2 100%); min-height: 100vh; padding: 20px; }11 .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; }12 .header { background: #2980b9; color: white; padding: 30px; }13 .header h1 { margin: 0 0 10px 0; font-size: 2em; font-weight: 300; }14 .header-meta { display: flex; flex-wrap: wrap; gap: 15px; margin-top: 15px; font-size: 0.9em; opacity: 0.9; }15 .meta-item { background: rgba(255,255,255,0.2); padding: 5px 12px; border-radius: 20px; }16 .nav-links { padding: 15px 30px; background: #f8f9fa; border-bottom: 1px solid #ecf0f1; }17 .nav-links a { color: #2980b9; text-decoration: none; margin-right: 20px; font-weight: 500; }18 .nav-links a:hover { text-decoration: underline; }19 .content { padding: 30px; }20 .description { margin-bottom: 30px; }21 .description h2 { color: #2c3e50; margin-bottom: 15px; }22 .flowchart-container { margin: 30px 0; }23 .flowchart-container h2 { color: #2c3e50; margin-bottom: 15px; }24 .mermaid { background: white; padding: 20px; border-radius: 10px; border: 1px solid #ecf0f1; overflow-x: auto; }25 .color-legend { background: #f8f9fa; padding: 20px; border-radius: 10px; margin: 30px 0; }26 .color-legend h3 { color: #2c3e50; margin-bottom: 15px; }27 .color-grid { display: grid; grid-template-columns: repeat(auto-fit, minmax(180px, 1fr)); gap: 15px; }28 .color-item { display: flex; align-items: center; gap: 10px; padding: 10px; background: white; border-radius: 5px; }29 .color-box { width: 30px; height: 30px; border-radius: 4px; border: 1px solid #ddd; }30 .info-card { background: #f8f9fa; padding: 20px; border-radius: 10px; margin-top: 20px; }31 .info-card h3 { color: #2c3e50; margin-bottom: 15px; }32 .info-card ul { list-style: none; padding: 0; }33 .info-card li { padding: 8px 0; border-bottom: 1px solid #ecf0f1; }34 .info-card li:last-child { border-bottom: none; }35 </style>36</head>37<body>38 <div class="container">39 <div class="header">40 <h1>Classical Partial Differential Equations</h1>41 <div class="header-meta">42 <span class="meta-item">Partial Differential Equations</span>43 <span class="meta-item">Mathematics</span>44 </div>45 </div>46 <div class="nav-links">47 <a href="../../mathematics_index.html">← Back to Mathematics Index</a>48 <a href="partial_differential_equations-index.html">Partial Differential Equations Index</a>49 <a href="https://arxiv.org/list/math.AP/recent" target="_blank">arXiv: Analysis of PDEs (math.AP)</a>50 </div>51 <div class="content">52 <div class="description">53 <h2>Description</h2>54 <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>55 </div>56 <div class="flowchart-container">57 <h2>Dependency Flowchart</h2>58 <div class="mermaid">59graph TD60 D1["D1 Laplace operator\nΔu = Σ ∂²u/∂xᵢ²"]61 D2["D2 Heat operator\n∂u/∂t − αΔu = 0"]62 D3["D3 Wave operator\n∂²u/∂t² − c²Δu = 0"]63 D4["D4 Classification\nElliptic, Parabolic, Hyperbolic"]64 65 T1["T1 Existence of Laplace solutions\nDirichlet problem has solution"]66 T2["T2 Uniqueness for heat equation\nMaximum principle"]67 T3["T3 Uniqueness for wave equation\nEnergy method"]68 T4["T4 Regularity\nSmooth data ⇒ smooth solution"]69 T5["T5 Harnack inequality\nFor nonnegative harmonic functions"]70 71 D1 --> D272 D1 --> D373 D1 --> D474 D2 --> T275 D3 --> T376 D1 --> T177 D1 --> T578 D2 --> T479 D3 --> T480 D4 --> T481 T1 --> T582 83 classDef definition fill:#3498db,color:#fff,stroke:#2980b984 classDef theorem fill:#1abc9c,color:#fff,stroke:#16a08585 class D1,D2,D3,D4 definition86 class T1,T2,T3,T4,T5 theorem87 </div>88 </div>89 <div class="color-legend">90 <h3>Color Scheme</h3>91 <div class="color-grid">92 <div class="color-item"><div class="color-box" style="background:#3498db"></div><div><strong>Blue</strong><br><small>Definitions (D1–D4)</small></div></div>93 <div class="color-item"><div class="color-box" style="background:#1abc9c"></div><div><strong>Teal</strong><br><small>Theorems (T1–T5)</small></div></div>94 </div>95 </div>96 <div class="info-card">97 <h3>Info</h3>98 <ul>99 <li><strong>Subcategory:</strong> partial_differential_equations</li>100 <li><strong>Keywords:</strong> Laplace equation, heat equation, wave equation, elliptic, parabolic, hyperbolic</li>101 <li><strong>Research frontier:</strong> <a href="https://arxiv.org/list/math.AP/recent" target="_blank">arXiv math.AP</a></li>102 </ul>103 </div>104 </div>105 </div>106 <script>107 mermaid.initialize({ startOnLoad: true, theme: 'default', flowchart: { useMaxWidth: true, htmlLabels: true, curve: 'step', nodeSpacing: 25, rankSpacing: 40 } });108 </script>109</body>110</html>111