1.2 Results

The current number of partial differential equations in this report is [ 2012 ].

Mathematica solved [ 1460 ] and Maple solved [ 1801 ].

Table 1.1:Percentage solved by each CAS for all problems


Mathematica Maple


72.56% 89.51%


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Table 1.2:Pecentage of solved by each CAS broken down by each chapter




chapter name Number of Problems Mathematica Maple




Miscellaneous PDE’s 149 75.17% 91.28%




Parabolic PDE’s (Diffusion) 130 91.54% 86.15%




Elliptic PDE’s (Laplace, Poisson, Helmholtz) 53 71.7% 75.47%




Hyperbolic PDE’s (Wave) 83 91.57% 85.54%




Handbook of first order partial differential equations 1585 70.28% 90.66%




Handbook of nonlinear partial differential equations 12 8.33% 41.67%




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Table 1.3:Percentage of failed due to time out among all problems that could not be solved


Mathematica Maple


7.25% 11.85%


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Table 1.4:Total real time used to solve all problems


Mathematica Maple


14.47 (hours) 8.89 (hours)


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Table 1.5:Average real time used to solve one problem


Mathematica Maple


25.882 (sec) 15.904 (sec)


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List of problems not solved by Maple nor by Mathematica [175] (8.7%)

32, 48, 57, 61, 119, 126, 131, 135, 136, 137, 169, 170, 272, 277, 278, 279, 307, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 332, 361, 362, 413, 454, 455, 456, 458, 474, 487, 488, 489, 492, 508, 516, 520, 521, 525, 526, 528, 530, 531, 543, 549, 557, 559, 561, 563, 564, 567, 620, 622, 623, 627, 628, 629, 650, 672, 673, 695, 698, 711, 712, 720, 723, 724, 735, 736, 746, 747, 748, 757, 758, 760, 761, 765, 766, 768, 770, 771, 772, 775, 776, 777, 779, 780, 783, 784, 785, 786, 787, 791, 802, 803, 808, 815, 816, 821, 822, 823, 824, 825, 826, 827, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 1029, 1172, 1179, 1194, 1366, 1377, 1378, 1380, 1383, 1384, 1385, 1386, 1387, 1413, 1416, 1417, 1421, 1428, 1429, 1450, 1451, 1452, 1453, 1457, 1462, 1463, 1555, 1574, 1577, 1578, 1628, 1629, 1632, 1726, 1736, 1748, 1749, 1800, 1801, 1906, 1907, 1914, 1915, 1916, 1918, 1924, 2001, 2003, 2008, 2009, 2010, 2011, 2012

List of problems solved by Maple but not by Mathematica [377] (18.74%)

16, 29, 34, 40, 67, 78, 100, 101, 102, 103, 104, 105, 107, 108, 116, 117, 118, 120, 121, 124, 127, 128, 130, 132, 133, 138, 140, 174, 182, 204, 214, 230, 297, 304, 310, 359, 394, 395, 408, 429, 449, 453, 457, 459, 460, 464, 465, 466, 469, 476, 480, 482, 484, 485, 486, 490, 493, 494, 499, 504, 505, 506, 507, 518, 519, 522, 524, 527, 529, 532, 537, 542, 551, 554, 558, 560, 570, 574, 576, 577, 578, 579, 583, 584, 589, 590, 591, 592, 593, 598, 599, 600, 601, 612, 615, 618, 621, 625, 634, 635, 636, 642, 643, 644, 646, 647, 648, 649, 652, 656, 657, 658, 659, 660, 661, 662, 663, 664, 668, 669, 671, 675, 676, 677, 678, 679, 682, 683, 684, 691, 696, 697, 699, 700, 701, 702, 705, 707, 709, 710, 717, 719, 721, 722, 729, 731, 733, 741, 743, 745, 753, 755, 756, 759, 769, 774, 778, 782, 788, 790, 792, 793, 795, 796, 798, 800, 801, 804, 805, 806, 807, 809, 810, 811, 812, 813, 814, 817, 818, 819, 820, 829, 830, 875, 884, 885, 901, 902, 907, 917, 929, 994, 1001, 1004, 1018, 1019, 1036, 1037, 1057, 1058, 1065, 1066, 1079, 1084, 1107, 1178, 1180, 1181, 1195, 1208, 1209, 1210, 1211, 1234, 1235, 1262, 1267, 1279, 1352, 1360, 1365, 1374, 1375, 1381, 1382, 1390, 1410, 1414, 1420, 1422, 1423, 1425, 1436, 1437, 1441, 1446, 1447, 1454, 1455, 1456, 1459, 1461, 1468, 1469, 1475, 1480, 1481, 1486, 1487, 1490, 1491, 1492, 1493, 1511, 1512, 1523, 1527, 1529, 1558, 1559, 1563, 1575, 1576, 1586, 1607, 1608, 1618, 1630, 1648, 1650, 1655, 1664, 1666, 1667, 1671, 1678, 1684, 1689, 1695, 1697, 1698, 1699, 1700, 1703, 1704, 1709, 1711, 1714, 1715, 1719, 1720, 1727, 1729, 1730, 1731, 1732, 1733, 1737, 1738, 1740, 1742, 1744, 1745, 1746, 1747, 1757, 1778, 1779, 1783, 1789, 1791, 1799, 1803, 1809, 1814, 1818, 1820, 1824, 1826, 1829, 1835, 1836, 1837, 1840, 1844, 1845, 1847, 1851, 1853, 1858, 1863, 1865, 1866, 1867, 1868, 1871, 1872, 1874, 1877, 1878, 1880, 1883, 1884, 1888, 1889, 1898, 1899, 1900, 1901, 1902, 1905, 1909, 1911, 1913, 1917, 1931, 1938, 1948, 1951, 1958, 1960, 1963, 1964, 1966, 1969, 1970, 1974, 1979, 1984, 1989, 1993, 1998, 1999, 2004, 2005, 2006, 2007

List of problems solved by Mathematica but not by Maple [36] (1.79%)

86, 94, 95, 166, 203, 226, 264, 265, 267, 268, 269, 270, 271, 275, 276, 300, 364, 393, 399, 401, 403, 409, 410, 411, 412, 1202, 1409, 1424, 1533, 1638, 1644, 1705, 1706, 1839, 1873, 1904

List of problems not solved by Mathematica [552] (27.44%)

16, 29, 32, 34, 40, 48, 57, 61, 67, 78, 100, 101, 102, 103, 104, 105, 107, 108, 116, 117, 118, 119, 120, 121, 124, 126, 127, 128, 130, 131, 132, 133, 135, 136, 137, 138, 140, 169, 170, 174, 182, 204, 214, 230, 272, 277, 278, 279, 297, 304, 307, 310, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 332, 359, 361, 362, 394, 395, 408, 413, 429, 449, 453, 454, 455, 456, 457, 458, 459, 460, 464, 465, 466, 469, 474, 476, 480, 482, 484, 485, 486, 487, 488, 489, 490, 492, 493, 494, 499, 504, 505, 506, 507, 508, 516, 518, 519, 520, 521, 522, 524, 525, 526, 527, 528, 529, 530, 531, 532, 537, 542, 543, 549, 551, 554, 557, 558, 559, 560, 561, 563, 564, 567, 570, 574, 576, 577, 578, 579, 583, 584, 589, 590, 591, 592, 593, 598, 599, 600, 601, 612, 615, 618, 620, 621, 622, 623, 625, 627, 628, 629, 634, 635, 636, 642, 643, 644, 646, 647, 648, 649, 650, 652, 656, 657, 658, 659, 660, 661, 662, 663, 664, 668, 669, 671, 672, 673, 675, 676, 677, 678, 679, 682, 683, 684, 691, 695, 696, 697, 698, 699, 700, 701, 702, 705, 707, 709, 710, 711, 712, 717, 719, 720, 721, 722, 723, 724, 729, 731, 733, 735, 736, 741, 743, 745, 746, 747, 748, 753, 755, 756, 757, 758, 759, 760, 761, 765, 766, 768, 769, 770, 771, 772, 774, 775, 776, 777, 778, 779, 780, 782, 783, 784, 785, 786, 787, 788, 790, 791, 792, 793, 795, 796, 798, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 811, 812, 813, 814, 815, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 875, 884, 885, 901, 902, 907, 917, 929, 994, 1001, 1004, 1018, 1019, 1029, 1036, 1037, 1057, 1058, 1065, 1066, 1079, 1084, 1107, 1172, 1178, 1179, 1180, 1181, 1194, 1195, 1208, 1209, 1210, 1211, 1234, 1235, 1262, 1267, 1279, 1352, 1360, 1365, 1366, 1374, 1375, 1377, 1378, 1380, 1381, 1382, 1383, 1384, 1385, 1386, 1387, 1390, 1410, 1413, 1414, 1416, 1417, 1420, 1421, 1422, 1423, 1425, 1428, 1429, 1436, 1437, 1441, 1446, 1447, 1450, 1451, 1452, 1453, 1454, 1455, 1456, 1457, 1459, 1461, 1462, 1463, 1468, 1469, 1475, 1480, 1481, 1486, 1487, 1490, 1491, 1492, 1493, 1511, 1512, 1523, 1527, 1529, 1555, 1558, 1559, 1563, 1574, 1575, 1576, 1577, 1578, 1586, 1607, 1608, 1618, 1628, 1629, 1630, 1632, 1648, 1650, 1655, 1664, 1666, 1667, 1671, 1678, 1684, 1689, 1695, 1697, 1698, 1699, 1700, 1703, 1704, 1709, 1711, 1714, 1715, 1719, 1720, 1726, 1727, 1729, 1730, 1731, 1732, 1733, 1736, 1737, 1738, 1740, 1742, 1744, 1745, 1746, 1747, 1748, 1749, 1757, 1778, 1779, 1783, 1789, 1791, 1799, 1800, 1801, 1803, 1809, 1814, 1818, 1820, 1824, 1826, 1829, 1835, 1836, 1837, 1840, 1844, 1845, 1847, 1851, 1853, 1858, 1863, 1865, 1866, 1867, 1868, 1871, 1872, 1874, 1877, 1878, 1880, 1883, 1884, 1888, 1889, 1898, 1899, 1900, 1901, 1902, 1905, 1906, 1907, 1909, 1911, 1913, 1914, 1915, 1916, 1917, 1918, 1924, 1931, 1938, 1948, 1951, 1958, 1960, 1963, 1964, 1966, 1969, 1970, 1974, 1979, 1984, 1989, 1993, 1998, 1999, 2001, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012

List of problems not solved by Maple [211] (10.49%)

32, 48, 57, 61, 86, 94, 95, 119, 126, 131, 135, 136, 137, 166, 169, 170, 203, 226, 264, 265, 267, 268, 269, 270, 271, 272, 275, 276, 277, 278, 279, 300, 307, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 332, 361, 362, 364, 393, 399, 401, 403, 409, 410, 411, 412, 413, 454, 455, 456, 458, 474, 487, 488, 489, 492, 508, 516, 520, 521, 525, 526, 528, 530, 531, 543, 549, 557, 559, 561, 563, 564, 567, 620, 622, 623, 627, 628, 629, 650, 672, 673, 695, 698, 711, 712, 720, 723, 724, 735, 736, 746, 747, 748, 757, 758, 760, 761, 765, 766, 768, 770, 771, 772, 775, 776, 777, 779, 780, 783, 784, 785, 786, 787, 791, 802, 803, 808, 815, 816, 821, 822, 823, 824, 825, 826, 827, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 1029, 1172, 1179, 1194, 1202, 1366, 1377, 1378, 1380, 1383, 1384, 1385, 1386, 1387, 1409, 1413, 1416, 1417, 1421, 1424, 1428, 1429, 1450, 1451, 1452, 1453, 1457, 1462, 1463, 1533, 1555, 1574, 1577, 1578, 1628, 1629, 1632, 1638, 1644, 1705, 1706, 1726, 1736, 1748, 1749, 1800, 1801, 1839, 1873, 1904, 1906, 1907, 1914, 1915, 1916, 1918, 1924, 2001, 2003, 2008, 2009, 2010, 2011, 2012

List of problems that are hand solved [195] (9.69%)

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 24, 26, 27, 28, 29, 30, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 91, 92, 109, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 166, 168, 171, 172, 175, 176, 177, 178, 180, 181, 183, 184, 185, 191, 194, 195, 198, 199, 200, 201, 202, 203, 206, 207, 208, 209, 210, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 232, 234, 235, 236, 237, 247, 252, 253, 258, 259, 262, 263, 264, 267, 268, 269, 270, 271, 272, 274, 275, 280, 281, 282, 287, 289, 290, 291, 292, 306, 307, 309, 310, 312, 315, 326, 328, 332, 333, 334, 335, 336, 337, 342, 347, 348, 349, 352, 353, 354, 355, 356, 357, 358, 359, 363, 364, 365, 372, 373, 382, 387, 388, 390, 391, 392, 394, 395, 396, 397, 401, 404, 406, 409, 410, 411, 412, 413, 418, 420, 422, 423, 424, 425, 426, 427, 428, 1367, 1368, 1369, 1370, 1514, 1515, 1584, 1585

List of problems that are animated [60] (2.98%)

2, 3, 4, 5, 6, 7, 8, 9, 10, 151, 152, 154, 159, 161, 163, 176, 181, 191, 199, 202, 208, 209, 210, 219, 222, 227, 229, 235, 237, 259, 263, 264, 268, 269, 271, 272, 274, 275, 334, 335, 336, 349, 354, 356, 357, 358, 363, 372, 373, 382, 387, 388, 390, 391, 395, 396, 397, 406, 411, 412