Numerical Solution of Heat Equation in Polar Cylindrical Coordinates by the Meshless Method of Lines

<div>Maximum absolute error when <span class="nowrap"><svg height="9.25986pt" id="M128" style="vertical-align:-0.2455397pt" version="1.1" viewbox="-0.0498162 -9.01432 46.4705 9.25986" width="46.4705pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M569 131C584 203 591 291 596 391C603 524 609 615 622 658L609 669C545 642 461 523 357 312C267 129 194 14 119 14C104 14 91 21 88 36H91C96 33 101 32 107 32C134 32 149 54 149 77C149 104 128 128 97 128C65 128 38 101 38 62C38 19 71 -15 118 -15C207 -15 274 84 391 315C446 423 501 518 534 563L535 562C518 501 511 401 508 297C504 160 495 63 476 -13H496C588 65 656 224 712 363C783 536 828 621 863 640H866C866 608 894 595 913 595C937 595 956 613 956 641C956 670 935 687 904 687C875 687 853 675 830 650C782 596 735 496 692 394C649 290 605 184 570 131H569Z"></path></g><g transform="matrix(.013,0,0,-0.013,16.242,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,27.505,0)"><path d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z"></path></g><g transform="matrix(.013,0,0,-0.013,33.745,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"></path></g><g transform="matrix(.013,0,0,-0.013,39.985,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"></path></g></svg>,</span> <svg height="8.6335pt" id="M129" style="vertical-align:-0.2846804pt" version="1.1" viewbox="-0.0498162 -8.34882 41.812 8.6335" width="41.812pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M279 468H265C237 462 212 452 191 438C134 403 97 369 81 338S57 259 57 196C57 95 89 23 153 -18H167L338 48L329 71C297 54 267 45 239 45C215 45 197 53 185 70C168 93 159 140 159 212V272C159 332 166 371 179 388C183 393 190 399 199 404L260 328L344 384L279 468Z"></path></g><g transform="matrix(.013,0,0,-0.013,8.637,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,19.9,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"></path></g><g transform="matrix(.013,0,0,-0.013,26.14,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"></path></g><g transform="matrix(.013,0,0,-0.013,29.104,0)"><path d="M137 343C167 482 260 545 321 574C357 591 397 603 429 609L423 641C382 634 335 622 295 608C189 570 37 457 37 238C37 84 125 -12 242 -12C362 -12 447 89 447 209C447 311 374 393 267 393C247 393 226 386 204 376L137 343ZM227 337C318 337 361 256 361 173C361 105 336 22 258 22C176 22 126 120 126 240C126 266 127 291 132 310C155 323 189 337 227 337Z"></path></g><g transform="matrix(.013,0,0,-0.013,35.344,0)"><path d="M249 635C141 635 70 555 70 471C70 401 114 353 179 316C143 294 106 267 90 252C68 231 45 202 45 157C45 50 130 -12 237 -12C322 -12 435 52 435 169C435 256 372 304 303 343C349 374 375 398 383 407C401 429 411 458 411 487C411 569 344 635 249 635ZM238 603C285 603 337 567 337 482C337 422 310 385 276 358C205 393 145 426 145 500C145 552 179 603 238 603ZM248 20C183 20 125 70 125 163C125 218 158 268 206 300C284 261 355 217 355 143C355 66 308 20 248 20Z"></path></g></svg> corresponding to Example <a href="https://static.hindawi.com/articles/jmath/volume-2021/8862139/figures/#ex3" target="_blank">3</a>.</div>

Journal of Mathematics

tab4

Table 4

Table 4: Numerical Solution of Heat Equation in Polar Cylindrical Coordinates by the Meshless Method of Lines 