A Numerical Algorithm for Solving Higher-Order Nonlinear BVPs with an Application on Fluid Flow over a Shrinking Permeable Infinite Long Cylinder

<table class="fixed-width table-group" id="tab2"><tr><td><table class="table"><colgroup><col style="width:3.01em"/><col style="width:4.22em"/><col style="width:9.39em"/></colgroup><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td class="align_left"><span style="width: 23.1773ptpx;"><svg height="9.25658pt" id="M212" style="vertical-align:-3.295671pt" version="1.1" viewbox="-0.0498162 -5.96091 23.1773 9.25658" width="23.1773pt" 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="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 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112C41 47 92 -12 163 -12C195 -12 243 25 283 51C288 35 293 22 308 7C319 -4 339 -12 354 -12L442 39ZM281 87C262 68 226 51 199 51C169 51 129 74 129 126C129 162 151 181 190 198C211 208 259 229 281 240V87Z" id="g58-95"></path></g><g transform="matrix(.0091,0,0,-0.0091,18.109,3.132)"><path d="M482 0V30C420 38 407 48 370 103L273 248C308 299 331 332 352 357C387 397 401 404 463 409V439H277V409C318 405 324 400 307 369C290 337 270 307 251 279L194 367C175 396 180 402 220 409V439H18V409C74 401 87 395 117 349L204 213C173 168 147 130 119 95C78 46 67 38 5 30V0H195V30C144 38 143 47 162 79C181 112 202 149 225 181L297 70C314 43 307 35 264 30V0H482Z" id="g58-118"></path></g></svg></span></td><td class="align_center"><span style="width: 21.4039ptpx;"><svg height="11.9316pt" id="M213" style="vertical-align:-3.2957pt" version="1.1" viewbox="-0.0498162 -8.6359 21.4039 11.9316" width="21.4039pt" 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="M593 650H142L136 622C219 616 224 611 209 531L130 120C114 39 108 34 23 28L17 0H501C518 32 557 130 568 158L540 169C519 132 495 93 469 70C439 43 409 35 332 35C272 35 241 37 225 50C211 61 211 89 222 149L253 320H350C441 320 443 312 441 239H471L510 434H481C455 366 448 358 360 358H260L303 572C311 613 312 615 348 615H431C509 615 522 603 532 581C544 555 546 531 545 491L574 493C583 555 592 631 593 650Z" id="g113-70"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.943,3.132)"><path d="M308 0V33C225 38 217 44 217 128V521C217 605 225 612 308 617V650H39V617C122 612 130 605 130 521V128C130 44 122 38 39 33V0H308Z" id="g58-71"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.11,3.132)"><path d="M46 650V618C122 613 130 611 130 522V128C130 46 123 40 40 32V0H319V32C227 40 217 42 217 128V279L292 260C303 259 322 258 338 261C436 272 534 337 534 467C534 534 508 580 469 609C428 640 371 650 299 650H46ZM217 558C217 585 220 596 228 603C235 610 256 615 284 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transform="matrix(.013,0,0,-0.013,0,0)"><path d="M537 353L508 390C487 357 455 333 417 333C389 333 344 353 309 370C265 391 212 414 172 414C122 414 79 382 50 340L81 307C101 339 134 364 172 364C198 364 245 343 280 326C324 305 375 283 417 283C466 283 507 314 537 353ZM537 168L508 205C487 172 455 148 417 148C389 148 344 168 309 185C265 206 212 229 172 229C122 229 79 197 50 155L81 122C101 154 134 179 172 179C198 179 245 158 280 141C324 120 375 98 417 98C466 98 507 129 537 168Z" id="g117-132"></path></g><g transform="matrix(.013,0,0,-0.013,11.263,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g><g transform="matrix(.013,0,0,-0.013,17.503,0)"><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" id="g113-49"></path></g><g transform="matrix(.0091,0,0,-0.0091,23.743,-5.741)"><path d="M556 236V289H56V236H556Z" id="g54-33"></path></g><g transform="matrix(.0091,0,0,-0.0091,29.303,-5.741)"><path d="M158 548H390L417 615L410 623H122L83 318C105 326 143 337 185 337C296 337 350 275 350 188C350 116 308 42 225 42C164 42 122 74 100 93C90 101 82 99 72 92C60 82 51 68 50 59C48 46 52 38 66 24C82 9 125 -12 172 -12C225 -11 292 15 346 59C408 108 437 166 437 226C437 309 371 397 242 397C214 397 170 382 133 369L158 548Z" id="g50-54"></path></g></svg></span></td><td class="align_center">0.0156001</td></tr><tr><td class="align_left">4</td><td class="align_center"><span style="width: 34.4452ptpx;"><svg height="11.6425pt" id="M215" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -11.4361 34.4452 11.6425" width="34.4452pt" 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="M537 353L508 390C487 357 455 333 417 333C389 333 344 353 309 370C265 391 212 414 172 414C122 414 79 382 50 340L81 307C101 339 134 364 172 364C198 364 245 343 280 326C324 305 375 283 417 283C466 283 507 314 537 353ZM537 168L508 205C487 172 455 148 417 148C389 148 344 168 309 185C265 206 212 229 172 229C122 229 79 197 50 155L81 122C101 154 134 179 172 179C198 179 245 158 280 141C324 120 375 98 417 98C466 98 507 129 537 168Z" id="g117-132"></path></g><g transform="matrix(.013,0,0,-0.013,11.263,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g><g transform="matrix(.013,0,0,-0.013,17.503,0)"><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" id="g113-49"></path></g><g transform="matrix(.0091,0,0,-0.0091,23.743,-5.741)"><path d="M556 236V289H56V236H556Z" id="g54-33"></path></g><g transform="matrix(.0091,0,0,-0.0091,29.303,-5.741)"><path d="M453 623H65C60 580 56 530 46 472H80C103 538 110 548 175 548H389C310 380 195 168 90 0L98 -12L175 -2C273 206 366 408 462 610L453 623Z" id="g50-56"></path></g></svg></span></td><td class="align_center">0.0312002</td></tr><tr><td class="align_left">5</td><td class="align_center"><span style="width: 34.4452ptpx;"><svg height="11.7978pt" id="M216" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -11.5914 34.4452 11.7978" width="34.4452pt" 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="M537 353L508 390C487 357 455 333 417 333C389 333 344 353 309 370C265 391 212 414 172 414C122 414 79 382 50 340L81 307C101 339 134 364 172 364C198 364 245 343 280 326C324 305 375 283 417 283C466 283 507 314 537 353ZM537 168L508 205C487 172 455 148 417 148C389 148 344 168 309 185C265 206 212 229 172 229C122 229 79 197 50 155L81 122C101 154 134 179 172 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325C330 309 297 297 262 297C179 297 127 369 127 456C127 515 156 598 231 598Z" id="g50-58"></path></g></svg></span></td><td class="align_center">0.0780005</td></tr><tr><td class="align_left">6</td><td class="align_center"><span style="width: 38.8935ptpx;"><svg height="11.7978pt" id="M217" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -11.5914 38.8935 11.7978" width="38.8935pt" 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="M537 353L508 390C487 357 455 333 417 333C389 333 344 353 309 370C265 391 212 414 172 414C122 414 79 382 50 340L81 307C101 339 134 364 172 364C198 364 245 343 280 326C324 305 375 283 417 283C466 283 507 314 537 353ZM537 168L508 205C487 172 455 148 417 148C389 148 344 168 309 185C265 206 212 229 172 229C122 229 79 197 50 155L81 122C101 154 134 179 172 179C198 179 245 158 280 141C324 120 375 98 417 98C466 98 507 129 537 168Z" id="g117-132"></path></g><g transform="matrix(.013,0,0,-0.013,11.263,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g><g transform="matrix(.013,0,0,-0.013,17.503,0)"><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" id="g113-49"></path></g><g transform="matrix(.0091,0,0,-0.0091,23.743,-5.741)"><path d="M556 236V289H56V236H556Z" id="g54-33"></path></g><g transform="matrix(.0091,0,0,-0.0091,29.303,-5.741)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path></g><g transform="matrix(.0091,0,0,-0.0091,33.735,-5.741)"><path d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z" id="g50-49"></path></g></svg></span></td><td class="align_center">0.156001</td></tr><tr><td class="align_left">7</td><td class="align_center"><span style="width: 38.8935ptpx;"><svg height="11.7978pt" id="M218" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -11.5914 38.8935 11.7978" width="38.8935pt" 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="M537 353L508 390C487 357 455 333 417 333C389 333 344 353 309 370C265 391 212 414 172 414C122 414 79 382 50 340L81 307C101 339 134 364 172 364C198 364 245 343 280 326C324 305 375 283 417 283C466 283 507 314 537 353ZM537 168L508 205C487 172 455 148 417 148C389 148 344 168 309 185C265 206 212 229 172 229C122 229 79 197 50 155L81 122C101 154 134 179 172 179C198 179 245 158 280 141C324 120 375 98 417 98C466 98 507 129 537 168Z" id="g117-132"></path></g><g transform="matrix(.013,0,0,-0.013,11.263,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g><g transform="matrix(.013,0,0,-0.013,17.503,0)"><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" id="g113-49"></path></g><g transform="matrix(.0091,0,0,-0.0091,23.743,-5.741)"><path d="M556 236V289H56V236H556Z" id="g54-33"></path></g><g transform="matrix(.0091,0,0,-0.0091,29.303,-5.741)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path></g><g transform="matrix(.0091,0,0,-0.0091,33.735,-5.741)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g></svg></span></td><td class="align_center">0.296402</td></tr><tr><td class="align_left">8</td><td class="align_center"><span style="width: 38.8935ptpx;"><svg height="11.7978pt" id="M219" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -11.5914 38.8935 11.7978" width="38.8935pt" 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="M537 353L508 390C487 357 455 333 417 333C389 333 344 353 309 370C265 391 212 414 172 414C122 414 79 382 50 340L81 307C101 339 134 364 172 364C198 364 245 343 280 326C324 305 375 283 417 283C466 283 507 314 537 353ZM537 168L508 205C487 172 455 148 417 148C389 148 344 168 309 185C265 206 212 229 172 229C122 229 79 197 50 155L81 122C101 154 134 179 172 179C198 179 245 158 280 141C324 120 375 98 417 98C466 98 507 129 537 168Z" id="g117-132"></path></g><g transform="matrix(.013,0,0,-0.013,11.263,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g><g transform="matrix(.013,0,0,-0.013,17.503,0)"><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" id="g113-49"></path></g><g transform="matrix(.0091,0,0,-0.0091,23.743,-5.741)"><path d="M556 236V289H56V236H556Z" id="g54-33"></path></g><g transform="matrix(.0091,0,0,-0.0091,29.303,-5.741)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path></g><g transform="matrix(.0091,0,0,-0.0091,33.735,-5.741)"><path d="M462 177V227H365V632H320C217 496 116 350 21 208V177H284V109C284 43 280 38 190 31V0H451V31C369 38 365 43 365 108V177H462ZM284 227H88C155 336 218 430 282 519H284V227Z" id="g50-53"></path></g></svg></span></td><td class="align_center">0.546004</td></tr><tr class="table-tr"><td colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>The upper bound of the error for the first solution at <svg height="11.7782pt" id="M209" style="vertical-align:-3.42938pt" version="1.1" viewbox="-0.0498162 -8.34882 27.8238 11.7782" width="27.8238pt" 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="M478 372C478 418 458 448 431 448C409 448 389 431 389 410C389 404 391 400 394 395C398 388 406 371 406 348C406 253 308 122 251 51H249C254 122 249 257 231 336C212 421 189 448 159 448C126 448 75 412 23 327L48 306C83 354 103 371 115 371C125 371 134 360 144 334C185 224 192 64 183 -19C146 -100 116 -202 110 -244L125 -261C154 -259 208 -234 222 -220C222 -194 225 -84 235 -23C247 -3 273 36 308 79C379 165 478 288 478 372Z" id="g113-225"></path></g><g transform="matrix(.013,0,0,-0.013,10.148,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,21.411,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z" id="g113-51"></path></g></svg> and <svg height="8.98583pt" id="M210" style="vertical-align:-0.3499308pt" version="1.1" viewbox="-0.0498162 -8.6359 35.105 8.98583" width="35.105pt" 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="M449 634C442 637 425 643 405 650C376 660 341 666 307 666C181 666 98 590 98 485C98 400 170 343 215 310L246 288C307 243 343 204 343 147C343 67 291 18 219 18C104 18 61 124 51 202L23 199C28 124 27 71 27 47C47 22 122 -16 204 -16C324 -16 428 60 428 174C428 256 379 309 307 360L276 382C223 419 179 455 179 516C179 576 221 632 293 632C379 632 410 564 418 487L448 490C446 536 446 592 449 634Z" id="g113-84"></path></g><g transform="matrix(.013,0,0,-0.013,9.768,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,21.031,0)"><path d="M535 230V280H52V230H535Z" id="g117-33"></path></g><g transform="matrix(.013,0,0,-0.013,28.662,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g></svg> versus the CPU time at different values of <span class="nowrap"><svg height="9.25658pt" id="M211" style="vertical-align:-3.295671pt" version="1.1" viewbox="-0.0498162 -5.96091 23.1773 9.25658" width="23.1773pt" 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="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z" id="g113-111"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.461,3.132)"><path d="M810 0V30C751 35 744 40 744 106V296C744 396 694 451 614 451C583 451 558 438 537 425C512 409 485 391 454 368C434 420 390 451 342 451C309 451 285 439 260 424C228 405 206 387 185 372V453C141 434 90 421 43 413V386C102 378 105 373 105 309V106C105 41 96 35 29 30V0H243V30C194 35 185 41 185 106V335C216 360 256 389 297 389C357 389 384 348 384 275V106C384 41 374 35 312 30V0H530V30C475 35 465 41 465 104V294C465 313 464 324 463 336C502 367 539 389 576 389C637 389 663 346 663 274V109C663 41 653 35 593 30V0H810Z" id="g58-107"></path></g><g transform="matrix(.0091,0,0,-0.0091,14.023,3.132)"><path d="M442 39L431 68C420 63 406 57 394 57C377 57 359 71 359 118V301C359 353 349 393 314 424C291 443 261 451 230 451C171 438 107 401 79 381C59 366 46 355 46 340C46 316 72 295 91 295C106 295 117 303 122 319C131 350 139 371 152 384C163 397 178 404 197 404C246 404 281 365 281 293V278C260 262 186 232 124 213C69 194 41 161 41 112C41 47 92 -12 163 -12C195 -12 243 25 283 51C288 35 293 22 308 7C319 -4 339 -12 354 -12L442 39ZM281 87C262 68 226 51 199 51C169 51 129 74 129 126C129 162 151 181 190 198C211 208 259 229 281 240V87Z" id="g58-95"></path></g><g transform="matrix(.0091,0,0,-0.0091,18.109,3.132)"><path d="M482 0V30C420 38 407 48 370 103L273 248C308 299 331 332 352 357C387 397 401 404 463 409V439H277V409C318 405 324 400 307 369C290 337 270 307 251 279L194 367C175 396 180 402 220 409V439H18V409C74 401 87 395 117 349L204 213C173 168 147 130 119 95C78 46 67 38 5 30V0H195V30C144 38 143 47 162 79C181 112 202 149 225 181L297 70C314 43 307 35 264 30V0H482Z" id="g58-118"></path></g></svg>.</span></div>

Complexity

tab2

Table 2

Table 2: A Numerical Algorithm for Solving Higher-Order Nonlinear BVPs with an Application on Fluid Flow over a Shrinking Permeable Infinite Long Cylinder 

Table 2 | A Numerical Algorithm for Solving Higher-Order Nonlinear BVPs with an Application on Fluid Flow over a Shrinking Permeable Infinite Long Cylinder 