Reconstruction of a Robin Coefficient by a Predictor-Corrector Method

<div>The exact solution <svg height="11.4781pt" id="M166" style="vertical-align:-2.18977pt" version="1.1" viewbox="-0.0498162 -9.28833 21.4865 11.4781" width="21.4865pt" 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="M558 454L532 470C512 451 497 442 465 442C421 442 376 450 318 450C146 450 24 313 24 164C24 46 91 -12 183 -12C317 -12 439 105 439 247C439 306 428 346 382 385L383 389C414 384 450 379 485 379C517 379 539 412 558 454ZM357 273C357 191 313 28 205 28C149 28 111 79 111 160C111 266 175 396 280 396C298 396 328 386 341 357C353 331 357 315 357 273Z" id="g50-240"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="576" vert-adv-y="576"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,7.49,0)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z" id="g50-41"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,12.144,0)"><path d="M329 433H203L239 587L230 596L147 534L123 433H57L30 395L34 388H115L61 129C37 16 59 -12 85 -12C147 -12 222 58 260 98L241 125C212 95 160 62 144 62C132 62 127 71 138 126L192 386L305 394L329 433Z" id="g50-117"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="347" vert-adv-y="347"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,16.655,0)"><path d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z" id="g50-42"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g></svg> and its approximations <svg height="12.5951pt" id="M167" style="vertical-align:-0.2064009pt" version="1.1" viewbox="-0.0498162 -12.3887 12.7631 12.5951" width="12.7631pt" 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="M558 454L532 470C512 451 497 442 465 442C421 442 376 450 318 450C146 450 24 313 24 164C24 46 91 -12 183 -12C317 -12 439 105 439 247C439 306 428 346 382 385L383 389C414 384 450 379 485 379C517 379 539 412 558 454ZM357 273C357 191 313 28 205 28C149 28 111 79 111 160C111 266 175 396 280 396C298 396 328 386 341 357C353 331 357 315 357 273Z" id="g50-240"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="576" vert-adv-y="576"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,7.49,-5.872)"><path d="M485 418C485 434 469 451 441 451C389 451 312 387 255 335C221 304 194 278 159 242H157L259 678C264 698 264 710 255 710C242 710 183 681 96 673L90 643L128 642C165 641 171 640 162 602L24 -5L31 -12C51 -4 81 4 110 9L145 181C154 195 180 220 205 242C233 170 263 106 291 53C318 1 335 -12 360 -12C383 -12 425 0 483 82L463 105C437 79 411 57 400 57C388 57 378 68 355 110C329 156 288 244 268 298C299 333 355 376 405 376C426 376 440 373 448 368C453 365 465 368 468 373C477 386 485 405 485 418Z" id="g50-108"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="509" vert-adv-y="509"></glyph.data></g></svg> with (a) <svg height="9.8079pt" id="M168" style="vertical-align:-0.5195704pt" version="1.1" viewbox="-0.0498162 -9.28833 38.8847 9.8079" width="38.8847pt" 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="M501 517C491 587 426 710 310 710C240 710 175 662 175 601C175 561 201 517 250 452C219 438 187 423 157 404C88 360 24 280 24 178C24 71 90 -12 195 -12C250 -12 297 6 334 32C412 87 452 170 452 249C452 335 408 396 333 481C259 565 229 601 229 621C229 642 245 652 275 652C359 652 427 583 490 499L501 517ZM364 233C364 142 323 33 222 33C173 33 119 81 119 178C119 265 156 335 193 374C213 395 242 413 272 425C308 384 364 319 364 233Z" id="g50-226"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="525" vert-adv-y="525"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,10.502,0)"><path d="M556 331V384H55V331H556ZM556 140V193H55V140H556Z" id="g54-34"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,22.12,0)"><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><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,28.451,0)"><path d="M611 626L575 651L170 -9L204 -35L611 626ZM200 635C93 635 50 532 50 449C50 365 94 264 199 264C306 264 349 365 349 450C349 534 304 635 200 635ZM199 601C264 601 273 501 273 450S264 298 200 298C139 298 126 398 126 450S139 601 199 601ZM590 359C485 359 440 256 440 173C440 89 485 -12 589 -12C694 -12 738 89 738 173C738 256 694 359 590 359ZM589 325C653 325 662 225 662 173C662 124 653 22 590 22C529 22 516 122 516 174C516 227 529 325 589 325Z" id="g58-35"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="786" vert-adv-y="786"></glyph.data></g></svg> and (b) <svg height="11.0084pt" id="M169" style="vertical-align:-1.72007pt" version="1.1" viewbox="-0.0498162 -9.28833 60.7936 11.0084" width="60.7936pt" 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="M501 517C491 587 426 710 310 710C240 710 175 662 175 601C175 561 201 517 250 452C219 438 187 423 157 404C88 360 24 280 24 178C24 71 90 -12 195 -12C250 -12 297 6 334 32C412 87 452 170 452 249C452 335 408 396 333 481C259 565 229 601 229 621C229 642 245 652 275 652C359 652 427 583 490 499L501 517ZM364 233C364 142 323 33 222 33C173 33 119 81 119 178C119 265 156 335 193 374C213 395 242 413 272 425C308 384 364 319 364 233Z" id="g50-226"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="525" vert-adv-y="525"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,10.502,0)"><path d="M556 331V384H55V331H556ZM556 140V193H55V140H556Z" id="g54-34"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,22.12,0)"><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><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,28.451,0)"><path d="M611 626L575 651L170 -9L204 -35L611 626ZM200 635C93 635 50 532 50 449C50 365 94 264 199 264C306 264 349 365 349 450C349 534 304 635 200 635ZM199 601C264 601 273 501 273 450S264 298 200 298C139 298 126 398 126 450S139 601 199 601ZM590 359C485 359 440 256 440 173C440 89 485 -12 589 -12C694 -12 738 89 738 173C738 256 694 359 590 359ZM589 325C653 325 662 225 662 173C662 124 653 22 590 22C529 22 516 122 516 174C516 227 529 325 589 325Z" id="g58-35"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="786" vert-adv-y="786"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,38.669,0)"><path d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z" id="g50-45"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="237" vert-adv-y="237"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,43.955,0)"><path d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z" id="g50-52"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,50.286,0)"><path d="M611 626L575 651L170 -9L204 -35L611 626ZM200 635C93 635 50 532 50 449C50 365 94 264 199 264C306 264 349 365 349 450C349 534 304 635 200 635ZM199 601C264 601 273 501 273 450S264 298 200 298C139 298 126 398 126 450S139 601 199 601ZM590 359C485 359 440 256 440 173C440 89 485 -12 589 -12C694 -12 738 89 738 173C738 256 694 359 590 359ZM589 325C653 325 662 225 662 173C662 124 653 22 590 22C529 22 516 122 516 174C516 227 529 325 589 325Z" id="g58-35"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="786" vert-adv-y="786"></glyph.data></g></svg>, and <svg height="9.20766pt" id="M170" style="vertical-align:-0.5195694pt" version="1.1" viewbox="-0.0498162 -8.68809 16.7107 9.20766" width="16.7107pt" 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="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><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.331,0)"><path d="M611 626L575 651L170 -9L204 -35L611 626ZM200 635C93 635 50 532 50 449C50 365 94 264 199 264C306 264 349 365 349 450C349 534 304 635 200 635ZM199 601C264 601 273 501 273 450S264 298 200 298C139 298 126 398 126 450S139 601 199 601ZM590 359C485 359 440 256 440 173C440 89 485 -12 589 -12C694 -12 738 89 738 173C738 256 694 359 590 359ZM589 325C653 325 662 225 662 173C662 124 653 22 590 22C529 22 516 122 516 174C516 227 529 325 589 325Z" id="g58-35"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="786" vert-adv-y="786"></glyph.data></g></svg> for Example <a href="https://static.hindawi.com/articles/mpe/volume-2015/496587/figures/#ex2">2</a>.</div>

Mathematical Problems in Engineering

fig2

Figure 2

Figure 2: Reconstruction of a Robin Coefficient by a Predictor-Corrector Method 