Mikhlin’s Integral Equation and the Integral Equation with the Generalized Neumann Kernel on Simply Connected Domains

<div>The curves (a) <svg height="13.3154pt" id="M603" style="vertical-align:-4.679501pt" version="1.1" viewbox="-0.0498162 -8.6359 19.9607 13.3154" width="19.9607pt" 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="M485 501C480 550 475 615 475 650H42V622C119 616 127 606 127 525V126C127 44 119 34 41 28V0H311V28C220 34 212 43 212 126V584C212 612 216 615 244 615H318C382 615 404 608 420 587C435 568 445 541 455 497L485 501Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.486,3.132)"><path d="M583 449L551 460C535 436 529 433 494 433C409 433 311 438 228 438C97 438 53 381 24 341L44 316C85 355 124 372 183 372C161 259 87 53 26 5L34 -12C53 -12 94 -4 118 12C166 92 211 256 236 371L394 369C375 301 352 212 334 121C329 96 326 75 326 58C326 5 349 -12 381 -12C431 -12 482 25 519 66L504 97C480 80 448 64 432 64C419 64 410 80 422 154C432 217 450 303 466 365C499 365 536 366 546 369C557 384 570 409 583 449Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.956,3.132)"><path d="M381 704H317L48 -163H112L381 704Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,14.863,3.132)"><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"></path></g></svg> and (b) <svg height="13.3154pt" id="M604" style="vertical-align:-4.679501pt" version="1.1" viewbox="-0.0498162 -8.6359 24.409 13.3154" width="24.409pt" 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="M485 501C480 550 475 615 475 650H42V622C119 616 127 606 127 525V126C127 44 119 34 41 28V0H311V28C220 34 212 43 212 126V584C212 612 216 615 244 615H318C382 615 404 608 420 587C435 568 445 541 455 497L485 501Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.486,3.132)"><path d="M583 449L551 460C535 436 529 433 494 433C409 433 311 438 228 438C97 438 53 381 24 341L44 316C85 355 124 372 183 372C161 259 87 53 26 5L34 -12C53 -12 94 -4 118 12C166 92 211 256 236 371L394 369C375 301 352 212 334 121C329 96 326 75 326 58C326 5 349 -12 381 -12C431 -12 482 25 519 66L504 97C480 80 448 64 432 64C419 64 410 80 422 154C432 217 450 303 466 365C499 365 536 366 546 369C557 384 570 409 583 449Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.956,3.132)"><path d="M381 704H317L48 -163H112L381 704Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,14.863,3.132)"><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"></path></g><g transform="matrix(.0091,0,0,-0.0091,19.295,3.132)"><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"></path></g></svg> in Example <a href="https://static.hindawi.com/articles/cmm/volume-2022/6488709/figures/#ex4">8</a>.</div>

Computational and Mathematical Methods

fig8

Figure 8

Figure 8: Mikhlin’s Integral Equation and the Integral Equation with the Generalized Neumann Kernel on Simply Connected Domains 