Spectral Collocation Methods for Fractional Integro-Differential Equations with Weakly Singular Kernels

<div>The <svg height="10.0062pt" id="M193" style="vertical-align:-0.04980946pt" version="1.1" viewbox="-0.0498162 -9.95639 17.2949 10.0062" width="17.2949pt" 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="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.185,-5.741)"><path d="M1000 227C1000 351 905 451 780 451C658 451 590 380 530 279C465 385 388 451 261 451C168 451 38 376 38 210C38 98 121 -12 261 -12C381 -12 448 59 508 160C573 53 652 -12 780 -12C879 -12 1000 64 1000 227ZM485 199C434 109 370 25 271 25C188 25 133 117 133 241C133 355 189 414 249 414C357 414 426 302 485 199ZM903 204C903 70 852 25 790 25C681 25 611 136 552 240C603 330 669 414 768 414C855 414 903 321 903 204Z"></path></g></svg> and <svg height="15.535pt" id="M194" style="vertical-align:-3.9436pt" version="1.1" viewbox="-0.0498162 -11.5914 32.7338 15.535" width="32.7338pt" 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="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.185,-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"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.294,3.784)"><path d="M624 292C624 400 569 451 537 451C520 451 497 442 490 436L489 427C521 404 550 351 550 280C550 170 513 51 423 51C383 51 361 87 361 150C361 197 384 301 397 339L391 343L310 328C307 297 301 249 293 212C270 102 223 56 182 56C144 56 114 93 114 172C114 278 161 369 243 430L223 457C99 392 24 290 24 156C24 43 81 -12 141 -12C187 -12 247 32 286 88C295 31 325 -12 380 -12C431 -12 475 16 511 49C581 113 624 195 624 292Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,14.192,.587)"><path d="M578 242V297H59V242H578Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,18.332,.587)"><path d="M554 97L534 128C507 93 465 66 455 66C448 66 444 75 451 110C474 229 500 342 528 454H519L440 426C428 363 415 308 407 276C359 198 251 61 193 61C169 61 164 108 171 138L238 447L228 454L154 432L62 19C44 -62 25 -156 25 -181C25 -232 44 -253 69 -253C99 -253 128 -231 136 -221L135 -211C120 -202 102 -152 102 -101C102 -57 114 -1 121 26H123C135 1 142 -12 167 -12C187 -12 207 -5 242 22C288 58 334 111 388 178H390L374 108C349 -1 373 -12 395 -12C429 -12 499 29 554 97Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,22.057,.587)"><path d="M202 22C202 97 141 138 100 138C74 138 46 121 46 92C46 74 56 66 61 64C97 55 127 31 127 -5C127 -43 97 -68 44 -90L59 -129C127 -110 202 -64 202 22Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,23.656,.587)"><path d="M578 242V297H59V242H578Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,27.797,.587)"><path d="M554 97L534 128C507 93 465 66 455 66C448 66 444 75 451 110C474 229 500 342 528 454H519L440 426C428 363 415 308 407 276C359 198 251 61 193 61C169 61 164 108 171 138L238 447L228 454L154 432L62 19C44 -62 25 -156 25 -181C25 -232 44 -253 69 -253C99 -253 128 -231 136 -221L135 -211C120 -202 102 -152 102 -101C102 -57 114 -1 121 26H123C135 1 142 -12 167 -12C187 -12 207 -5 242 22C288 58 334 111 388 178H390L374 108C349 -1 373 -12 395 -12C429 -12 499 29 554 97Z"></path></g></svg> errors for Example <a href="https://static-preview.hindawi.com/articles/jmath/volume-2022/3767559/figures/#ex2">2</a>.</div>

Journal of Mathematics

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Figure 2

Figure 2: Spectral Collocation Methods for Fractional Integro-Differential Equations with Weakly Singular Kernels 