Effective Analytical Computational Technique for Conformable Time-Fractional Nonlinear Gardner Equation and Cahn-Hilliard Equations of Fourth and Sixth Order Emerging in Dispersive Media

<div>Absolute errors <svg height="12.5794pt" id="M367" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 45.0317 12.5794" width="45.0317pt" 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="M162 -163V703H101V-163H162Z"></path></g><g transform="matrix(.013,0,0,-0.013,3.419,0)"><path d="M670 190C569 71 508 22 463 22C442 22 434 34 434 53C434 68 438 92 457 127L617 424H539C427 190 279 19 184 19C164 19 151 34 151 51C151 75 160 106 173 129L332 424H257C220 381 156 326 90 326C73 326 57 330 40 338L30 312C47 301 67 296 88 296C131 296 177 313 213 342L145 216C116 161 99 116 99 77C99 17 133 -11 176 -11C244 -11 311 36 387 123L390 122C387 109 382 74 382 63C382 23 401 -8 452 -8C525 -8 597 54 693 173L670 190Z"></path></g><g transform="matrix(.013,0,0,-0.013,16.114,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,26.65,0)"><path d="M670 190C569 71 508 22 463 22C442 22 434 34 434 53C434 68 438 92 457 127L617 424H539C427 190 279 19 184 19C164 19 151 34 151 51C151 75 160 106 173 129L332 424H257C220 381 156 326 90 326C73 326 57 330 40 338L30 312C47 301 67 296 88 296C131 296 177 313 213 342L145 216C116 161 99 116 99 77C99 17 133 -11 176 -11C244 -11 311 36 387 123L390 122C387 109 382 74 382 63C382 23 401 -8 452 -8C525 -8 597 54 693 173L670 190Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,36.439,3.132)"><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"></path></g><g transform="matrix(.013,0,0,-0.013,41.385,0)"><path d="M162 -163V703H101V-163H162Z"></path></g></svg> for different values of fractional order <svg height="6.1673pt" id="M368" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 7.51131 6.1673" width="7.51131pt" 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="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"></path></g></svg> at <svg height="10.2124pt" id="M369" style="vertical-align:-1.576501pt" version="1.1" viewbox="-0.0498162 -8.6359 40.6898 10.2124" width="40.6898pt" 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="M532 526H387L459 658H380L309 526H206L193 496H295L256 423C221 378 163 306 98 306C81 306 57 310 40 318L30 292C47 281 67 276 88 276C129 276 168 293 205 326L155 234C123 174 98 115 98 70C98 15 131 -11 174 -11C243 -11 311 50 411 175L390 192C291 80 234 19 183 19C163 19 150 32 150 51C150 66 153 88 164 109L373 496H519L532 526Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.795,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.058,0)"><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,-0.013,28.298,0)"><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,-0.013,31.262,0)"><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"></path></g><g transform="matrix(.013,0,0,-0.013,37.502,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g></svg> <svg height="9.49473pt" id="M370" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 28.4893 9.49473" width="28.4893pt" 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="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.806,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.068,0)"><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"></path></g></svg> of FGEs (<a href="https://static-preview.hindawi.com/articles/jfs/volume-2022/4422186/figures/#EEq15" target="_blank">21</a>) and (<a href="https://static-preview.hindawi.com/articles/jfs/volume-2022/4422186/figures/#EEq16" target="_blank">22</a>).</div>

Journal of Function Spaces

tab2

Table 2

Table 2: Effective Analytical Computational Technique for Conformable Time-Fractional Nonlinear Gardner Equation and Cahn-Hilliard Equations of Fourth and Sixth Order Emerging in Dispersive Media 

Table 2 | Effective Analytical Computational Technique for Conformable Time-Fractional Nonlinear Gardner Equation and Cahn-Hilliard Equations of Fourth and Sixth Order Emerging in Dispersive Media 