Analysis of Fractional-Order Regularized Long-Wave Models via a Novel Transform

<div>Example 3 solution graph (a) exact solution, (b) analytical solution at <svg height="10.8649pt" id="M112" style="vertical-align:-1.57657pt" version="1.1" viewbox="-0.0498162 -9.28833 31.0077 10.8649" width="31.0077pt" 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="M494 514C482 587 419 712 303 712C238 712 174 667 174 603C174 561 205 514 249 449C219 438 187 422 162 407C93 366 23 283 23 177C23 69 87 -12 190 -12C244 -12 288 5 328 33C406 87 444 170 444 249C444 329 404 391 331 475C265 550 222 605 222 627C222 647 238 657 267 657C355 657 421 585 484 499L494 514ZM359 234C359 143 319 30 219 30C172 30 114 75 114 178C114 275 163 343 195 378C212 397 241 415 269 425C305 382 359 313 359 234Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.354,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,21.617,0)"><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><g transform="matrix(.013,0,0,-0.013,27.857,0)"><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> (c) analytical solution at various fractional-orders of <span class="nowrap"><svg height="9.49473pt" id="M113" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 6.84582 9.49473" width="6.84582pt" 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="M494 514C482 587 419 712 303 712C238 712 174 667 174 603C174 561 205 514 249 449C219 438 187 422 162 407C93 366 23 283 23 177C23 69 87 -12 190 -12C244 -12 288 5 328 33C406 87 444 170 444 249C444 329 404 391 331 475C265 550 222 605 222 627C222 647 238 657 267 657C355 657 421 585 484 499L494 514ZM359 234C359 143 319 30 219 30C172 30 114 75 114 178C114 275 163 343 195 378C212 397 241 415 269 425C305 382 359 313 359 234Z"></path></g></svg>,</span> and (d) <span class="nowrap"><svg height="9.2729pt" id="M114" style="vertical-align:-0.2063894pt" version="1.1" viewbox="-0.0498162 -9.06651 37.7539 9.2729" width="37.7539pt" 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="M522 671C503 656 484 642 463 631C450 624 436 617 421 617C402 617 385 625 368 633C318 657 256 686 214 686C175 686 138 673 112 644C95 625 83 602 72 579C56 545 42 510 30 474L48 463C63 502 83 540 112 569C133 591 162 601 192 601C212 601 230 596 249 590C278 580 335 559 378 543C435 573 486 611 533 655L522 671ZM43 -16L96 21C123 -3 155 -27 192 -27C274 -27 348 17 405 74C450 119 464 183 464 245C464 293 455 348 448 399C444 420 444 442 442 463C442 481 451 498 460 514C474 537 492 557 512 575L496 580C445 548 402 507 362 463C357 421 363 378 368 335C373 293 378 250 378 207C378 177 376 147 367 118C361 96 352 74 336 58C316 38 291 21 261 21C237 21 214 31 197 48C168 77 169 157 144 117C105 82 68 43 38 0L43 -16Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.834,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.097,0)"><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,-0.013,28.337,0)"><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,-0.013,31.301,0)"><path d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"></path></g></svg>.</span></div>

Journal of Function Spaces

fig3

Figure 3

Figure 3: Analysis of Fractional-Order Regularized Long-Wave Models via a Novel Transform 