The Analysis of Fractional-Order Nonlinear Systems of Third Order KdV and Burgers Equations via a Novel Transform

<div>Exact solution and analytical solution for <svg height="12.2089pt" id="M129" style="vertical-align:-2.503pt" version="1.1" viewbox="-0.0498162 -9.7059 33.4261 12.2089" width="33.4261pt" 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="M442 569C442 628 407 660 362 660C328 660 292 648 255 616C200 568 165 508 123 334L106 264C80 242 52 222 23 202L30 176C54 191 76 205 100 222L87 166C73 103 79 43 95 19C108 0 139 -12 169 -12C242 -12 318 38 369 106L354 129C322 93 260 49 215 49C173 49 147 81 170 187L189 274C293 344 442 452 442 569ZM380 565C380 478 267 377 196 322L216 406C254 556 282 623 332 623C364 623 380 596 380 565Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.047,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.545,0)"><path d="M425 682C425 693 422 699 407 699C369 699 308 677 260 646C213 651 191 674 191 701C191 715 195 726 200 734L183 740C153 730 118 709 118 681C118 644 163 619 218 615C168 582 115 534 115 463C115 418 147 382 180 369C102 329 23 261 23 163C23 74 74 38 158 14C183 7 201 2 221 -2C279 -15 290 -33 290 -58C290 -94 251 -145 230 -169L252 -189C280 -171 367 -103 367 -25C367 20 333 51 266 68C245 73 218 80 191 89C139 106 107 135 107 192C107 261 161 329 221 355C248 350 281 347 314 347C330 347 370 388 370 406C370 418 366 422 352 422C321 422 280 413 255 407C218 413 192 448 192 493C192 544 229 588 266 610C309 610 354 615 376 623C397 631 425 651 425 682Z"></path></g><g transform="matrix(.013,0,0,-0.013,16.37,0)"><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><g transform="matrix(.013,0,0,-0.013,21.514,0)"><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,-0.013,28.716,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg> problem 2.</div>

Complexity

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Figure 5

Figure 5: The Analysis of Fractional-Order Nonlinear Systems of Third Order KdV and Burgers Equations via a Novel Transform 