Exponentially Fitted and Trigonometrically Fitted Explicit Modified Runge-Kutta Type Methods for Solving <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.526799pt" id="M1" height="19.6225pt" version="1.1" viewBox="-0.0657574 -15.0957 135.584 19.6225" width="135.584pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-122" d="M556 393C556 426 537 448 514 448C496 448 478 435 471 420C468 414 462 401 466 394C473 382 480 368 480 346C480 268 392 143 338 67H336C329 163 319 253 309 330S286 448 254 448C215 448 160 383 127 337L143 311C167 344 200 373 208 373S222 365 229 320C246 214 264 66 268 -20C219 -83 131 -171 17 -239L25 -261L137 -235C248 -110 273 -79 335 8C384 77 481 215 520 287C544 332 556 363 556 393Z"/></g><g transform="matrix(.012,0,0,-0.012,9.938,-7.578)"><path id="g50-31" d="M310 541L304 571C290 586 211 619 185 610L80 76L131 52L310 541Z"/></g><g transform="matrix(.012,0,0,-0.012,13.95,-7.578)"><use xlink:href="#g50-31"/></g><g transform="matrix(.012,0,0,-0.012,17.96,-7.578)"><use xlink:href="#g50-31"/></g><g transform="matrix(.017,0,0,-0.017,23.002,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,28.94,0)"><path id="g113-121" d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"/></g><g transform="matrix(.017,0,0,-0.017,38.533,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g><g transform="matrix(.017,0,0,-0.017,49.264,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.017,0,0,-0.017,64.131,0)"><path id="g113-103" d="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"/></g><g transform="matrix(.017,0,0,-0.017,75.156,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.017,0,0,-0.017,81.093,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.017,0,0,-0.017,90.687,0)"><path id="g113-45" 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"/></g><g transform="matrix(.017,0,0,-0.017,97.475,0)"><use xlink:href="#g113-122"/></g><g transform="matrix(.017,0,0,-0.017,107.414,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.017,0,0,-0.017,114.202,0)"><use xlink:href="#g113-122"/></g><g transform="matrix(.012,0,0,-0.012,124.141,-7.578)"><use xlink:href="#g50-31"/></g><g transform="matrix(.017,0,0,-0.017,129.151,0)"><use xlink:href="#g113-42"/></g></svg>

<table class="fixed-width table-group" id="tab2"><tr><td><table class="table"><colgroup><col style="width:2.50em"/><col style="width:7.12em"/><col style="width:6.64em"/><col style="width:6.72em"/><col style="width:6.67em"/><col style="width:6.71em"/><col style="width:6.73em"/><col style="width:6.71em"/></colgroup><tr><td class="thead-hr" colspan="8"><hr/></td></tr><tr class="thead"><td class="align_left"><span style="width: 7.39387ptpx;"><svg height="6.1673pt" id="M208" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 7.39387 6.1673" width="7.39387pt" 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="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z" id="g113-121"></path></g></svg></span></td><td class="align_center">Exact Solution</td><td class="align_center">RK5B</td><td class="align_center">RKF5</td><td class="align_center">RKT5</td><td class="align_center">EMFRKT5</td><td class="align_center">TFRK</td><td class="align_center">TFMRKT5</td></tr><tr><td class="thead-hr" colspan="8"><hr/></td></tr><tr><td class="align_left">0.0</td><td class="align_center">1.000000000</td><td class="align_center">1.000000000</td><td class="align_center">1.000000000</td><td class="align_center">1.000000000</td><td class="align_center">1.000000000</td><td class="align_center">1.000000000</td><td class="align_center">1.000000000</td></tr><tr><td class="align_left">0.2</td><td class="align_center">1.221211030</td><td class="align_center">1.2212100068</td><td class="align_center">1.2212100097</td><td class="align_center">1.2212100039</td><td class="align_center">1.2212100052</td><td class="align_center">1.2212100218</td><td class="align_center">1.2212100052</td></tr><tr><td class="align_left">0.4</td><td class="align_center">1.488834893</td><td class="align_center">1.4888347851</td><td class="align_center">1.4888347895</td><td class="align_center">1.4888347797</td><td class="align_center">1.4888347885</td><td class="align_center">1.4888348090</td><td class="align_center">1.4888347885</td></tr><tr><td class="align_left">0.6</td><td class="align_center">1.807361404</td><td class="align_center">1.8073614063</td><td class="align_center">1.8073614114</td><td class="align_center">1.8073613988</td><td class="align_center">1.8073614237</td><td class="align_center">1.8073614357</td><td class="align_center">1.8073614237</td></tr><tr><td class="align_left">0.8</td><td class="align_center">2.179819234</td><td class="align_center">2.1798192463</td><td class="align_center">2.1798192513</td><td class="align_center">2.1798192371</td><td class="align_center">2.1798192873</td><td class="align_center">2.1798192788</td><td class="align_center">2.1798192873</td></tr><tr><td class="align_left">1.0</td><td class="align_center">2.608275822</td><td class="align_center">2.6082748841</td><td class="align_center">2.6082748883</td><td class="align_center">2.6082748735</td><td class="align_center">2.6082749587</td><td class="align_center">2.6082749176</td><td class="align_center">2.6082749587</td></tr><tr class="table-tr"><td colspan="8"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Numerical results for problem in Thin Film Flow (<a href="https://static.hindawi.com/articles/jam/volume-2018/4029371/figures/#EEq34" target="_blank">58</a>) taking <svg height="9.49473pt" id="M206" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 37.258 9.49473" width="37.258pt" 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="M499 88L487 113C459 86 422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z" id="g113-105"></path></g><g transform="matrix(.013,0,0,-0.013,10.34,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,21.603,0)"><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" id="g113-49"></path></g><g transform="matrix(.013,0,0,-0.013,27.843,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z" id="g113-47"></path></g><g transform="matrix(.013,0,0,-0.013,30.807,0)"><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" id="g113-50"></path></g></svg> and <span class="nowrap"><svg height="9.49473pt" id="M207" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 27.8499 9.49473" width="27.8499pt" 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="M480 416C480 431 465 448 438 448C388 448 312 383 252 330C217 299 188 273 155 237H153L257 680C262 700 263 712 253 712C240 712 183 684 97 674L92 648L126 647C166 646 172 645 163 606L23 -6L29 -12C51 -5 77 2 107 8C115 62 130 128 142 180C153 193 179 220 204 241C231 170 259 106 288 54C317 0 336 -12 358 -12C381 -12 423 2 477 80L460 100C434 74 408 54 398 54C385 54 374 65 351 107C326 154 282 241 263 299C296 332 351 377 403 377C424 377 436 372 445 368C449 366 456 368 462 375C472 386 480 402 480 416Z" id="g113-108"></path></g><g transform="matrix(.013,0,0,-0.013,10.171,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,21.434,0)"><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" id="g113-51"></path></g></svg>.</span></div>

Journal of Applied Mathematics

tab2

Table 2

Table 2: Exponentially Fitted and Trigonometrically Fitted Explicit Modified Runge-Kutta Type Methods for Solving <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.526799pt" id="M1" height="19.6225pt" version="1.1" viewBox="-0.0657574 -15.0957 135.584 19.6225" width="135.584pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-122" d="M556 393C556 426 537 448 514 448C496 448 478 435 471 420C468 414 462 401 466 394C473 382 480 368 480 346C480 268 392 143 338 67H336C329 163 319 253 309 330S286 448 254 448C215 448 160 383 127 337L143 311C167 344 200 373 208 373S222 365 229 320C246 214 264 66 268 -20C219 -83 131 -171 17 -239L25 -261L137 -235C248 -110 273 -79 335 8C384 77 481 215 520 287C544 332 556 363 556 393Z"/></g><g transform="matrix(.012,0,0,-0.012,9.938,-7.578)"><path id="g50-31" d="M310 541L304 571C290 586 211 619 185 610L80 76L131 52L310 541Z"/></g><g transform="matrix(.012,0,0,-0.012,13.95,-7.578)"><use xlink:href="#g50-31"/></g><g transform="matrix(.012,0,0,-0.012,17.96,-7.578)"><use xlink:href="#g50-31"/></g><g transform="matrix(.017,0,0,-0.017,23.002,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,28.94,0)"><path id="g113-121" d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"/></g><g transform="matrix(.017,0,0,-0.017,38.533,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g><g transform="matrix(.017,0,0,-0.017,49.264,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.017,0,0,-0.017,64.131,0)"><path id="g113-103" d="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"/></g><g transform="matrix(.017,0,0,-0.017,75.156,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.017,0,0,-0.017,81.093,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.017,0,0,-0.017,90.687,0)"><path id="g113-45" 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"/></g><g transform="matrix(.017,0,0,-0.017,97.475,0)"><use xlink:href="#g113-122"/></g><g transform="matrix(.017,0,0,-0.017,107.414,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.017,0,0,-0.017,114.202,0)"><use xlink:href="#g113-122"/></g><g transform="matrix(.012,0,0,-0.012,124.141,-7.578)"><use xlink:href="#g50-31"/></g><g transform="matrix(.017,0,0,-0.017,129.151,0)"><use xlink:href="#g113-42"/></g></svg>