Efficient Two-Derivative Runge-Kutta-Nyström Methods for Solving General Second-Order Ordinary Differential Equations <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 131.557 19.6225" width="131.557pt"><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(.017,0,0,-0.017,18.976,0)"><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,-0.017,24.914,0)"><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,-0.017,34.506,0)"><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,-0.017,45.238,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.017,0,0,-0.017,60.105,0)"><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,-0.017,71.13,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.017,0,0,-0.017,77.067,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.017,0,0,-0.017,86.659,0)"><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,-0.017,93.448,0)"><use xlink:href="#g113-122"/></g><g transform="matrix(.017,0,0,-0.017,103.386,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.017,0,0,-0.017,110.175,0)"><use xlink:href="#g113-122"/></g><g transform="matrix(.012,0,0,-0.012,120.115,-7.578)"><use xlink:href="#g50-31"/></g><g transform="matrix(.017,0,0,-0.017,125.124,0)"><use xlink:href="#g113-42"/></g></svg>

<div>The time curves for all methods for Problem <a href="https://static.hindawi.com/articles/ddns/volume-2018/2393015/figures/#problem2">2</a>.</div>

Discrete Dynamics in Nature and Society

fig4

Figure 4

Figure 4: Efficient Two-Derivative Runge-Kutta-Nyström Methods for Solving General Second-Order Ordinary Differential Equations <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 131.557 19.6225" width="131.557pt"><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(.017,0,0,-0.017,18.976,0)"><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,-0.017,24.914,0)"><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,-0.017,34.506,0)"><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,-0.017,45.238,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.017,0,0,-0.017,60.105,0)"><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,-0.017,71.13,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.017,0,0,-0.017,77.067,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.017,0,0,-0.017,86.659,0)"><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,-0.017,93.448,0)"><use xlink:href="#g113-122"/></g><g transform="matrix(.017,0,0,-0.017,103.386,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.017,0,0,-0.017,110.175,0)"><use xlink:href="#g113-122"/></g><g transform="matrix(.012,0,0,-0.012,120.115,-7.578)"><use xlink:href="#g50-31"/></g><g transform="matrix(.017,0,0,-0.017,125.124,0)"><use xlink:href="#g113-42"/></g></svg>