One-Step Family of Three Optimized Second-Derivative Hybrid Block Methods for Solving First-Order Stiff Problems

<table class="table-group" id="tab1"><tr><td><table class="table"><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr class="thead"><td class="align_left"><span style="width: 10.8257ptpx;"><svg height="6.25863pt" id="M36" style="vertical-align:-0.1933298pt" version="1.1" viewbox="-0.0498162 -6.0653 10.8257 6.25863" width="10.8257pt" 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="M801 97L785 128C766 109 732 86 722 86C717 86 712 92 717 114L759 300C790 439 749 454 720 454C700 454 681 447 653 434C628 422 565 377 487 276H485L490 301C520 439 483 454 451 454C427 454 404 445 375 429C353 417 294 378 208 268H206L223 335C243 412 241 454 204 454C161 454 89 411 21 347L36 314C51 331 92 357 103 357C112 357 110 348 106 331C80 211 56 108 24 -2L33 -12C64 -4 113 6 152 9C165 74 180 143 198 196C231 251 330 364 363 364C379 364 379 333 367 275C349 188 325 94 298 -4L307 -12C339 -4 385 6 430 9C440 73 452 131 468 194C510 259 602 364 637 364C647 364 651 342 641 301L590 88C571 8 590 -12 623 -12C657 -12 732 23 801 97Z"></path></g></svg></span></td><td class="align_center"><b>Off-step points</b></td></tr><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr><td class="align_left">1</td><td class="align_center"><span style="width: 5.60619ptpx;"><svg height="6.1673pt" id="M37" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 5.60619 6.1673" width="5.60619pt" 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="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z"></path></g></svg></span></td></tr><tr><td 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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,-0.013,19.777,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,30.313,0)"><path d="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z"></path></g></svg></span></td></tr><tr><td class="align_left">3</td><td class="align_center"><span style="width: 49.8096ptpx;"><svg height="26.1319pt" id="M39" style="vertical-align:-9.275301pt" version="1.1" viewbox="-0.0498162 -16.8566 49.8096 26.1319" width="49.8096pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g 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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,-0.013,19.286,0)"><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,-0.013,24.429,0)"><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,-0.013,33.574,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,44.111,0)"><path d="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z"></path></g></svg></span></td></tr><tr class="table-tr"><td colspan="2"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Distribution of the first three off-step points.</div>

Journal of Applied Mathematics

tab1

Table 1

Table 1: One-Step Family of Three Optimized Second-Derivative Hybrid Block Methods for Solving First-Order Stiff Problems 