Initial Value Problems with Generalized Fractional Derivatives and Their Solutions via Generalized Laplace Decomposition Method

<div>Plots of approximate solutions for Example <a href="https://static.hindawi.com/articles/amp/volume-2022/3586802/figures/#ex1">1</a>, when <svg height="11.1258pt" id="M152" style="vertical-align:-2.77698pt" version="1.1" viewbox="-0.0498162 -8.34882 40.6115 11.1258" width="40.6115pt" 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="M362 285L435 424L363 423C326 380 261 326 196 326C179 326 163 330 146 338L136 312C153 301 173 296 194 296C237 296 283 305 319 342L23 -209H100L241 53C356 266 508 411 595 411C620 411 631 395 631 371C631 279 519 97 461 65C440 69 423 69 404 69C348 69 314 44 314 21C314 4 335 -11 386 -11C417 -11 448 0 479 17C530 13 556 -8 603 -49L625 -29C598 7 560 30 520 46C613 110 694 248 694 342C694 417 646 441 602 441C535 441 460 395 365 283L362 285Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.681,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.944,0)"><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,-0.013,31.184,0)"><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,-0.013,34.148,0)"><path d="M249 635C141 635 70 555 70 471C70 401 114 353 179 316C143 294 106 267 90 252C68 231 45 202 45 157C45 50 130 -12 237 -12C322 -12 435 52 435 169C435 256 372 304 303 343C349 374 375 398 383 407C401 429 411 458 411 487C411 569 344 635 249 635ZM238 603C285 603 337 567 337 482C337 422 310 385 276 358C205 393 145 426 145 500C145 552 179 603 238 603ZM248 20C183 20 125 70 125 163C125 218 158 268 206 300C284 261 355 217 355 143C355 66 308 20 248 20Z"></path></g></svg> (dashed), <svg height="11.1258pt" id="M153" style="vertical-align:-2.77698pt" version="1.1" viewbox="-0.0498162 -8.34882 31.373 11.1258" width="31.373pt" 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="M362 285L435 424L363 423C326 380 261 326 196 326C179 326 163 330 146 338L136 312C153 301 173 296 194 296C237 296 283 305 319 342L23 -209H100L241 53C356 266 508 411 595 411C620 411 631 395 631 371C631 279 519 97 461 65C440 69 423 69 404 69C348 69 314 44 314 21C314 4 335 -11 386 -11C417 -11 448 0 479 17C530 13 556 -8 603 -49L625 -29C598 7 560 30 520 46C613 110 694 248 694 342C694 417 646 441 602 441C535 441 460 395 365 283L362 285Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.681,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.944,0)"><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></svg> (red), and <svg height="11.1258pt" id="M154" style="vertical-align:-2.77698pt" version="1.1" viewbox="-0.0498162 -8.34882 40.6115 11.1258" width="40.6115pt" 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="M362 285L435 424L363 423C326 380 261 326 196 326C179 326 163 330 146 338L136 312C153 301 173 296 194 296C237 296 283 305 319 342L23 -209H100L241 53C356 266 508 411 595 411C620 411 631 395 631 371C631 279 519 97 461 65C440 69 423 69 404 69C348 69 314 44 314 21C314 4 335 -11 386 -11C417 -11 448 0 479 17C530 13 556 -8 603 -49L625 -29C598 7 560 30 520 46C613 110 694 248 694 342C694 417 646 441 602 441C535 441 460 395 365 283L362 285Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.681,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.944,0)"><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,-0.013,31.184,0)"><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,-0.013,34.148,0)"><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"></path></g></svg> (thick).</div>

Advances in Mathematical Physics

fig2

Figure 2

Figure 2: Initial Value Problems with Generalized Fractional Derivatives and Their Solutions via Generalized Laplace Decomposition Method 