It is known that power series expansion of certain functions such as <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M2" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 39.3372 11.5564" width="39.3372pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g121-113"/></g><g transform="matrix(.013,0,0,-0.013,4.875,0)"><use xlink:href="#g121-99"/></g><g transform="matrix(.013,0,0,-0.013,10.466,0)"><use xlink:href="#g121-97"/></g><g transform="matrix(.013,0,0,-0.013,15.886,0)"><use xlink:href="#g121-102"/></g><g transform="matrix(.013,0,0,-0.013,22.914,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,27.412,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.013,0,0,-0.013,34.679,0)"><use xlink:href="#g113-42"/></g></svg> diverges beyond a finite radius of convergence. We present here an iterative power series expansion (IPS) to obtain a power series representation of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M3" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 39.3372 11.5564" width="39.3372pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g121-113"/></g><g transform="matrix(.013,0,0,-0.013,4.875,0)"><use xlink:href="#g121-99"/></g><g transform="matrix(.013,0,0,-0.013,10.466,0)"><use xlink:href="#g121-97"/></g><g transform="matrix(.013,0,0,-0.013,15.886,0)"><use xlink:href="#g121-102"/></g><g transform="matrix(.013,0,0,-0.013,22.914,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,27.412,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.013,0,0,-0.013,34.679,0)"><use xlink:href="#g113-42"/></g></svg> that is convergent for all <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M4" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 7.39387 6.1673" width="7.39387pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-121"/></g></svg>.</span> The convergent series is a sum of the Taylor series of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M5" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 39.3372 11.5564" width="39.3372pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g121-113"/></g><g transform="matrix(.013,0,0,-0.013,4.875,0)"><use xlink:href="#g121-99"/></g><g transform="matrix(.013,0,0,-0.013,10.466,0)"><use xlink:href="#g121-97"/></g><g transform="matrix(.013,0,0,-0.013,15.886,0)"><use xlink:href="#g121-102"/></g><g transform="matrix(.013,0,0,-0.013,22.914,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,27.412,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.013,0,0,-0.013,34.679,0)"><use xlink:href="#g113-42"/></g></svg> and a complementary series that cancels the divergence of the Taylor series for <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="18.53pt" style="vertical-align:-2.15067pt" id="M6" height="11.439pt" version="1.1" viewBox="-0.0498162 -9.28833 18.53 11.439"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.013,0,0,-0.013,10.899,0)"><path id="g117-94" d="M531 285V335L57 550V494L473 310L57 127V71L531 285ZM531 -40V10H57V-40H531Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="13.04pt" style="vertical-align:-2.15067pt" height="11.439pt" version="1.1" viewBox="22.1121838 -9.28833 13.04 11.439"><g transform="matrix(.013,0,0,-0.013,22.162,0)"><path id="g113-238" d="M574 449L545 460C526 432 516 430 487 430C404 430 311 435 226 435C104 435 56 379 25 341L43 318C81 354 121 372 181 372C161 246 87 53 23 3L30 -12C48 -12 88 -4 113 11C157 75 207 248 232 371L386 367L326 109C321 86 318 66 318 50C318 4 339 -12 366 -12C410 -12 461 21 505 69L492 96C467 79 434 60 418 60C406 60 400 78 411 147C422 217 439 300 457 366C487 366 524 367 536 370C547 385 558 408 574 449Z"/></g><g transform="matrix(.013,0,0,-0.013,29.846,0)"><path id="g113-48" d="M368 703H309L44 -163H104L368 703Z"/></g></svg><span class="isep"/><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="6.462pt" style="vertical-align:-2.15067pt" height="11.439pt" version="1.1" viewBox="35.1571838 -9.28833 6.462 11.439"><g transform="matrix(.013,0,0,-0.013,35.207,0)"><path id="g113-51" 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"/></g></svg></span>. The method is general and can be applied to other functions known to have finite radius of convergence, such as <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="11.596pt" style="vertical-align:-2.2681pt" id="M7" height="13.8595pt" version="1.1" viewBox="-0.0498162 -11.5914 11.596 13.8595"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><use xlink:href="#g113-48"/></g></svg><span class="isep"/><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="21.275pt" style="vertical-align:-2.2681pt" height="13.8595pt" version="1.1" viewBox="11.5511838 -11.5914 21.275 13.8595"><g transform="matrix(.013,0,0,-0.013,11.601,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,16.099,0)"><use xlink:href="#g113-50"/></g><g transform="matrix(.013,0,0,-0.013,25.245,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g></svg><span class="ibiop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="16.962pt" style="vertical-align:-2.2681pt" height="13.8595pt" version="1.1" viewBox="35.7311838 -11.5914 16.962 13.8595"><g transform="matrix(.013,0,0,-0.013,35.781,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.0091,0,0,-0.0091,43.048,-5.741)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g><g transform="matrix(.013,0,0,-0.013,47.995,0)"><use xlink:href="#g113-42"/></g></svg></span>. A straightforward application of this method is to solve analytically nonlinear differential equations, which we also illustrate here. The method provides also a robust and very efficient numerical algorithm for solving nonlinear differential equations numerically. A detailed comparison with the fourth-order Runge-Kutta method and extensive analysis of the behavior of the error and CPU time are performed.

U. Al Khawaja

Qasem M. Al-Mdallal

Convergent Power Series of <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 39.3372 11.5564" width="39.3372pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g121-113" d="M319 325C317 349 306 409 297 431C277 440 250 449 209 449C117 449 57 389 57 319C57 243 122 209 182 182C232 159 261 135 261 91C261 48 227 21 190 21C130 21 85 79 68 145L41 140C41 104 51 36 58 22C75 7 121 -12 172 -12C252 -12 337 35 337 126C337 195 286 231 210 262C166 281 126 304 126 348C126 388 152 417 191 417C240 417 274 378 294 318L319 325Z" /></g><g transform="matrix(.013,0,0,-0.013,4.875,0)"><path id="g121-99" d="M380 106C343 72 306 56 265 56C195 56 116 112 115 248C235 252 361 262 377 265C396 269 400 277 400 297C400 374 333 449 250 449H249C198 449 144 421 103 376S37 269 37 201C37 88 109 -12 232 -12C263 -12 332 6 395 84L380 106ZM225 412C281 412 315 364 314 312C314 297 308 292 290 292C232 290 176 289 120 289C135 370 180 412 225 412Z" /></g><g transform="matrix(.013,0,0,-0.013,10.466,0)"><path id="g121-97" d="M390 111C344 68 312 56 269 56C212 56 118 102 118 241C118 346 175 401 241 401C277 401 312 388 342 360C350 352 355 349 361 349C372 349 394 371 394 392C394 403 391 411 378 422C362 436 329 449 288 449H287C250 449 190 432 138 392C71 341 37 274 37 197C37 90 112 -12 238 -12C297 -12 363 32 407 90L390 111Z" /></g><g transform="matrix(.013,0,0,-0.013,15.886,0)"><path id="g121-102" d="M513 0V26C455 31 447 37 447 103V278C447 398 390 449 309 449C255 449 202 412 166 376V712C127 700 67 684 19 677V653C85 647 87 643 87 580V103C87 37 79 31 19 26L18 0H231V26C171 32 166 39 166 103V341C194 373 232 392 270 392C337 392 368 351 368 269V103C368 38 360 32 302 26V0H513Z" /></g><g transform="matrix(.013,0,0,-0.013,22.914,0)"><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(.013,0,0,-0.013,27.412,0)"><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(.013,0,0,-0.013,34.679,0)"><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></svg> and Solutions to Nonlinear Differential Equations

The aim of this paper is to study the early stage of HBV infection and impact delay in the infection process on the adaptive immune response, which includes cytotoxic T-lymphocytes and antibodies. In this stage, the growth of the healthy hepatocyte cells is logistic while the growth of the infected ones is linear. To investigate the role of the treatment at this stage, we also consider two types of treatment: interferon-<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M1" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 7.51131 6.1673" width="7.51131pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g></svg> (IFN) and nucleoside analogues (NAs). To find the best strategy to use this treatment, an optimal control approach is developed to find the possibility of having a functional cure to HBV.

Karam Allali

Adil Meskaf

Abdessamad Tridane

Mathematical Modeling of the Adaptive Immune Responses in the Early Stage of the HBV Infection

We propose a generalized virus dynamics model with distributed delays and both modes of transmission, one by virus-to-cell infection and the other by cell-to-cell transfer. In the proposed model, the distributed delays describe (i) the time needed for infected cells to produce new virions and (ii) the time necessary for the newly produced virions to become mature and infectious. In addition, the infection transmission process is modeled by general incidence functions for both modes. Furthermore, the qualitative analysis of the model is rigorously established and many known viral infection models with discrete and distributed delays are extended and improved.

Khalid Hattaf

Noura Yousfi

Qualitative Analysis of a Generalized Virus Dynamics Model with Both Modes of Transmission and Distributed Delays

We discuss the existence of global solutions to the magnetohydrodynamics (MHD) equations, where the effects of finite Larmor radius corrections are taken into account. Unlike the usual MHD, the pressure is a tensor and it depends on not only the density but also the magnetic field. We show the existence of global solutions by the energy methods. Our techniques of proof are based on the existence of local solution by semigroups theory and a priori estimate.

Fariha Elsrrawi

Harumi Hattori

Existence of Global Solutions for Nonlinear Magnetohydrodynamics with Finite Larmor Radius Corrections

In this article, the linearization problem of fifth-order ordinary differential equation is presented by using the generalized Sundman transformation. The necessary and sufficient conditions which allow the nonlinear fifth-order ordinary differential equation to be transformed to the simplest linear equation are found. There is only one case in the part of sufficient conditions which is surprisingly less than the number of cases in the same part for order 2, 3, and 4. Moreover, the derivations of the explicit forms for the linearizing transformation are exhibited. Examples for the main results are included.

Supaporn Suksern

Kwanpaka Naboonmee

Linearization of Fifth-Order Ordinary Differential Equations by Generalized Sundman Transformations

We define and study a tritrophic bioeconomic model of Lotka-Volterra with a prey, middle predator, and top predator populations. These fish populations are exploited by two fishermen. We study the existence and the stability of the equilibrium points by using eigenvalues analysis and Routh-Hurwitz criterion. We determine the equilibrium point that maximizes the profit of each fisherman by solving the Nash equilibrium problem. Finally, following some numerical simulations, we observe that if the price varies, then the profit behavior of each fisherman will be changed; also, we conclude that the price change mechanism improves the fishing effort of the fishermen.

Meriem Bentounsi

Imane Agmour

Naceur Achtaich

Youssef El Foutayeni

The Impact of Price on the Profits of Fishermen Exploiting Tritrophic Prey-Predator Fish Populations

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International Journal of Differential Equations | Hindawi

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International Journal of Differential Equations publishes research on differential equations, and related integral equations, from all scientists who use differential equations as tools within their own discipline.

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International Journal of Differential Equations

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International Journal of Differential Equations