We compute explicitly the oscillation constant for Euler type half-linear second-order differential equation having multi-different periodic coefficients.

Adil Misir

Banu Mermerkaya

Critical Oscillation Constant for Euler Type Half-Linear Differential Equation Having Multi-Different Periodic Coefficients

For an approximately controllable semilinear system, the problem of computing control for a given target state is converted into an equivalent problem of solving operator equation which is ill-posed. We exhibit a sequence of regularized controls which steers the semilinear control system from an arbitrary initial state <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M1" height="11.7978pt" version="1.1" viewBox="-0.0498162 -11.5914 12.3403 11.7978" width="12.3403pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><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(.0091,0,0,-0.0091,7.267,-5.741)"><path id="g50-49" d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z"/></g></svg> to an <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M2" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 5.63229 6.1673" width="5.63229pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-227" d="M401 397C401 420 368 448 302 448C245 448 169 416 122 377C62 327 23 254 23 169C23 45 83 -12 181 -12C252 -12 323 29 374 85L358 107C305 62 257 43 210 43C147 43 110 98 110 189V214L313 208L321 256L115 250C132 342 190 405 253 405C291 405 323 389 346 360C356 348 364 348 377 357C392 367 401 384 401 397Z"/></g></svg> neighbourhood of the target state <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.29111pt" id="M3" height="9.25202pt" version="1.1" viewBox="-0.0498162 -5.96091 12.3847 9.25202" width="12.3847pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.0091,0,0,-0.0091,7.202,3.132)"><path id="g50-241" d="M487 458L455 460C436 434 425 433 397 433C331 433 279 437 225 437C103 437 52 374 24 338L49 312C100 363 152 379 229 374C210 308 184 213 163 117C158 94 156 74 156 58C156 6 180 -12 211 -12C261 -12 313 27 348 68L332 99C307 76 285 64 267 64C252 64 236 86 250 148L301 373C320 373 344 369 364 369C399 369 435 370 450 372C458 386 470 411 487 458Z"/></g></svg> at time <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.6370001pt" id="M4" height="8.98582pt" version="1.1" viewBox="-0.0498162 -8.34882 27.5889 8.98582" width="27.5889pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-241" d="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z"/></g><g transform="matrix(.013,0,0,-0.013,9.914,0)"><path id="g117-92" d="M512 230V281L75 514V456L453 256V254L75 55V-3L512 230Z"/></g><g transform="matrix(.013,0,0,-0.013,21.177,0)"><path id="g113-49" 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"/></g></svg> under the assumption that the nonlinear function <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42947pt" id="M5" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 8.47692 12.7178" width="8.47692pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><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></svg> is Lipschitz continuous. The convergence of the sequences of regularized controls and the corresponding mild solutions are shown under some assumptions on the system operators. It is also proved that the target state corresponding to the regularized control is close to the actual state to be attained.

Ravinder Katta

N. Sukavanam

Approximate Controllability of Semilinear Control System Using Tikhonov Regularization

In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.

S&#xfc;leyman Cengizci

An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations

In this paper, we study the existence and uniqueness of solutions for the following boundary value problem of nonlinear fractional differential equation: <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.127501pt" id="M1" height="16.0477pt" version="1.1" viewBox="-0.0498162 -11.9202 119.202 16.0477" width="119.202pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><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(.0091,0,0,-0.0091,4.498,-5.741)"><path id="g50-68" d="M690 629C653 643 577 667 483 667C225 667 24 528 24 292C24 92 174 -17 367 -17C448 -17 534 1 570 10C592 34 642 131 658 174C648 180 639 186 629 192C562 89 499 21 382 21C223 21 130 138 130 298C130 467 255 627 462 627C582 627 643 589 650 462L686 466C689 517 688 580 690 629Z"/></g><g transform="matrix(.013,0,0,-0.013,11.18,0)"><path id="g113-69" d="M743 375C743 570 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The main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle.

Cheng-Min Su

Jian-Ping Sun

Ya-Hong Zhao

Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation

Fernando Bernal-V&#xed;lchis

Nakao Hayashi

Pavel I. Naumkin

Asymptotics for the Ostrovsky-Hunter Equation in the Critical Case

In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nystr&#xf6;m method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHTRKNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

F. F. Ngwane

S. N. Jator

A Trigonometrically Fitted Block Method for Solving Oscillatory Second-Order Initial Value Problems and Hamiltonian Systems

A family of boundary value methods (BVMs) with continuous coefficients is derived and used to obtain methods which are applied via the block unification approach. The methods obtained from these continuous BVMs are weighted the same and are used to simultaneously generate approximations to the exact solution of systems of second-order boundary value problems (BVPs) on the entire interval of integration. The convergence of the methods is analyzed. Numerical experiments were performed to show efficiency and accuracy advantages.

T. A. Biala

A Family of Boundary Value Methods for Systems of Second-Order Boundary Value Problems

A six-compartment mathematical model is formulated to investigate the role of media campaigns in Ebola transmission dynamics. The model includes tweets or messages sent by individuals in different compartments. The media campaigns reproduction number is computed and used to discuss the stability of the disease states. The presence of a backward bifurcation as well as a forward bifurcation is shown together with the existence and local stability of the endemic equilibrium. Results show that messages sent through media have a more significant beneficial effect on the reduction of Ebola cases if they are more effective and spaced out.

Sylvie Diane Djiomba Njankou

Farai Nyabadza

Modelling the Potential Role of Media Campaigns in Ebola Transmission Dynamics

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


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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