In this article, a numerical solution is proposed for singularly perturbed delay parabolic reaction-diffusion problem with mixed-type boundary conditions. The problem is discretized by the implicit Euler method on uniform mesh in time and extended cubic B-spline collocation method on a Shishkin mesh in space. The parameter-uniform convergence of the method is given, and it is shown to be <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="M1" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 5.44961 6.1673" width="5.44961pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-247" d="M387 375C387 402 357 448 257 448C172 448 82 404 82 326C82 289 108 255 156 241V239C85 223 23 181 23 116C23 39 89 -12 182 -12C265 -12 336 31 378 91L361 114C320 73 269 47 216 47C157 47 115 82 115 137C115 191 160 219 218 219C243 219 262 218 272 217L304 259L302 266C295 265 281 264 255 264C195 264 163 294 163 335C163 377 200 416 249 416C293 416 321 389 329 342C331 332 335 329 341 329C355 329 387 352 387 375Z"/></g></svg>-</span>uniformly convergent of <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="37.182pt" style="vertical-align:-2.2681pt" id="M2" height="14.0618pt" version="1.1" viewBox="-0.0498162 -11.7937 37.182 14.0618"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-80" d="M699 369C699 549 575 666 407 666C186 666 23 488 23 278C23 101 145 -16 312 -16C535 -16 699 153 699 369ZM600 373C600 210 500 19 321 19C186 19 120 129 120 272C120 450 232 631 399 631C541 631 600 522 600 373Z"/></g><g transform="matrix(.013,0,0,-0.013,9.387,0)"><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,-0.013,13.885,0)"><path id="g113-133" d="M600 0V24L339 665L311 657L43 24V0H600ZM497 50H107L303 539L497 50Z"/></g><g transform="matrix(.013,0,0,-0.013,22.212,0)"><path id="g113-117" d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"/></g><g transform="matrix(.013,0,0,-0.013,29.551,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g></svg><span class="ibiop"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="52.557pt" style="vertical-align:-2.2681pt" height="14.0618pt" version="1.1" viewBox="40.0371838 -11.7937 52.557 14.0618"><g transform="matrix(.013,0,0,-0.013,40.087,0)"><path id="g113-79" d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z"/></g><g transform="matrix(.0091,0,0,-0.0091,50.998,-5.741)"><path id="g54-33" d="M556 236V289H56V236H556Z"/></g><g transform="matrix(.0091,0,0,-0.0091,56.558,-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,-0.013,61.525,0)"><path id="g121-106" d="M238 0V26C174 32 166 38 166 104V712C132 700 70 683 18 677V653C81 647 87 645 87 577V104C87 38 78 32 15 26V0H238Z"/></g><g transform="matrix(.013,0,0,-0.013,64.814,0)"><path id="g121-108" d="M524 0V26C466 32 460 36 460 104V297C460 393 411 449 331 449C302 449 276 437 248 419C223 402 201 387 181 372V451C137 432 90 420 42 411V388C96 378 102 374 102 310V104C102 38 97 33 29 26V0H246V26C187 32 181 36 181 104V339C211 365 250 390 290 390C357 390 381 345 381 276V109C381 40 374 32 315 26V0H524Z"/></g><g transform="matrix(.0091,0,0,-0.0091,71.964,-5.944)"><use xlink:href="#g50-51"/></g><g transform="matrix(.013,0,0,-0.013,76.91,0)"><use xlink:href="#g113-79"/></g><g transform="matrix(.013,0,0,-0.013,87.82,0)"><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>,</span></span> where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M3" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 12.9135 8.8423" width="12.9135pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-133"/></g><g transform="matrix(.013,0,0,-0.013,8.327,0)"><use xlink:href="#g113-117"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M4" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 11.0475 8.8423" width="11.0475pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-79"/></g></svg> denote the step size in time and number of mesh intervals in space, respectively. The proposed method gives accurate results by choosing suitable value of the free parameter <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M5" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 7.30254 9.49473" width="7.30254pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-233" d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"/></g></svg>.</span> Some numerical results are carried out to support the theory.

Fasika Wondimu Gelu

Gemechis File Duressa

A Uniformly Convergent Collocation Method for Singularly Perturbed Delay Parabolic Reaction-Diffusion Problem

In this paper, we present the notion of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42947pt" id="M2" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 35.1006 12.7178" width="35.1006pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-230"/></g><g transform="matrix(.013,0,0,-0.013,9.381,0)"><use xlink:href="#g117-33"/></g><g transform="matrix(.013,0,0,-0.013,19.917,0)"><use xlink:href="#g113-243"/></g><g transform="matrix(.013,0,0,-0.013,27.263,0)"><use xlink:href="#g117-33"/></g></svg>expansive mapping in complete rectangular metric spaces and study various fixed point theorems for such mappings. The presented theorems extend, generalize, and improve many existing results in the literature.

Mohamed Rossafi

Abdelkarim Kari

El Miloudi Marhrani

Mohamed Aamri

Fixed Point Theorems for Generalized <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 35.1006 12.7178" width="35.1006pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-230" d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z" /></g><g transform="matrix(.013,0,0,-0.013,9.381,0)"><path id="g117-33" d="M535 230V280H52V230H535Z" /></g><g transform="matrix(.013,0,0,-0.013,19.917,0)"><path id="g113-243" d="M542 242C542 369 469 429 379 468C370 472 372 496 376 513L416 681L400 691L344 648L219 51C167 69 113 128 113 217C113 324 166 387 266 428L252 459C124 418 23 336 23 202C23 95 90 13 205 -8L182 -132C165 -222 191 -254 209 -261L215 -258L270 -7C415 -4 542 98 542 242ZM469 232C469 126 403 33 279 39L355 409C399 394 469 331 469 232Z" /></g><g transform="matrix(.013,0,0,-0.013,27.263,0)"><path id="g117-33" d="M535 230V280H52V230H535Z" /></g></svg>Expansive Mapping in Rectangular Metric Spaces

In this article, a numerical method based on the shifted Chebyshev functions for the numerical approximation of the coupled nonlinear variable-order fractional sine-Gordon equations is shown. The variable-order fractional derivative is considered in the sense of Caputo-Prabhakar. To solve the problem, first, we obtain the operational matrix of the Caputo-Prabhakar fractional derivative of shifted Chebyshev polynomials. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional sine-Gordon equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate the accuracy and efficiency of the proposed method.

MohammadHossein Derakhshan

A Numerical Scheme Based on the Chebyshev Functions to Find Approximate Solutions of the Coupled Nonlinear Sine-Gordon Equations with Fractional Variable Orders

In this work, we develop an a posteriori error analysis of a conforming mixed finite element method for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium on isotropic meshes in <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04979992pt" id="M1" height="12.308pt" version="1.1" viewBox="-0.0498162 -12.2582 15.4968 12.308" width="15.4968pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g197-29" d="M712 0L497 285C582 312 628 378 628 466C628 609 518 662 390 662H70V0H236V271H310L505 0H712ZM491 604C551 577 584 532 584 464C584 401 550 345 491 321V604ZM447 320C428 317 411 315 392 315H236V618H397C412 618 432 613 447 610V320ZM626 44H528L363 271C392 271 422 270 451 275L626 44ZM192 44H114V618H192V44Z"/></g><g transform="matrix(.0091,0,0,-0.0091,9.711,-5.741)"><path id="g50-101" d="M538 684C542 703 539 710 529 710C512 710 455 682 363 674L361 644H396C444 644 448 642 438 594L405 438C373 450 347 451 336 451C288 451 200 416 147 374C69 312 24 207 24 113C24 29 60 -12 94 -12C126 -12 156 4 196 30C231 53 296 100 346 166H348L328 81C311 10 325 -12 348 -12C380 -12 450 24 512 98L495 125C466 97 431 70 417 70C408 70 408 80 411 98L538 684ZM390 373L361 240C336 196 210 56 145 56C132 56 113 73 113 131C113 216 156 338 220 379C243 394 266 402 303 402C331 402 375 388 390 373Z"/></g></svg>,</span> <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="17.711pt" style="vertical-align:-2.26807pt" id="M2" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 17.711 11.5564"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-101" d="M530 686C535 705 530 712 521 712C504 712 448 684 359 674L358 648H393C437 648 439 646 429 593L400 435C372 447 345 448 332 448C286 448 194 414 144 373C68 311 23 203 23 111C23 26 57 -12 91 -12C120 -12 147 3 188 29C227 54 290 102 341 170H343L322 71C308 6 320 -12 341 -12C373 -12 442 27 501 96L485 120C455 91 422 67 408 67C401 67 401 76 404 91C440 294 479 473 530 686ZM387 375L355 241C326 187 200 53 142 53C126 53 109 73 109 130C109 217 154 337 218 381C240 396 265 404 297 404S372 390 387 375Z"/></g><g transform="matrix(.013,0,0,-0.013,10.847,0)"><path id="g117-173" d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="13.715pt" style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="21.2931838 -9.28833 13.715 11.5564"><g transform="matrix(.013,0,0,-0.013,21.343,0)"><path id="g113-124" d="M293 -169V-141C218 -131 209 -90 209 -44C209 19 222 85 222 152C222 207 198 255 139 269V273C199 286 222 334 222 388C222 454 209 523 209 588C209 632 218 671 293 681V709C234 709 148 695 148 577C147 513 155 438 155 372C155 337 152 291 64 285V256C152 250 155 204 155 169C156 105 148 31 148 -41C148 -157 234 -169 293 -169Z"/></g><g transform="matrix(.013,0,0,-0.013,25.854,0)"><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><g transform="matrix(.013,0,0,-0.013,32.094,0)"><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></svg><span class="ipunc"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="10.995pt" style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="37.1871838 -9.28833 10.995 11.5564"><g transform="matrix(.013,0,0,-0.013,37.237,0)"><path id="g113-52" d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z"/></g><g transform="matrix(.013,0,0,-0.013,43.477,0)"><path id="g113-126" d="M283 255V284C195 290 192 337 192 375C192 436 200 508 200 580C200 696 116 709 54 709V681C127 671 139 634 139 588C138 524 125 452 125 387C125 333 149 286 209 272V268C148 253 125 206 125 151C126 87 139 16 139 -47C139 -92 127 -131 54 -141V-169C115 -169 200 -152 200 -41C200 32 192 104 192 164C192 202 195 249 283 255Z"/></g></svg>.</span></span> The approach utilizes a Lagrange multiplier method to impose weakly the interface conditions [Ilona Ambartsumyan et al., Numerische Mathematik, 140 (2): 513-553, 2018]. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. The proof of reliability makes use of suitable auxiliary problems, diverse continuous inf-sup conditions satisfied by the bilinear forms involved, Helmholtz decomposition, and local approximation properties of the Clément interpolant. On the other hand, inverse inequalities and the localization technique based on simplexe-bubble and face-bubble functions are the main tools for proving the efficiency of the estimator. Up to minor modifications, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.

Hou&#x00E9;danou Koffi Wilfrid

A Posteriori Error Analysis for a Lagrange Multiplier Method for a Stokes/Biot Fluid-Poroelastic Structure Interaction Model

The primary purpose of this research is to demonstrate an efficient replacement of double transform called the double Laplace&#x2013;Sumudu transform (DLST) and prove some related theorems of the new double transform. Also, we will discuss the fundamental properties of the double Laplace&#x2013;Sumudu transform of some basic functions. Then, by utilizing those outcomes, we will apply it to the partial differential equations to show its simplicity, efficiency, and high accuracy.

Shams A. Ahmed

Applications of New Double Integral Transform (Laplace&#x2013;Sumudu Transform) in Mathematical Physics

We give different types of new characterizations for the boundedness and essential norms of generalized weighted composition operators between Zygmund-type spaces. Consequently, we obtain new characterizations for the compactness of such operators.

Mostafa Hassanlou

Amir H. Sanatpour

New Characterizations for the Essential Norms of Generalized Weighted Composition Operators between Zygmund-Type Spaces

In this article, a numerical technique based on the Chebyshev cardinal functions (CCFs) and the Lagrange multiplier technique for the numerical approximation of the variable-order fractional integrodifferential equations are shown. The variable-order fractional derivative is considered in the sense of regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives. To solve the problem, first, we obtain the operational matrix of the regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives of CCFs. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional integrodifferential equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate its accuracy and efficiency.

New Numerical Algorithm to Solve Variable-Order Fractional Integrodifferential Equations in the Sense of Hilfer-Prabhakar Derivative

In this paper, I consider the Cauchy problem for the incompressible Navier-Stokes equations in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.053209pt" id="M1" height="14.0096pt" version="1.1" viewBox="-0.0498162 -9.95639 15.9261 14.0096" width="15.9261pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g197-29" d="M712 0L497 285C582 312 628 378 628 466C628 609 518 662 390 662H70V0H236V271H310L505 0H712ZM491 604C551 577 584 532 584 464C584 401 550 345 491 321V604ZM447 320C428 317 411 315 392 315H236V618H397C412 618 432 613 447 610V320ZM626 44H528L363 271C392 271 422 270 451 275L626 44ZM192 44H114V618H192V44Z"/></g><g transform="matrix(.0091,0,0,-0.0091,9.711,-5.741)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g><g transform="matrix(.0091,0,0,-0.0091,9.711,3.784)"><path id="g54-36" d="M556 236V289H337V504H275V289H56V236H275V-4H337V236H556Z"/></g></svg> for <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="17.789pt" style="vertical-align:-1.11981pt" id="M2" height="9.46863pt" version="1.1" viewBox="-0.0498162 -8.34882 17.789 9.46863"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g><g transform="matrix(.013,0,0,-0.013,10.158,0)"><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="6.416pt" style="vertical-align:-1.11981pt" height="9.46863pt" version="1.1" viewBox="21.3711838 -8.34882 6.416 9.46863"><g transform="matrix(.013,0,0,-0.013,21.421,0)"><path id="g113-52" d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z"/></g></svg></span> with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial data. This paper is a continuation of my work in my previous papers, where the initial data are considered in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04980946pt" id="M3" height="10.0062pt" version="1.1" viewBox="-0.0498162 -9.95639 13.3907 10.0062" width="13.3907pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g197-31" d="M622 618V662H20V618H238V0H404V618H622ZM360 44H282V618H360V44Z"/></g><g transform="matrix(.0091,0,0,-0.0091,8.091,-5.741)"><use xlink:href="#g50-111"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04980946pt" id="M4" height="10.0062pt" version="1.1" viewBox="-0.0498162 -9.95639 15.0218 10.0062" width="15.0218pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g197-29"/></g><g transform="matrix(.0091,0,0,-0.0091,9.711,-5.741)"><use xlink:href="#g50-111"/></g></svg> respectively. In this paper, because of the nonempty boundary in our domain of interest, the details in obtaining the desired result are significantly different and more challenging than the work of my previous papers. This challenges arise due to the possible noncommutativity nature of the Leray projector with the derivatives in the direction of normal to the boundary of the domain of interest. Therefore, we only consider one derivative of the velocity field in that direction.

Santosh Pathak

Maximum Norm Estimates of the Solution of the Navier-Stokes Equations in the Halfspace with Bounded Initial Data

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<p><em>Abstract and Applied Analysis</em> publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.</p>


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<p>Chief Editor, Dr Wong, is an associate professor at Nanyang Technological University, Singapore. Her research interests include differential equations, difference equations, integral equations, and numerical mathematics.</p>


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Abstract and Applied Analysis - Table of contents 2021

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Abstract and Applied Analysis publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.

Abstract and Applied Analysis