In this work, we investigate various types of solutions for the generalised resonant dispersive nonlinear Schr&#xf6;dinger equation (GRD-NLSE) with power law nonlinearity. Based on simple mathematical techniques, the complicated form of the GRD-NLSE is reduced to an ordinary differential equation (ODE) which has a variety of solutions. The analytic solution of the resulting ODE gives rise to bright soliton, singular soliton, peaked soliton, compacton solutions, solitary pattern solutions, rational solution, Weierstrass elliptic periodic type solutions, and some other types of solutions. Constraint conditions for the existence of solitons and other solutions are given.

K. S. Al-Ghafri

Different Physical Structures of Solutions for a Generalized Resonant Dispersive Nonlinear Schr&#xf6;dinger Equation with Power Law Nonlinearity

In this paper we present a mathematical model for the transmission dynamics of Contagious Bovine Pleuropneumonia (CBPP) by considering antibiotic treatment and vaccination. The model is comprised of susceptible, vaccinated, exposed, infectious, persistently infected, and recovered compartments. We analyse the model by deriving a formula for the control reproduction number <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.29108pt" id="M1" height="12.3054pt" version="1.1" viewBox="-0.0498162 -9.01432 16.6699 12.3054" width="16.6699pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g198-19" d="M852 669C820 657 792 643 768 628C705 667 627 687 547 687C310 687 154 548 154 412C154 332 212 282 296 282C434 282 515 426 523 557H503C474 402 410 312 298 312C241 312 195 352 195 408C195 535 341 657 546 657C613 657 683 644 738 609C640 538 574 431 486 271C374 71 314 16 189 16C118 16 69 60 69 95H71C78 84 93 83 101 83C129 83 144 106 144 131C144 160 121 179 93 179C60 179 34 151 34 108C34 42 100 -15 207 -15C364 -15 491 73 603 304C614 297 639 293 650 293C681 293 683 254 683 245C683 181 599 150 599 70C599 21 645 -12 692 -12C747 -12 804 30 868 103L851 119C789 52 741 18 706 18C685 18 674 29 674 52C674 92 750 157 750 226C750 257 727 287 696 310V312C812 315 876 394 876 473C876 525 849 577 793 612C812 627 832 639 859 650L852 669ZM814 486C814 417 765 345 686 337C669 348 649 354 626 354C673 451 708 533 766 588C799 561 814 525 814 486Z"/></g><g transform="matrix(.0091,0,0,-0.0091,12.272,3.132)"><path id="g50-100" d="M387 400C387 425 348 451 303 451C247 451 176 414 132 376C69 322 24 228 24 148C24 43 74 -12 147 -12C211 -12 301 33 363 103L346 128C319 99 249 51 193 51C148 51 112 84 112 165C112 230 130 287 154 330C170 359 199 400 243 400C277 400 304 383 326 354C333 345 343 343 354 348C378 360 387 382 387 400Z"/></g></svg> and prove that, for <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="27.787pt" style="vertical-align:-3.29108pt" id="M2" height="12.3054pt" version="1.1" viewBox="-0.0498162 -9.01432 27.787 12.3054"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g198-19"/></g><g transform="matrix(.0091,0,0,-0.0091,12.272,3.132)"><use xlink:href="#g50-100"/></g><g transform="matrix(.013,0,0,-0.013,20.156,0)"><path id="g117-91" d="M512 -3V55L134 254V256L512 456V514L75 281V230L512 -3Z"/></g></svg><span class="irelop"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="6.438pt" style="vertical-align:-3.29108pt" height="12.3054pt" version="1.1" viewBox="31.369183800000002 -9.01432 6.438 12.3054"><g transform="matrix(.013,0,0,-0.013,31.419,0)"><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></svg>,</span></span> the disease free equilibrium is globally asymptotically stable; thus CBPP dies out, whereas for <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="27.787pt" style="vertical-align:-3.29108pt" id="M3" height="12.3054pt" version="1.1" viewBox="-0.0498162 -9.01432 27.787 12.3054"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g198-19"/></g><g transform="matrix(.0091,0,0,-0.0091,12.272,3.132)"><use xlink:href="#g50-100"/></g><g transform="matrix(.013,0,0,-0.013,20.156,0)"><path id="g117-92" d="M512 230V281L75 514V456L453 256V254L75 55V-3L512 230Z"/></g></svg><span class="irelop"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="6.438pt" style="vertical-align:-3.29108pt" height="12.3054pt" version="1.1" viewBox="31.369183800000002 -9.01432 6.438 12.3054"><g transform="matrix(.013,0,0,-0.013,31.419,0)"><use xlink:href="#g113-50"/></g></svg>,</span></span> the unique endemic equilibrium is globally asymptotically stable and hence the disease persists. Thus, <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="27.787pt" style="vertical-align:-3.29108pt" id="M4" height="12.3054pt" version="1.1" viewBox="-0.0498162 -9.01432 27.787 12.3054"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g198-19"/></g><g transform="matrix(.0091,0,0,-0.0091,12.272,3.132)"><use xlink:href="#g50-100"/></g><g transform="matrix(.013,0,0,-0.013,20.156,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="6.438pt" style="vertical-align:-3.29108pt" height="12.3054pt" version="1.1" viewBox="31.369183800000002 -9.01432 6.438 12.3054"><g transform="matrix(.013,0,0,-0.013,31.419,0)"><use xlink:href="#g113-50"/></g></svg></span> acts as a sharp threshold between the disease dying out or causing an epidemic. As a result, the threshold of antibiotic treatment is <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="24.727pt" style="vertical-align:-3.9436pt" id="M5" height="14.0461pt" version="1.1" viewBox="-0.0498162 -10.1025 24.727 14.0461"><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><g transform="matrix(.0091,0,0,-0.0091,7.385,-5.741)"><path id="g50-43" d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 262C225 236 198 227 145 220C133 201 125 177 126 158C141 148 167 143 189 144C222 186 243 205 290 236C288 179 282 152 262 102C272 82 289 63 306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.059,3.784)"><path id="g50-117" d="M329 433H203L239 587L230 596L147 534L123 433H57L30 395L34 388H115L61 129C37 16 59 -12 85 -12C147 -12 222 58 260 98L241 125C212 95 160 62 144 62C132 62 127 71 138 126L192 386L305 394L329 433Z"/></g><g transform="matrix(.013,0,0,-0.013,17.096,0)"><use xlink:href="#g117-34"/></g></svg><span class="irelop"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="34.447pt" style="vertical-align:-3.9436pt" height="14.0461pt" version="1.1" viewBox="28.309183800000003 -10.1025 34.447 14.0461"><g transform="matrix(.013,0,0,-0.013,28.359,0)"><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><g transform="matrix(.013,0,0,-0.013,34.599,0)"><path id="g113-47" d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"/></g><g transform="matrix(.013,0,0,-0.013,37.563,0)"><use xlink:href="#g113-50"/></g><g transform="matrix(.013,0,0,-0.013,43.803,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.013,0,0,-0.013,50.043,0)"><path id="g113-53" d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z"/></g><g transform="matrix(.013,0,0,-0.013,56.283,0)"><path id="g113-58" d="M244 635C114 635 38 519 38 422C38 317 111 240 217 240C236 240 255 244 277 256L345 292C311 140 203 39 59 15L64 -15C89 -15 150 -5 204 17C339 72 440 202 440 386C440 521 368 635 244 635ZM228 602C326 602 352 479 352 390C352 370 351 347 348 324C327 308 293 296 258 296C174 296 124 369 124 458C124 517 152 602 228 602Z"/></g></svg>.</span></span> Thus, without using vaccination, more than <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4673595pt" id="M6" height="8.85534pt" version="1.1" viewBox="-0.0498162 -8.38798 37.9932 8.85534" width="37.9932pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-57" 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"/></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path id="g113-54" d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"/></g><g transform="matrix(.013,0,0,-0.013,12.48,0)"><use xlink:href="#g113-47"/></g><g transform="matrix(.013,0,0,-0.013,15.444,0)"><use xlink:href="#g113-53"/></g><g transform="matrix(.013,0,0,-0.013,21.684,0)"><use xlink:href="#g113-54"/></g><g transform="matrix(.013,0,0,-0.013,28.029,0)"><path id="g121-35" d="M594 629L561 652L159 -9L190 -32L594 629ZM194 635C92 635 49 534 49 453C49 374 92 272 193 272C295 272 338 374 338 454S295 635 194 635ZM193 605C257 605 265 504 265 453C265 405 257 302 194 302C133 302 122 403 122 454S133 605 193 605ZM567 351C465 351 421 250 421 169C421 90 465 -12 566 -12S710 90 710 170S667 351 567 351ZM566 322C628 322 637 221 637 169C637 121 628 18 567 18S494 119 494 170C494 223 506 322 566 322Z"/></g></svg> of the infectious cattle should receive antibiotic treatment or the period of infection should be reduced to less than 8.15 days to control the disease. Similarly, the threshold of vaccination is <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="24.352pt" style="vertical-align:-3.4295pt" id="M7" height="13.532pt" version="1.1" viewBox="-0.0498162 -10.1025 24.352 13.532"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-239" d="M516 277C516 378 460 448 376 448C335 448 288 434 240 401C158 345 127 277 107 159C53 -161 38 -221 23 -247L28 -261C43 -260 95 -250 114 -226C122 -216 129 -199 146 -75C152 -29 157 0 164 31C182 9 206 -12 237 -12C267 -12 325 8 376 46C451 102 516 179 516 277ZM432 259C432 134 366 40 259 40C236 40 204 48 171 76L195 203C219 327 253 400 326 400C398 400 432 336 432 259Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.01,-5.741)"><use xlink:href="#g50-43"/></g><g transform="matrix(.013,0,0,-0.013,16.721,0)"><use xlink:href="#g117-34"/></g></svg><span class="irelop"></span><span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="34.443pt" style="vertical-align:-3.4295pt" height="13.532pt" version="1.1" viewBox="27.934183800000003 -10.1025 34.443 13.532"><g transform="matrix(.013,0,0,-0.013,27.984,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.013,0,0,-0.013,34.224,0)"><use xlink:href="#g113-47"/></g><g transform="matrix(.013,0,0,-0.013,37.188,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.013,0,0,-0.013,43.428,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.013,0,0,-0.013,49.668,0)"><use xlink:href="#g113-57"/></g><g transform="matrix(.013,0,0,-0.013,55.908,0)"><use xlink:href="#g113-53"/></g></svg>.</span></span> Therefore, we have to vaccinate at least <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4673595pt" id="M8" height="8.85534pt" version="1.1" viewBox="-0.0498162 -8.38798 22.4913 8.85534" width="22.4913pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-57"/></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.013,0,0,-0.013,12.527,0)"><use xlink:href="#g121-35"/></g></svg> of susceptible cattle in less than 49.5 days, to control the disease. Using both vaccination and antibiotic treatment, the threshold value of vaccination depends on the rate of antibiotic treatment, <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="13.828pt" style="vertical-align:-3.29111pt" id="M9" height="9.25202pt" version="1.1" viewBox="-0.0498162 -5.96091 13.828 9.25202"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-223"/></g><g transform="matrix(.0091,0,0,-0.0091,7.059,3.132)"><use xlink:href="#g50-117"/></g><g transform="matrix(.013,0,0,-0.013,10.727,0)"><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> and is denoted by <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.34099pt" id="M10" height="11.3019pt" version="1.1" viewBox="-0.0498162 -5.96091 14.6848 11.3019" width="14.6848pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-239"/></g><g transform="matrix(.0091,0,0,-0.0091,6.162,3.132)"><path id="g50-223" d="M558 106L533 129C501 84 476 67 464 67C446 67 435 116 414 239C457 297 508 367 550 440L539 451L484 438C458 390 429 338 404 300H402C375 407 351 451 285 451C168 451 24 312 24 153C24 56 66 -12 132 -12C206 -12 284 64 345 156H347C367 24 386 -12 422 -12C457 -12 503 7 558 106ZM338 206C272 99 215 56 173 56C140 56 116 96 116 170C116 303 194 405 255 405C304 405 324 299 338 206Z"/></g><g transform="matrix(.0065,0,0,-0.0065,11.231,5.233)"><path id="g176-117" d="M336 436H211L246 592L238 601L150 538L128 436H60L31 395L37 389H121L66 133C41 17 64 -12 95 -12C158 -12 231 58 270 100L250 132C217 97 166 64 151 64C139 64 134 71 146 128L199 383L312 395L336 436Z"/></g></svg>.</span> Hence, if <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4673595pt" id="M11" height="8.85534pt" version="1.1" viewBox="-0.0498162 -8.38798 22.4913 8.85534" width="22.4913pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-54"/></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.013,0,0,-0.013,12.527,0)"><use xlink:href="#g121-35"/></g></svg> of infectious cattle receive antibiotic treatment, then at least <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4673595pt" id="M12" height="8.85534pt" version="1.1" viewBox="-0.0498162 -8.38798 22.4913 8.85534" width="22.4913pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-54"/></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.013,0,0,-0.013,12.527,0)"><use xlink:href="#g121-35"/></g></svg> of susceptible cattle should get vaccination in less than 73.8 days in order to control the disease.

Achamyelesh Amare Aligaz

Justin Manango W. Munganga

Mathematical Modelling of the Transmission Dynamics of Contagious Bovine Pleuropneumonia with Vaccination and Antibiotic Treatment

In this paper, a hybrid iterative algorithm is proposed for finding a common element of the set of common fixed points of finite family of nonexpansive mappings and the set of common solutions of the variational inequality for an inverse strongly monotone mapping on the real Hilbert space. We establish the strong convergence of the proposed method for approximating a common element of the above defined sets under some suitable conditions. The results presented in this paper extend and improve some well-known corresponding results in the earlier and recent literature.

Shamshad Husain

Nisha Singh

An Iterative Method for Finding Common Solution of the Fixed Point Problem of a Finite Family of Nonexpansive Mappings and a Finite Family of Variational Inequality Problems in Hilbert Space

In this paper, optimal homotopy asymptotic method (OHAM) and its implementation on subinterval, called multistage optimal homotopy asymptotic method (MOHAM), are presented for solving linear and nonlinear systems of Volterra integral equations of the second kind. To illustrate these approaches two examples are presented. The results confirm the efficiency and ability of these methods for such equations. The results will be compared to find out which method is more accurate. Advantages of applying MOHAM are also illustrated.

Jafar Biazar

Roya Montazeri

Optimal Homotopy Asymptotic and Multistage Optimal Homotopy Asymptotic Methods for Solving System of Volterra Integral Equations of the Second Kind

In this work, the kinetically reduced local Navier-Stokes equations are applied to the simulation of two- and three-dimensional unsteady viscous incompressible flow problems. The reduced differential transform method is used to find the new approximate analytical solutions of these flow problems. The new technique has been tested by using four selected multidimensional unsteady flow problems: two- and three-dimensional Taylor decaying vortices flow, Kovasznay flow, and three-dimensional Beltrami flow. The convergence analysis was discussed for this approach. The numerical results obtained by this approach are compared with other results that are available in previous works. Our results show that this method is efficient to provide new approximate analytic solutions. Moreover, we found that it has highly precise solutions with good convergence, less time consuming, being easily implemented for high Reynolds numbers, and low Mach numbers.

Abdul-Sattar J. Al-Saif

Assma J. Harfash

A New Approximate Analytical Solutions for Two- and Three-Dimensional Unsteady Viscous Incompressible Flows by Using the Kinetically Reduced Local Navier-Stokes Equations

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<p><em>Journal of Applied Mathematics</em> publishes original research papers and review articles in all areas of applied, computational, and industrial mathematics.</p>


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<p>Chief Editor, Professor Theodore E. Simos, is based at Ulyanovsk State Technical University, Russia. His main research interest is the numerical analysis of differential equations.</p>


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Journal of Applied Mathematics publishes original research papers and review articles in all areas of applied, computational, and industrial mathematics.

Journal of Applied Mathematics