The determinant of a matrix is very powerful tool that helps in establishing properties of matrices. Indisputably, its importance in various engineering and applied science problems has made it a mathematical area of increasing significance. From developed and existing methods of finding determinant of a matrix, basketweave method/Sarrus’ rule has been shown to be the simplest, easiest, very fast, accurate, and straightforward method for the computation of the determinant of 3 × 3 matrices. However, its gross limitation is that this method/rule does not work for matrices larger than 3 × 3 and this fact is well established in literatures. Therefore, the state-of-the-art methods for finding the determinants of 4 × 4 matrix and larger matrices are predominantly founded on non-basketweave method/non-Sarrus’ rule. In this work, extension of the simple, easy, accurate, and straightforward approach to the determinant of larger matrices is presented. The paper presents the developments of new method with different schemes based on the basketweave method/Sarrus’ rule for the computation of the determinant of 4 × 4. The potency of the new method is revealed in generalization of the basketweave method/non-Sarrus’ rule for the computation of the determinant of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3499303pt" id="M3" height="7.35473pt" version="1.1" viewBox="-0.0498162 -7.0048 26.6711 7.35473" width="26.6711pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.013,0,0,-0.013,9.432,0)"><use xlink:href="#g117-42"/></g><g transform="matrix(.013,0,0,-0.013,19.968,0)"><use xlink:href="#g113-111"/></g></svg> (<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 27.8368 8.98582" width="27.8368pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.013,0,0,-0.013,10.158,0)"><use xlink:href="#g117-92"/></g><g transform="matrix(.013,0,0,-0.013,21.421,0)"><use xlink:href="#g113-52"/></g></svg>) matrices. The new method is very efficient, very consistence for handy calculations, highly accurate, and fastest compared to other existing methods. 

M. G. Sobamowo

On the Extension of Sarrus&#x2019; Rule to <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 67.7963 11.5564" width="67.7963pt"><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,-0.013,9.432,0)"><path id="g117-42" d="M528 54L331 254L528 455L492 493L294 291L96 493L60 455L257 254L60 54L96 16L294 217L492 16L528 54Z" /></g><g transform="matrix(.013,0,0,-0.013,19.968,0)"><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,-0.013,30.853,0)"><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,-0.013,35.351,0)"><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,-0.013,45.509,0)"><path id="g117-92" d="M512 230V281L75 514V456L453 256V254L75 55V-3L512 230Z" /></g><g transform="matrix(.013,0,0,-0.013,56.772,0)"><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,-0.013,63.012,0)"><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> Matrices: Development of New Method for the Computation of the Determinant of <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-0.3499298pt" height="8.69875pt" version="1.1" viewBox="-0.0498162 -8.34882 26.097 8.69875" width="26.097pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><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,-0.013,9.146,0)"><path id="g117-42" d="M528 54L331 254L528 455L492 493L294 291L96 493L60 455L257 254L60 54L96 16L294 217L492 16L528 54Z" /></g><g transform="matrix(.013,0,0,-0.013,19.682,0)"><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></svg> Matrix

We prove some common fixed point results for a pair of mappings which satisfy generalized contractive conditions with rational expressions having point-dependent control functions as coefficients in complex valued <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M2" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.21949 9.49473" width="6.21949pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-99"/></g></svg>-metric spaces. The results of this paper generalize and extend the several known results in complex valued <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M3" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.21949 9.49473" width="6.21949pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-99"/></g></svg>-metric spaces. Finally, examples are provided to verify the effectiveness and to usability of our main results. 

Anil Kumar Dubey

Manjula Tripathi

Ravi Prakash Dubey

Various Fixed Point Theorems in Complex Valued <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-0.2063999pt" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.21949 9.49473" width="6.21949pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z" /></g></svg>-Metric Spaces

Reflection of longitudinal displacement waves in a generalized thermoelastic half space under the action of uniform magnetic field has been investigated. The magnetic field is applied in such a direction that the problem can be considered as a two-dimensional one. The discussion is based on the three theories of generalized thermoelasticity: Lord-Shulman (L-S), Green-Lindsay (G-L), and Green-Naghdi (G-N) with energy dissipation. We compute the possible wave velocities for different models. Amplitude ratios have been presented. The effects of magnetic field on various subjects of interest are discussed and shown graphically.

Narottam Maity

S. P. Barik

P. K. Chaudhuri

Reflection of Plane Waves in Generalized Thermoelastic Half Space under the Action of Uniform Magnetic Field

This study concerns the development of a straightforward numerical technique associated with Classical Newton&#x2019;s Method for providing a more accurate approximate solution of scalar nonlinear equations. The proposed procedure is based on some practical geometric rules and requires the knowledge of the local slope of the curve representing the considered nonlinear function. Therefore, this new technique uses, only as input data, the first-order derivative of the nonlinear equation in question. The relevance of this numerical procedure is tested, evaluated, and discussed through some examples.

Gr&#xe9;gory Antoni

An Efficient and Straightforward Numerical Technique Coupled to Classical Newton&#x2019;s Method for Enhancing the Accuracy of Approximate Solutions Associated with Scalar Nonlinear Equations

Regular perturbation technique is applied to analyze the fluid flow and heat transfer in a pipe containing third-grade fluid with temperature-dependent viscosities and heat generation under slip and no slip conditions. The obtained approximate solutions were used to investigate the effects of slip on the heat transfer characteristics of the laminar flow in a pipe under Reynolds&#x2019;s and Vogel&#x2019;s temperature-dependent viscosities. Also, the effects of parameters such as variable viscosity, non-Newtonian parameter, viscous dissipation, and pressure gradient at various values were established. The results of this work were compared with the numerical results found in literature and good agreements were established. The results can be used to advance the analysis and study of the behavior of third-grade fluid flow and steady state heat transfer processes such as those found in coal slurries, polymer solutions, textiles, ceramics, catalytic reactors, and oil recovery applications.

B. Y. Ogunmola

A. T. Akinshilo

Perturbation Solutions for Hagen-Poiseuille Flow and Heat Transfer of Third-Grade Fluid with Temperature-Dependent Viscosities and Internal Heat Generation

This paper deals with the study of the stability and the bifurcation analysis of a Leslie-Gower predator-prey model with Michaelis-Menten type predator harvesting. It is shown that the proposed model exhibits the bistability for certain parametric conditions. Dulac&#x2019;s criterion has been adopted to obtain the sufficient conditions for the global stability of the model. Moreover, the model exhibits different kinds of bifurcations (e.g., the saddle-node bifurcation, the subcritical and supercritical Hopf bifurcations, Bogdanov-Takens bifurcation, and the homoclinic bifurcation) whenever the values of parameters of the model vary. The analytical findings and numerical simulations reveal far richer and complex dynamics in comparison to the models with no harvesting and with constant-yield predator harvesting.

Manoj Kumar Singh

B. S. Bhadauria

Brajesh Kumar Singh

Qualitative Analysis of a Leslie-Gower Predator-Prey System with Nonlinear Harvesting in Predator

This work focuses on the identification of optimal model parameters related to Abrasive Waterjet Milling (AWJM) process. The evenly movement as well as variations of the jet feed speed was taken into account and studied in terms of 3D time dependent AWJM model. This gives us the opportunity to predict the shape of the milled trench surfaces. The required trench profile could be obtained with high precision in lack of knowledge about the model parameters and based only on the experimental measurements. We use the adjoint approach to identify the AWJM model parameters. The complexity of inverse problem paired with significant amount of unknowns makes it reasonable to use automatic differentiation software to obtain the adjoint statement. The interest in investigating this problem is caused by needs of industrial milling applications to predict the behavior of the process. This study proposes the possibility of identifying the AWJM model parameters with sufficiently high accuracy and predicting the shapes formation relying on self-generated data or on experimental measurements for both evenly jets movement and arbitrary changes of feed speed. We provide the results acceptable in the production and estimate the suitable parameters taking into account different types of model and measurement errors.

Didier Auroux

Vladimir Groza

Sensitivity Studies and Parameters Identification for Noisy 3D Moving AWJM Model

This paper deals with a new numerical iterative method for finding the approximate solutions associated with both scalar and vector nonlinear equations. The iterative method proposed here is an extended version of the numerical procedure originally developed in previous works. The present study proposes to show that this new root-finding algorithm combined with a stationary-type iterative method (e.g., Gauss-Seidel or Jacobi) is able to provide a longer accurate solution than classical Newton-Raphson method. A numerical analysis of the developed iterative method is addressed and discussed on some specific equations and systems.

A New Accurate and Efficient Iterative Numerical Method for Solving the Scalar and Vector Nonlinear Equations: Approach Based on Geometric Considerations

Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M1" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 9.97754 8.68572" width="9.97754pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-153" d="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z"/></g></svg> be a bounded domain in a real Euclidean space. We consider the equation <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.616401pt" id="M2" height="17.4036pt" version="1.1" viewBox="-0.0498162 -11.7872 388.827 17.4036" width="388.827pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-222" d="M471 379C471 433 467 511 427 578C379 658 305 712 208 712C154 712 98 699 70 684C70 668 95 634 111 634C141 651 180 665 234 665C306 665 350 627 372 575C391 530 393 480 393 442V415C371 436 344 448 313 448C280 448 227 437 176 411C100 372 23 279 23 148C23 67 64 -12 180 -12C293 -12 360 46 407 129C450 205 471 310 471 379ZM382 335C382 142 289 50 238 35C226 31 213 28 200 28C139 28 110 91 110 164C110 310 191 382 223 395C245 404 259 409 280 409C305 409 332 404 359 388C379 376 382 362 382 335Z"/></g><g transform="matrix(.013,0,0,-0.013,6.555,0)"><path id="g113-118" d="M515 96L502 119C471 88 431 62 423 62C416 62 411 70 417 101C440 223 469 341 497 448H486L412 422L380 277C330 188 210 57 152 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and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M4" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 30.0856 11.5564" width="30.0856pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-76"/></g><g transform="matrix(.013,0,0,-0.013,9.817,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,14.315,0)"><use xlink:href="#g113-46"/></g><g transform="matrix(.013,0,0,-0.013,17.279,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.013,0,0,-0.013,22.422,0)"><use xlink:href="#g113-46"/></g><g transform="matrix(.013,0,0,-0.013,25.386,0)"><use xlink:href="#g113-42"/></g></svg> are matrix-valued functions and <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 20.0251 11.5564" width="20.0251pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-71"/></g><g transform="matrix(.013,0,0,-0.013,7.895,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,12.393,0)"><use xlink:href="#g113-46"/></g><g transform="matrix(.013,0,0,-0.013,15.357,0)"><use xlink:href="#g113-42"/></g></svg> is a nonlinear mapping. Conditions for the exponential stability of the steady state are established. Our approach is based on a norm estimate for operator commutators.

Michael Gil&#x2019;

On Stability of Vector Nonlinear Integrodifferential Equations

A numerical method is proposed to study the laminar boundary layer about a flat plate in a uniform stream of fluid. The presented method is based on the quartic B-spline approximations with minimizing the error <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.1815pt" id="M1" height="11.8174pt" version="1.1" viewBox="-0.0498162 -8.6359 13.3712 11.8174" width="13.3712pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-77" d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z"/></g><g transform="matrix(.0091,0,0,-0.0091,8.294,3.132)"><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></svg>-norm. Theoretical considerations are discussed. The computed results are compared with some numerical results to show the efficiency of the proposed approach.

Hossein Aminikhah

Somayyeh Kazemi

Numerical Solution of the Blasius Viscous Flow Problem by Quartic B-Spline Method

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International Journal of Engineering Mathematics - Table of contents 2016

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International Journal of Engineering Mathematics

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International Journal of Engineering Mathematics