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In the study of boundary layer regions, it is in practice to dimensionless the governing equations and grouping variables together into dimensionless numbers in order to curtail the total number of variables. In non-similar flows, in contrast to similar flows the basic quantities change in the flow direction. Flows across porous media are important because of its applications in biological and environmental systems such as tissues, bones soils, etc. In Darcy-Forchheimer-Brinkman framework the basic quantities change in the stream wise direction. Therefore, non-similar boundary layer model is developed for forced convection in the steady flow of Casson nano-fluid over horizontal plate installed in non-Darcy porous media. The governing equations describing Darcy-Forchheimer-Brinkman model are transformed into dimensionless non-similar model by making use of appropriate non-similar transformations. The dimensionless partial differential system is solved through local non-similarity via bvp4c. The impact of emerging non-dimensional parameters, namely, Casson parameter (β ), Schmidt number (Sc), porosity (ε), Prandtl number (Pr), 1st order porous resistance parameter (R1), thermophoresis parameter ( Nt), 2nd order porous resistance parameter (R2), Brownian motion parameter (Nb) and dimensionless viscous dissipation on the dimensionless velocity, temperature and concentration distributions are examined in detail. Furthermore, the impacts of these parameters on dimensionless friction coefficient and heat transfer rate are also explored. Finally, comparison of non-similar solutions with local similar solutions has been made with published data.
Umer Farooq; Muzamil Hussain; Muhammad Ahsan Ijaz; Waseem Asghar Khan; Fozia Bashir Farooq. Impact of non-similar modeling on Darcy-Forchheimer-Brinkman model for forced convection of Casson nano-fluid in non-Darcy porous media. International Communications in Heat and Mass Transfer 2021, 125, 105312 .
AMA StyleUmer Farooq, Muzamil Hussain, Muhammad Ahsan Ijaz, Waseem Asghar Khan, Fozia Bashir Farooq. Impact of non-similar modeling on Darcy-Forchheimer-Brinkman model for forced convection of Casson nano-fluid in non-Darcy porous media. International Communications in Heat and Mass Transfer. 2021; 125 ():105312.
Chicago/Turabian StyleUmer Farooq; Muzamil Hussain; Muhammad Ahsan Ijaz; Waseem Asghar Khan; Fozia Bashir Farooq. 2021. "Impact of non-similar modeling on Darcy-Forchheimer-Brinkman model for forced convection of Casson nano-fluid in non-Darcy porous media." International Communications in Heat and Mass Transfer 125, no. : 105312.
The authors regret in the published paper referenced above and agree with the discussion by Pantokratoras (2019, “Discussion: “Computational Analysis for Mixed Convective Flows of Viscous Fluids With Nanoparticles” (Farooq, U., Lu, D. C., Ahmed, S., and Ramzan, M., 2019, ASME J. Therm. Sci. Eng. Appl., 11(2), p. 021013),” ASME J. Therm. Sci. Eng. Appl., 11(5), p. 055503). In this Closure, the non-similar mathematical model is developed to describe the mixed convective nanofluid flow over vertical sheet which is stretching at an exponential rate. In the published article referenced above, similarity transformations are utilized to convert the governing nonlinear partial differential equations (PDEs) into ordinary differential equations (ODEs). The important physical numbers such as magnetic field (M2), Brownian motion parameter (Nb), thermophoresis (Nt), Eckert number (Ec), ratio of mass transfer Grashof to heat transfer Grashof (N), buoyancy parameter (λ), and Reynolds number (Re) appearing in the dimensionless ODEs are still functions of coordinate “x”; therefore, the problem is non-similar. In this corrigendum, the non-similar model is developed by using ξ(x) as non-similarity variable and η(x, y) as pseudo-similarity variable. The dimensionless non-similar model is numerically simulated by employing local non-similarity via bvp4c. The graphical results show no change in behavior. The important thermal and mass transport quantities such as Nusselt number and Sherwood number have been computed for the non-similar model, and results are compared with the published article.
Jifeng Cui; Umer Farooq; Raheela Razzaq; Waseem Asghar Khan; Mogtaba Ahmed Yousif. Closure to “Computational Analysis for Mixed Convective Flows of Viscous Fluids With Nanoparticles” (Farooq, U., Lu, D. C., Ahmed, S., and Ramzan, M., 2019, ASME J. Therm. Sci. Eng. Appl., 11(2), p. 021013). Journal of Thermal Science and Engineering Applications 2021, 13, 1 -6.
AMA StyleJifeng Cui, Umer Farooq, Raheela Razzaq, Waseem Asghar Khan, Mogtaba Ahmed Yousif. Closure to “Computational Analysis for Mixed Convective Flows of Viscous Fluids With Nanoparticles” (Farooq, U., Lu, D. C., Ahmed, S., and Ramzan, M., 2019, ASME J. Therm. Sci. Eng. Appl., 11(2), p. 021013). Journal of Thermal Science and Engineering Applications. 2021; 13 (6):1-6.
Chicago/Turabian StyleJifeng Cui; Umer Farooq; Raheela Razzaq; Waseem Asghar Khan; Mogtaba Ahmed Yousif. 2021. "Closure to “Computational Analysis for Mixed Convective Flows of Viscous Fluids With Nanoparticles” (Farooq, U., Lu, D. C., Ahmed, S., and Ramzan, M., 2019, ASME J. Therm. Sci. Eng. Appl., 11(2), p. 021013)." Journal of Thermal Science and Engineering Applications 13, no. 6: 1-6.
The hydraulic network study is a real-life engineering problem extensively base on applied and computational approaches. Several numerical schemes are tested for defining of hydraulic pipe networks solution, such as, Hardy Cross Method, Newton Method and Modified Newton method are presented in this paper. Convergence analysis are also compared and deliberated by using a multi-loop hydraulic network. As a novelty, a Modified Newton method tabulated data, graphical results and comparisons are presented with the Hardy-Cross method and 2nd -order Newton–Raphson method. On other hand, we applied Ancient Chinese Algorithm and its modification for value of friction factor by using Colebrook-White relation. Numerical efficiency and comparison of Ancient Chinese Algorithm with Newton method and Bubbfil Algorithm I and Algorthm-II is presented and discussed quantitative approach to demonstrate the feasible result.
Waseem Asghar Khan. Numerical and simulation analysis comparison of hydraulic network problem base on higher-order efficiency approach. Alexandria Engineering Journal 2021, 60, 4889 -4903.
AMA StyleWaseem Asghar Khan. Numerical and simulation analysis comparison of hydraulic network problem base on higher-order efficiency approach. Alexandria Engineering Journal. 2021; 60 (5):4889-4903.
Chicago/Turabian StyleWaseem Asghar Khan. 2021. "Numerical and simulation analysis comparison of hydraulic network problem base on higher-order efficiency approach." Alexandria Engineering Journal 60, no. 5: 4889-4903.
The aspiration of this research is to explore the impact of non-similar modeling for mixed convection in magnetized second grade nano-fluid flow. The flow is initiated by the stretching of a sheet at an exponential rate in the upward vertical direction. The buoyancy effects in terms of temperature and concentration differences are inserted in the x-momentum equation. The aspects of heat and mass transfer are studied by means of dimensionless thermophoresis, Schmidt, and Brownian motion parameters. The governing coupled partial differential system (PDEs) is remodeled into coupled non-similar nonlinear PDEs by introducing non-similar transformations. The numerical analysis for the dimensionless non-similar partial differential system is performed by using local non-similarity method via bvp4c. Finally, the quantitative effects of emerging dimensionless quantities on the non-dimensional velocity, temperature and mass concentration in the boundary layer are conferred graphically and inferences are drawn that important quantities of interest are substantially affected by these parameters. It is concluded non-similar modeling in contrast to similar models are more general and more accurate in convection studies in the presence of buoyancy effects for second grade non-Newtonian fluids.
Ammarah Raees; Umer Farooq; Muzamil Hussain; Waseem Asghar Khan; Fozia Bashir Farooq. Non-similar mixed convection analysis for magnetic flow of second-grade nanofluid over a vertically stretching sheet. Communications in Theoretical Physics 2021, 73, 065801 .
AMA StyleAmmarah Raees, Umer Farooq, Muzamil Hussain, Waseem Asghar Khan, Fozia Bashir Farooq. Non-similar mixed convection analysis for magnetic flow of second-grade nanofluid over a vertically stretching sheet. Communications in Theoretical Physics. 2021; 73 (6):065801.
Chicago/Turabian StyleAmmarah Raees; Umer Farooq; Muzamil Hussain; Waseem Asghar Khan; Fozia Bashir Farooq. 2021. "Non-similar mixed convection analysis for magnetic flow of second-grade nanofluid over a vertically stretching sheet." Communications in Theoretical Physics 73, no. 6: 065801.
The correlation coefficient between two variables is an important aspect of statistics. The accuracy of assessments of correlation relies on information from a set of discourses. Data collected in statistical studies are often full of exceptions. Pythagorean fuzzy soft sets (PFSS) are a parametrized family of extended Pythagorean fuzzy sets (PFS). They comprise a generalization of intuitionistic fuzzy soft sets which may be used to accurately assess deficiencies and uncertainties in evaluations. PFSS can accommodate uncertainty more competently than intuitionistic fuzzy soft sets and are the most important strategy when dealing with fuzzy information in decision-making processes. Herein, the concept and characteristics of correlation coefficients and the weighted correlation coefficients in PFSS are discussed. We also introduce the Pythagorean fuzzy soft weighted average (PFSWA) and Pythagorean fuzzy soft weighted geometric (PFSWG) operators and discuss their desirable characteristics. A prioritization technique for order preference by similarity to the ideal solution (TOPSIS) under the PFSS environment based on correlation coefficients and weighted correlation coefficients will be introduced. Through the proposed methodology, a technique for decision-making is developed. Additionally, an application of the proposed TOPSIS technique is presented for green supplier selection in green supply chain management (GSCM). The practicality, efficacy, and flexibility of the proposed approach is proved through comparative analyses, drawing upon existing studies.
Rana Zulqarnain; Xiao Xin; Imran Siddique; Waseem Asghar Khan; Mogtaba Yousif. TOPSIS Method Based on Correlation Coefficient under Pythagorean Fuzzy Soft Environment and Its Application towards Green Supply Chain Management. Sustainability 2021, 13, 1642 .
AMA StyleRana Zulqarnain, Xiao Xin, Imran Siddique, Waseem Asghar Khan, Mogtaba Yousif. TOPSIS Method Based on Correlation Coefficient under Pythagorean Fuzzy Soft Environment and Its Application towards Green Supply Chain Management. Sustainability. 2021; 13 (4):1642.
Chicago/Turabian StyleRana Zulqarnain; Xiao Xin; Imran Siddique; Waseem Asghar Khan; Mogtaba Yousif. 2021. "TOPSIS Method Based on Correlation Coefficient under Pythagorean Fuzzy Soft Environment and Its Application towards Green Supply Chain Management." Sustainability 13, no. 4: 1642.
The correlation coefficient between the two parameters plays a significant part in statistics. Furthermore, the exactness of the assessment of correlation depends upon information from the set of discourses. The data collected for various statistical studies are full of ambiguities. The idea of interval-valued intuitionistic fuzzy soft sets is an extension of intuitionistic fuzzy soft sets that is used to express insufficient evaluation, uncertainty, and anxiety in decision-making. Intuitionistic fuzzy soft sets consider two different types of information, such as membership degree and nonmembership degree. In this paper, the concepts and properties of the correlation coefficient and the weighted correlation coefficient of interval-valued intuitionistic fuzzy soft sets are proposed. A prioritization technique for order preference by similarity to the ideal solution based on interval-valued intuitionistic fuzzy soft sets of correlation coefficients and the weighted correlation coefficient is introduced. We also proposed interval-valued intuitionistic fuzzy soft weighted average and interval-valued intuitionistic fuzzy soft weighted geometric operators and developed decision-making techniques based on the proposed operators. By using the developed techniques, a method for solving decision-making problems is proposed. To ensure the applicability of the proposed methods, an illustrative example is given. Finally, we present a comparison of some existing methods with our proposed techniques.
Rana Muhammad Zulqarnain; Xiao Long Xin; Muhammad Saqlain; Waseem Asghar Khan. TOPSIS Method Based on the Correlation Coefficient of Interval-Valued Intuitionistic Fuzzy Soft Sets and Aggregation Operators with Their Application in Decision-Making. Journal of Mathematics 2021, 2021, 1 -16.
AMA StyleRana Muhammad Zulqarnain, Xiao Long Xin, Muhammad Saqlain, Waseem Asghar Khan. TOPSIS Method Based on the Correlation Coefficient of Interval-Valued Intuitionistic Fuzzy Soft Sets and Aggregation Operators with Their Application in Decision-Making. Journal of Mathematics. 2021; 2021 ():1-16.
Chicago/Turabian StyleRana Muhammad Zulqarnain; Xiao Long Xin; Muhammad Saqlain; Waseem Asghar Khan. 2021. "TOPSIS Method Based on the Correlation Coefficient of Interval-Valued Intuitionistic Fuzzy Soft Sets and Aggregation Operators with Their Application in Decision-Making." Journal of Mathematics 2021, no. : 1-16.
In this paper we suggest a series of iterative schemes for solution of nonlinear equations. We have derived these schemes using Adomian Decomposition Technique developed by Adomian in 1980s [22]. The methods we suggest have good efficiency and important thing is that these methods are of practical importance. The fourth order convergence of the method is proved and few examples are presented to illustrate the methods.
Waseem Asghar Khan; Khalida Inayat Noor; Kaleemulah Bhatti; Faryal Aijaz Ansari. A new fourth order Newton-type method for solution of system of nonlinear equations. Applied Mathematics and Computation 2015, 270, 724 -730.
AMA StyleWaseem Asghar Khan, Khalida Inayat Noor, Kaleemulah Bhatti, Faryal Aijaz Ansari. A new fourth order Newton-type method for solution of system of nonlinear equations. Applied Mathematics and Computation. 2015; 270 ():724-730.
Chicago/Turabian StyleWaseem Asghar Khan; Khalida Inayat Noor; Kaleemulah Bhatti; Faryal Aijaz Ansari. 2015. "A new fourth order Newton-type method for solution of system of nonlinear equations." Applied Mathematics and Computation 270, no. : 724-730.
Muhammad Aslam Noor; Waseem Asghar Khan. Global Error Minimization method for nonlinear oscillators with highly nonlinearity. Applied Mathematics & Information Sciences 2013, 7, 1557 -1561.
AMA StyleMuhammad Aslam Noor, Waseem Asghar Khan. Global Error Minimization method for nonlinear oscillators with highly nonlinearity. Applied Mathematics & Information Sciences. 2013; 7 (4):1557-1561.
Chicago/Turabian StyleMuhammad Aslam Noor; Waseem Asghar Khan. 2013. "Global Error Minimization method for nonlinear oscillators with highly nonlinearity." Applied Mathematics & Information Sciences 7, no. 4: 1557-1561.
Muhammad Aslam Noor; Waseem Asghar Khan. New iterative methods for solving nonlinear equation by using homotopy perturbation method. Applied Mathematics and Computation 2012, 219, 3565 -3574.
AMA StyleMuhammad Aslam Noor, Waseem Asghar Khan. New iterative methods for solving nonlinear equation by using homotopy perturbation method. Applied Mathematics and Computation. 2012; 219 (8):3565-3574.
Chicago/Turabian StyleMuhammad Aslam Noor; Waseem Asghar Khan. 2012. "New iterative methods for solving nonlinear equation by using homotopy perturbation method." Applied Mathematics and Computation 219, no. 8: 3565-3574.
Muhammad Aslam Noor; Waseem Asghar Khan; Akhtar Hussain. A new modified Halley method without second derivatives for nonlinear equation. Applied Mathematics and Computation 2007, 189, 1268 -1273.
AMA StyleMuhammad Aslam Noor, Waseem Asghar Khan, Akhtar Hussain. A new modified Halley method without second derivatives for nonlinear equation. Applied Mathematics and Computation. 2007; 189 (2):1268-1273.
Chicago/Turabian StyleMuhammad Aslam Noor; Waseem Asghar Khan; Akhtar Hussain. 2007. "A new modified Halley method without second derivatives for nonlinear equation." Applied Mathematics and Computation 189, no. 2: 1268-1273.
Muhammad Aslam Noor; Khalida Inayat Noor; Waseem Asghar Khan; Faizan Ahmad. On iterative methods for nonlinear equations. Applied Mathematics and Computation 2006, 183, 128 -133.
AMA StyleMuhammad Aslam Noor, Khalida Inayat Noor, Waseem Asghar Khan, Faizan Ahmad. On iterative methods for nonlinear equations. Applied Mathematics and Computation. 2006; 183 (1):128-133.
Chicago/Turabian StyleMuhammad Aslam Noor; Khalida Inayat Noor; Waseem Asghar Khan; Faizan Ahmad. 2006. "On iterative methods for nonlinear equations." Applied Mathematics and Computation 183, no. 1: 128-133.