Muhammad Usman

A difficult matter for researchers is to develop numerical methods to analyze the behavior of nonlinear dynamical systems that arise in mathematical physics due to their nonlinearity and multi-dimensional nature. In this affection, a novel and effective coupling of spectral and one-step approaches is suggested and used to inspect the robust solutions to the mathematical problems. This proposed method is grounded on the one-step sixth-order Runge–Kutta (RK) method and the shifted Vieta–Fibonacci (sVF) polynomial-based spectral approach. First, use the sixth-order one-step technique for the temporal approximation, then utilize the sVF polynomials to estimate the spatial variables. The sVF polynomial-based new operational matrices are used to calculate the derivative terms of the mathematical models. Related theorems are presented in the paper to provide a mathematical validation of the technique. The suggested semi-spectral techniques transform the investigated nonlinear models into a set of linear algebraic equations, making them simpler to solve. We conduct theoretical and numerical research into the stability, convergence, and error-bound analysis to support the computational algorithm's mathematical formulation. Several mathematical models, including the Hartmann numbers ranging from 0 to 106, are taken into consideration to demonstrate the precision and efficacy of the proposed computational technique. A thorough comparison analysis illustrates the recommended computational approach's validity, correctness, and dependability. The suggested construction of sixth-order in the temporal direction and exponential order around 3.5545e0.1515M in the spatial direction is determined to be exceptional in handling nonlinear problems and examining the exact solutions.

I must reject this paper because the theoretical study does not justify the proposed method.

This study deals with the analysis of the heat and velocity profile of the fractional-order Oldroyd-B bio-nanofluid within a bounded channel. The study has a wide range of scope in modern fields of basic science such as medicine, the food industry, electrical appliances, nuclear as well as industrial cooling systems, reducing pollutants, fluids used in the brake systems of vehicles, etc. Oldroyd-B fluid is taken as a bio-nanofluid composed of base fluid (blood) and copper as nanoparticles. Using the fractional-order Oldroyd-B parameter, the governing equation is generalized from an integer to a non-integer form. A strong approach, i.e., a finite difference scheme, is applied to discretize the model, because the fractional approach can well address the physical phenomena and memory effect of the flow regime. Therefore, a Caputo fractional differentiation operator is used for the purpose. The transformations for the channel flow are utilized to transfigure the fractional-order partial differential equations (PDEs) into non-dimension PDEs. The graphical outcomes for non-integer ordered Oldroyd-B bio-nanofluid dynamics and temperature profiles are navigated using the numerical technique. These results are obtained under some very important physical conditions applied as a magnetic field effect, variable thermal conductivity, permeable medium, and heat source/sink. The results show that the addition of (copper) nanoparticles to (blood) base fluids enhances the thermal conductivity. For a comparative study, the obtained results are compared with the built-in results using the mathematical software MAPLE 2016.

This article contains a study of mixed convection in micropolar nanofluid near an oblique stagnation point in the presence of a magnetic field. Similarity transformations are used to convert governing partial differential equations to non-linear ordinary differential equations. Modified Chebyshev collocation method in computational software Maple is used for the solution of governing nonlinear differential equations. A comparison of numerical results obtained by modified Chebyshev collocation method and finite difference method is made to show the accuracy of the method. Graphical results for velocity components, microrotation, temperature, and flow patterns are part of this study. Numerical values for free parameter (A), skin friction, and Nusselt numbers for different parameters are also calculated. It is found that microrotation profiles are enhanced by increasing the effect of stretching while decline with enhancing angle of strike γ. Also, the temperature of micropolar nanofluid is increased by increasing the value of the magnetic parameter and micropolar coefficient. The temperature gradient of nanofluid shows a decline when values of stretching parameter and the angle of the strike are increased.