Dual solutions for MHD flow of a water-based TiO2-Cu hybrid nanofluid over a continuously moving thin needle in presence of thermal radiation

In this analysis, the flow and heat transfer characteristics of an aqueous hybrid nanofluid with TiO2 and Cu as the nanoparticles past a horizontal slim needle in the presence of thermal radiation effect is investigated. We hope that the present research is applicable in fiber technology, polymer ejection, blood flow, etc. The Prandtl number of the base fluid is kept constant at 6.2. The needle is considered thin when its thickness does not exceed that of the boundary layer over it. Using the similarity transformation method, the governing PDEs are transformed to a set of non-linear ODEs. Then, the converted ODEs are numerically solved with help of bvp4c routine from MATLAB. Results indicate that the dual similarity solutions are obtained only when the slim needle moves in the opposite direction of the free stream. In addition, the first solutions are stable and physically realizable. Besides, the second nanoparticle's mass and also the magnetic parameter lead to decrease the range of the velocity ratio parameter for which the solution exists.

Casson fluid flow with heat and mass transfer in a channel using the differential transform method

In the present investigation, we consider the heat and mass transfer characteristics of steady, incompressible and electrically conducting Casson fluid flow in a channel. The effect of chemical reactions have also been considered. The differential transform method (DTM) is applied to a system of non-linear ODEs, and the results are obtained in the form of DTM series. The principal gain of this approach is that it applies to the non-linear ODEs without requiring any discretization, linearization or perturbation. The velocity, mass and heat transfer profiles thus obtained are in good agreement with those provided by the quasi-linearization method (QLM). Graphical results for velocity, concentration and temperature fields are presented for a certain range of values of the governing parameters.

Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation method, we propose a quantum algorithm for solving n-dimensional nonlinear dissipative ordinary differential equations (ODEs). Our algorithm first converts the original nonlinear ODEs into other nonlinear ODEs which can be embedded into finite-dimensional linear ODEs. Then we solve the embedded linear ODEs with quantum linear ODEs algorithm and obtain a state ε-close to the normalized exact solution of the original nonlinear ODEs with success probability Ω(1). The complexity of our algorithm is O(gηTpoly(log(nT/ε))), where η, g measure the decay of the solution. Our algorithm provides exponential improvement over the best classical algorithms or previous quantum algorithms in n or ε.

Heat enhancement analysis of the hybridized micropolar nanofluid with Cattaneo–Christov and stratification effects

In the present article, it is examined the steady bio-convective hybridized micropolar nanofluid flow with the stratification conditions above a vertical exponentially stretching surface. The SWCNT+MWCNT are used in a base fluid of water to formulate the Hybrid nanoparticles in the current article. To examine the mass and heat transfer rate, the activation energy and Cattaneo-Christov heat flux are factored into the equation. The relevant transformations are manipulated to transfer the flow model into the coupled non-linear ODEs. To answer the coupled equations, the Bvp4c Matlab approach is being used. The conclusion of various parameters is examined graphically. The physical numbers observed via graphs, such as friction factor, local Sherwood number and local microorganism number. It is worth noticing that the axial and angular velocity reduces close the boundary and enhances away from the boundary with the escalation of solid volume fraction of SWCNT and MWCNT. Further, as increases the Peclet number, microorganism stratification parameters, and bio-convection Schmidt number, the microorganism sketch declines.

Theory of Functional Connections Applied to Linear ODEs Subject to Integral Constraints and Linear Ordinary Integro-Differential Equations

This study shows how the Theory of Functional Connections (TFC) allows us to obtain fast and highly accurate solutions to linear ODEs involving integrals. Integrals can be constraints and/or terms of the differential equations (e.g., ordinary integro-differential equations). This study first summarizes TFC, a mathematical procedure to obtain constrained expressions. These are functionals representing all functions satisfying a set of linear constraints. These functionals contain a free function, g(x), representing the unknown function to optimize. Two numerical approaches are shown to numerically estimate g(x). The first models g(x) as a linear combination of a set of basis functions, such as Chebyshev or Legendre orthogonal polynomials, while the second models g(x) as a neural network. Meaningful problems are provided. In all numerical problems, the proposed method produces very fast and accurate solutions.

Effective Pedagogy of Guiding Undergraduate Engineering Students Solving First-Order Ordinary Differential Equations

An alternative pedagogical design is discussed that aims to guide engineering students to solve first-order ordinary differential equations (ODEs), and is based on students’ learning weaknesses identified from previous teaching and learning activities. This approach supported student’s self-enrichment through exploration of relevant resources in ODEs, and guided students towards the choice of their own effective ways for solving ODEs for different problems. This paper presents the practices on designing and delivering solution techniques for first-order linear ODEs using this approach for more than 400 undergraduate engineering students at a regional university in Australia during 2014–2017. The timeline involved initial experimentation in 2014 and 2015, followed by refinements to the pedagogy based on student’s feedback. The refined pedagogy was then used for the advanced mathematics course in 2016 and 2017. Significant improvements were made in student’s learning outcomes in effectively and accurately solving the first-order linear ODEs over this period.

The Magneto-Natural Convection Flow of a Micropolar Hybrid Nanofluid Over a Vertical Plate Saturated in a Porous Medium

In this study, we investigate the convective flow of a micropolar hybrid nanofluid through a vertical radiating permeable plate in a saturated porous medium. The impact of the presence or absence of the internal heat generation (IHG) in the medium is examined as well as the impacts of the magnetic field and thermal radiation. We apply similarity transformations to the non-dimensionalized equations and render them as a system of non-linear ODEs (Ordinary Differential Equations) subject to appropriate boundary conditions. This system of non-linear ODEs is solved by an adaptive mesh transformation Chebyshev differential quadrature method. The influence of the governing parameters on the temperature, microrotation and velocity is examined. The skin friction coefficient and the Nusselt number are tabulated. We determine that the skin friction coefficient and heat transport rate increase with the increment in the magnetic field. Moreover, the increment in the micropolarity and nanoparticle volume fraction enhances the skin friction coefficient and the Nusselt number. We also conclude that the IHG term improved the flow of the hybrid nanofluid. Finally, our results indicate that employing a hybrid nanofluid increases the heat transfer compared with that in pure water and a nanofluid.