Ordinary Boundary Value Problems Research Articles

Double Diffusive Nonlinear Convective MHD Unsteady Slip-Flow Regime in a Rectangular Channel

In the paper, we numerically explored the combined impacts of non-linear thermal and mixed convective unsteady flow in a channel with slip conditions. The flow is caused by a moving flat parallel surface and is also electrically conductive. We analyse the mechanisms of heat, and mass transfer by incorporating temperature and concentration jumps. To simplify the model problem, we apply appropriate similarity transformations, reducing the prevailing problem to a nonlinear coupled ordinary boundary value problem. The transformed problem is solved using the Chebyshev Collocation Approach (CCA). We performed a comparative analysis by comparing the CCA with the literature to verify the accuracy of our approach, and a good agreement is found. In addition, we conducted a comprehensive parametric study to analyze the trends in the solutions obtained. The study reveals that the parameters M, α1, α3, Pr, and Sc have about 20% stronger impact on the nonlinear system compared to the linear system on both surfaces of the horizontal channel.

Effects of Hall Current and Viscous Dissipation on Bioconvection Transport of Nanofluid over a Rotating Disk with Motile Microorganisms.

The study of rotating-disk heat-flow problems is relevant to computer storage devices, rotating machineries, heat-storage devices, MHD rotators, lubrication, and food-processing devices. Therefore, this study investigated the effects of a Hall current and motile microorganisms on nanofluid flow generated by the spinning of a disk under multiple slip and thermal radiation conditions. The Buongiorno model of a nonhomogeneous nanofluid under Brownian diffusion and thermophoresis was applied. Using the Taylor series, the effect of Resseland radiation was linearized and included in the energy equation. By implementing the appropriate transformations, the governing partial differential equations (PDEs) were simplified into a two-point ordinary boundary value problem. The classical Runge-Kutta dependent shooting method was used to find the numerical solutions, which were validated using the data available in the literature. The velocity, motile microorganism distribution, temperature, and concentration of nanoparticles were plotted and comprehensively analyzed. Moreover, the density number, Sherwood number, shear stresses, and Nusselt number were calculated. The radial and tangential velocity declined with varying values of magnetic numbers, while the concentration of nanoparticles, motile microorganism distribution, and temperature increased. There was a significant reduction in heat transfer, velocities, and motile microorganism distribution under the multiple slip conditions. The Hall current magnified the velocities and reduced the heat transfer. Thermal radiation improved the Nusselt number, while the thermal slip conditions reduced the Nusselt number.

Computational study of three-dimensional flow and heat transfer of 25 nm Cu–H2O nanoliquid with convective thermal condition and radiative heat flux using modified Buongiorno model

The three-dimensional steady flow of an incompressible 25 nm water-based copper nanoliquid over a bi-directional elongated surface with convective thermal boundary condition is studied. This work may meet various thermal engineering applications for instance thermal energy exchangers, solar energy collectors, geophysical transports, radiators, electronic cooling devices, and nuclear reactors. Corcione's model for effective thermal conductivity and dynamic viscosity is used in conjunction with correlations based on effective medium theory for density, specific heat, and electrical conductivity. The effects of radiative heat and convective thermal energy conditions are also taken into account. Modified Buongiorno inhomogeneous model-based governing equations transmuted to nonlinear ordinary boundary value problem. The subsequent system has been deciphered using computer algebraic system software. The method used is validated against the available data and found an exceptional agreement. Two dimensional charts are designed to analyze the impact of the key parameters involved in the model. Three dimensional surface plots are plotted for the study of the Nusselt number, friction coefficients, and Sherwood number values are tabulated for several key parameters. A regression analysis is also performed for the local Nusselt number.

ON A METHOD FOR SOLVING A FAMILY OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we consider a boundary value problem for a family of linear differential equations that obey a family of nonlinear two-point boundary conditions. For each fixed value of the family parameter, the boundary value problem under study is a nonlinear two-point boundary value problem for a system of ordinary differential equations. Non-local boundary value problems for systems of partial differential equations, in particular, non-local boundary value problems for systems of hyperbolic equations with mixed derivatives, can be reduced to the family of boundary value problems for ordinary differential equations. Therefore, the establishment of solvability conditions and the development of methods for solving a family of boundary value problems for differential equations are actual problems. In this paper, using the ideas of the parametrization method of D. S. Dzhumabaev, which was originally developed to establish the signs of unambiguous solvability of a linear two-point boundary value problem for a system of ordinary equations, a method for finding a numerical solution to the problem under consideration is proposed.

Numerical Solution of the Seventh Order Boundary Value Problems Using B-Spline Method

We develop a numerical method for solving the boundary value problem of The Linear Seventh Ordinary Boundary Value Problem by using seventh degree B-Spline function. Formulation is based on particular terms of order of seventh order boundary value problem. We obtain Septic B-Spline formulation and the Collocation B-spline Method is formulated as an approximation solution. We apply the presented method to solve an example of seventh-order boundary value problem which the results show that there is an agreement between approximate solutions and exact solutions. Resulting low absolute errors show that the presented numerical method is effective for solving high order boundary value problems. Finally, a general conclusion has been included.

Similarity solution of MHD slip with energy mass transport through chemically reacting stretching permeable surface in porous media with variable properties

In this paper, a mathematical model is presented for magnetohydrodynamic viscous incompressible boundary layer flow of an electrically conducting chemically reacting fluid over a moving permeable flat vertical surface in a porous medium with variable fluid properties. A linear momentum slip boundary condition is included. Temperature and concentration difference between the wall and the environment are assumed to vary with the axial distance according to power law forms. Furthermore, a concentration-dependent mass diffusivity is featured to provide for a more realistic simulation of relevance to materials processing systems. The governing equations with associated boundary conditions are transformed into similarity equations using appropriate variables furnished with Lie group algebraic methods. The resulting nonlinear differential ordinary boundary value problem is then solved numerically with a Runge–Kutta–Fehlberg fourth–fifth order numerical method. The effects of the governing parameters on the boundary layer flow characteristics (velocity, temperature, concentration, wall shear stress, heat and mass transfer rates) have been investigated, displayed graphically and discussed. The computations indicate that variable mass diffusivity plays a significant role on transport characteristics. It is also shown that concentration magnitudes increase whereas the rate of mass transfer at the wall decreases with increasing mass diffusivity. Comparison of numerical and analytical results with results from the open literature show excellent agreement. Further verification of the general model is conducted with the Spectral Relaxation Method (SRM). The study is relevant to electroconductive polymeric sheet processing in manufacturing operations and smart coating systems.

Applying new fixed point theorems on fractional and ordinary differential equations

In this paper, we consider a fixed point theorem that extends and unifies several existing results in the literature. We apply the proven fixed point results on the existence of solution of ordinary boundary value problems and fractional boundary value problems with integral type boundary conditions in the frame of some Caputo type fractional operators.

MHD flow of SWCNT and MWCNT nanoliquids past a rotating stretchable disk with thermal and exponential space dependent heat source

The main purpose of this investigation is to analyze the impacts of a novel exponential space dependent heat source on MHD slip flow of carbon nanoliquids past a stretchable rotating disk. The flow is created due to rotation and stretching of the disk. Aspects of the convective condition and cross-diffusion (Soret and Dufour effects) are also accounted. A comparative study of nanofluids made up of SWCNT’s (single-walled carbon nanotube) and MWCNT’s (multi-walled carbon nanotube) is presented. The governing partial differential equations system is reduced to nonlinear ordinary boundary value problem. The Runge–Kutta–Fehlberg is utilized for numerical simulations. Embedded dimensionless parameters on the flow fields are examined via graphical illustrations. The rate of heat mass transfer can be controlled by cross-diffusion, exponential space-based heat source and thermal-based heat source effects. It is also proved that A novel idea of the exponential space dependent heat source is implemented in the investigation of the slip flow over a rotating deformable disk under the effects of cross-diffusion, temperature based heat source and magnetic field for the first time. A comparison between two different fluids namely SWCNT-H2O nanoliquid and MWCNT-H2O nanoliquid are studied.

Variational Iteration Method for Solving Boundary Value Problems of Fractional Order

This paper presents an approximate solution of ordinary boundary value problems using the variational iteration method. The fractional derivatives are described in the Caputo sense, because it allows traditional initial and boundary conditions to be included in the formulation of the problem. The solutions of proposed fractional equations are calculated in the form of convergent series with easily computable components. Also, we construct the variational iteration formula and its convergence to the exact solution for solving system of fractional order differential equations. Some examples are solved as an illustration to the method, in order to show the accuracy of the method and comparison with results the exact solution for the given test problems, which are given for comparison purpose.

О первой смешанной задаче в банаховых пространствах для вырождающихся параболических уравнений с меняющимся направлением времени

The article is devoted to studying one of the sections of nonclassical differential equations, namely, matters concerned with solvability of parabolic equations with changing second-order time direction. As is known, in ordinary boundary-value problems for strictly parabolic equations, the smoothness of the initial and boundary conditions completely ensures that the solutions belong to the Holder spaces, but in the case of equations with changing time direction, the smoothness of the initial and boundary conditions does not ensure that the solutions belong to these spaces. S.A. Tersenov (for a model parabolic equation with changing time direction) and S.G. Pyatkov (for a more general second-order equation) obtained the necessary and sufficient conditions for solvability of the corresponding mixed problems in Holder spaces. In so doing, they always assumed the initial and boundary conditions being equal to zero. Cases in which the initial and boundary conditions belong to Banach spaces are considered. The functional spaces in which the solutions must be sought are introduced. Relevant a priori estimates, which make it possible to obtain the solvability conditions for these problems, are obtained. The properties of the obtained solutions have been studied. In particular, the equivalence of the Riesz and Littlewood-Paley conditions similar to the conditions for solutions of strictly elliptic and strictly parabolic second order equations is established. A unique solvability of the first mixed problem with boundary and initial functions from the Banach space has been proved.

Influence of temperature and magnetic field on the oblique stagnation-point flow for a nanofluid past a vertical stretching/shrinking sheet

Purpose This paper aims to consider the influence of the temperature and of an external magnetic field on the steady oblique stagnation-point flow for a Boussinesquian nanofluid past a stretching or shrinking sheet. Design/methodology/approach The flow is reduced through similarity transformations to an ordinary boundary value problem, which is solved numerically in MATLAB using the bvp4c function. The behavior of the solution is discussed physically, and some analytical considerations concerning existence of the solution and the occurrence of dual solutions are drawn. Findings The study of the influence of an external magnetic field on the oblique stagnation-point flow of a Buongiorno's Boussinesquian nanofluid is carried out. The fluid clashes on a vertical stretching or shrinking sheet. Dual solutions appear for suitable values of the parameters. Originality/value The present results are new and original.

Framing the Cattaneo–Christov Heat Flux Phenomena on CNT- Based Maxwell Nanofluid Along Stretching Sheet with Multiple Slips

A theoretical model on carbon nanotube-based upper convective Maxwell nanofluid flow with thermal and velocity slip boundary conditions is thoroughly presented here. To formulate the viscoelastic relaxation framework on boundary layer flow uniquely, Cattaneo–Christov heat flux model is employed in the momentum equation. The dominating equations and boundary conditions are turned into system of ordinary boundary value problem by using proper transformations, and hence, the similarity solutions are gained numerically by applying Runge–Kutta fourth-order method with shooting practice. The impacts of the governing parameters upon the flow and temperature are represented in tabular as well as in graphical form and discussed in detail. The findings elucidate that both the velocity and temperature are inverse functions of thermal relaxation time for single-walled and multi-walled carbon nanotube-based nanofluids. Fluid relaxation parameter is positively correlated with skin friction and rate of heat transport for both types of nanofluids, but the temperature slip makes a contrary with the same.

Marangoni convective MHD flow of SWCNT and MWCNT nanoliquids due to a disk with solar radiation and irregular heat source

Present study addresses the Marangoni transport of dissipating SWCNT and MWCNT nanofluids under the influence of magnetic force and radiation. A novel exponential space dependent heat source is considered. The flow is generated due to a disk with surface tension created by thermal gradient. The partial differential equations system governing the flow of carbon-water nanoliquids and heat transfer through Marangoni convection is established. Subsequent system is reduced to nonlinear ordinary boundary value problem via generalized Karman transformations. Numerical solutions are developed of the arising nonlinear problem via Runge-Kutta based shooting approach. Impacts of embedded parameters are focused on Nusselt number, velocity and heat transport distributions through graphical illustrations. Our simulations figured out that the heat transfer rate increased via Marangoni convection; however it is decayed by applied magnetic force. The temperature of SWCNT-H2O nanoliquid dominates MWCNT-H2O nanoliquid.

F-transform-based shooting method for nonlinear boundary value problems

We study a class of ordinary two-point boundary value problems for nonlinear second-order ordinary differential equations. We propose to use the framework of shooting method and apply the second-degree F-transform for solving the corresponding initial value problem. We make experiments that demonstrate effectiveness of the proposed F-transform-based shooting method in the class of second-order numerical methods. Moreover, we show stability of the proposed method and its ability to "ignore noise".

New Modified Adomian Decomposition Recursion Schemes for Solving Certain Types of Nonlinear Fractional Two-Point Boundary Value Problems

We apply new modified recursion schemes obtained by the Adomian decomposition method (ADM) to analytically solve specific types of two-point boundary value problems for nonlinear fractional order ordinary and partial differential equations. The new modified recursion schemes, which sometimes utilize the technique of Duan’s convergence parameter, are derived using the Duan-Rach modified ADM. The Duan-Rach modified ADM employs all of the given boundary conditions to compute the remaining unknown constants of integration, which are then embedded in the integral solution form before constructing recursion schemes for the solution components. New modified recursion schemes obtained by the method are generated in order to analytically solve nonlinear fractional order boundary value problems with a variety of two-point boundary conditions such as Robin and separated boundary conditions. Some numerical examples of such problems are demonstrated graphically. In addition, the maximal errors (MEn) or the error remainder functions (ERn(x)) of each problem are calculated.

Numerical solutions of a mathematical model of blood flow in the deforming porous channel using radial basis function collocation method

In this paper, a mathematical model describing blood flow in the deforming porous channel is solved numerically. The blood flow is considered as a micropolar fluid flow. The governing system of nonlinear partial differential equations is reduced to a nonlinear system of ordinary boundary value problem by using some suitable similar transformations. An efficient numerical technique based on the integration radial basis function process, IRBF, coupled with collocation method is performed to approximate the solution of the boundary value problem. Numerical results are presented through the figures and tables to demonstrate efficiency, accuracy, and convergence of the method. Moreover, the effects of various embedded model parameters such as expansion ratio, slip coefficient and others on flow velocity profiles have been explained graphically.

First-harmonic balance for fast evaluation of power-law fluid flow enhancement under periodic pressure gradient

This work considers rectilinear laminar flow of power-law liquids under a periodic pressure gradient. In contrast to Newtonian flow, previous studies have shown that the interaction between nonlinearities by non-Newtonian rheology and exogenous periodic forcing can yield important changes in the flow patterns, such as mean flowrate enhancement for shear-thinning fluid whereas flowrate reduction for shear-thickening fluid relative to non-forcing conditions. The aim of this work was to evaluate the ability of first-harmonic approximation for estimating mean flowrate enhancement/reductions under periodic pressure gradient oscillations. The first-harmonic balance involves the projection of nonlinearities on a first-harmonic Fourier basis, which reduces the governing partial differential equation into three ordinary boundary-value problems that should be solved numerically. The results showed that the predictions from the first-harmonic balance were in good agreement, both qualitative and quantitative, with direct numerical simulations of the governing partial differential equation. The advantage of the former over the latter is the computational burden since numerical computations for the first-harmonic equations does not depend of the forcing frequency.

Advances in the Adomian decomposition method for solving two-point nonlinear boundary value problems with Neumann boundary conditions

A new straightforward approach for solving ordinary and partial second-order boundary value problems with Neumann boundary conditions is introduced in this research. This approach depends mainly on the Adomian decomposition method with a new definition of the differential operator and its inverse, which has been modified for Neumann boundary conditions. The effectiveness of the proposed approach is verified by several linear and nonlinear examples.

Neumann Boundary Value Problems with not Coercive Potential

Existence results of positive solutions for an ordinary Neumann boundary value problem are established. No asymptotic condition on the nonlinear term either at zero or at infinity is requested. The approach is based on variational methods.

Existence of solutions and multiple solutions for a class of weighted p(r)-Laplacian system

This paper investigates the existence of solutions for a class of weighted p ( r ) -Laplacian ordinary system boundary value problems, the sufficient conditions for the existence of solutions have been given via Leray–Schauder's degree, and the existence of multiple solutions has been discussed via critical point theory. As an application, we discussed the existence of the radial solutions for the p ( x ) -Laplacian partial differential system boundary value problems.