Differential Boundary Value Problems Research Articles

Exponential heat source effects on the stagnation‐point heat transport of Williamson nanoliquid with nonlinear Boussinesq approximation

AbstractThe nonlinear two‐point partial differential boundary value problem associated with the nano‐pseudoplastic material flow and heat transport subject to nonlinear Boussinesq approximation is computed and explored statistically. Heat transportation features are analyzed by the consideration of an exponential space‐related heat source and the Buongiorno model of nanofluids. The boundary‐driven expressions of the physical phenomenon are coupled and highly complicated due to the consideration of nonlinear convection terms. Reasonable variables are employed to reform the partial differential equations into a system of ordinary differential expressions and are solved numerically. Furthermore, correlation and regression techniques are employed for the statistical evaluation of the phenomenon. The probable error is implemented to calculate the reliability of the computed correlation factors. The exponential index and Schmidt number are positively correlated with the reduced skin friction coefficient whereas the other parameters are negatively correlated with it. The heat transfer rate is improved predominantly by the nonlinear thermal convection parameter. The temperature is enhanced by the intensification of the exponential‐based heat source factor. The temperature and concentration profiles are boosted by incrementing the Biot number values.

On coupled system of nonlinear Ψ-Hilfer hybrid fractional differential equations

Abstract This paper is dedicated to investigating the existence of solutions to the initial value problem (IVP) for a coupled system of Ψ-Hilfer hybrid fractional differential equations (FDEs) and boundary value problem (BVP) for a coupled system of Ψ-Hilfer hybrid FDEs. Analysis of the current paper depends on the two fixed point theorems involving three operators characterized on Banach algebra. In the view of an application, we provided useful examples to exhibit the effectiveness of our achieved results.

On the solvability of a nonlinear difference boundary value problem in the case of parametric resonance

We construct necessary and sufficient conditions for the existence of solution of seminonlinear boundary value problem for a parametric excitation system of difference equations. The convergent iteration algorithms for the construction of the solutions of the semi-nonlinear boundary value problem for a parametric excitation system difference equations in the critical case have been found. The investigation of periodic and Noetherian boundary-value problems in the critical cases is traditionally performed under the assumption that the differential equation and boundary conditions are known and fixed. As a rule, the study of periodic problems in the case of parametric resonance is reduced to the investigation of the problems of stability. At the same time, due to numerous applications in electronics, geodesy, plasma theory, nonlinear optics, mechanics, and machine-building, the analysis of periodic boundary-value problems in the case of parametric resonance requires not only to find the solutions but also to determine the eigenfunctions of the corresponding differenсе equation. The investigation of autonomous Noetherian boundary-value problems is also reduced to the study of Noetherian boundary-value problems in the case of parametric resonance because the change of the independent variable in the critical case gives a nonautonomous boundary-value problem with an additional unknown quantity. The aim of the present paper is to construct the solutions of Noetherian boundary-value problems in the case of parametric resonance whose solvability is guaranteed by the corresponding choice of the eigenfunction of the analyzed boundary-value problem. The applied classification of Noetherian boundary-value prob\-lems in the case of parametric resonance depending on the simplicity or multiplicity of roots of the equation for generating constants noticeably differs from a similar classification of periodic problems in the case of parametric resonance and corresponds to the general classification of periodic and Noetherian boundary-value problems. The equation for generating constants obtained for the Noetherian boundary-value problems in the case of parametric resonance strongly differs from the conventional equation for generating constants in the absence of parametric resonance by the dependence both of the equation and of its roots on a small parameter, which leads to noticeable corrections of the approximate solutions as compared with the approximations obtained by the Poincare method. Using the convergent iteration algorithms we expand solution of seminonlinear two-point boundary value problem for a parametric excitation Mathieu type difference equation in the neighborhood of the generating solution. Estimates for the value of residual of the solutions of the seminonlinear two-point boundary value problem for a parametric excitation Mathieu type difference equation are found.

Two Stage Implicit Method on Hexagonal Grids for Approximating the First Derivatives of the Solution to the Heat Equation

The first type of boundary value problem for the heat equation on a rectangle is considered. We propose a two stage implicit method for the approximation of the first order derivatives of the solution with respect to the spatial variables. To approximate the solution at the first stage, the unconditionally stable two layer implicit method on hexagonal grids given by Buranay and Arshad in 2020 is used which converges with Oh2+τ2 of accuracy on the grids. Here, h and 32h are the step sizes in space variables x1 and x2, respectively and τ is the step size in time. At the second stage, we propose special difference boundary value problems on hexagonal grids for the approximation of first derivatives with respect to spatial variables of which the boundary conditions are defined by using the obtained solution from the first stage. It is proved that the given schemes in the difference problems are unconditionally stable. Further, for r=ωτh2≤37, uniform convergence of the solution of the constructed special difference boundary value problems to the corresponding exact derivatives on hexagonal grids with order Oh2+τ2 is shown. Finally, the method is applied on a test problem and the numerical results are presented through tables and figures.

Hermite-Hadamard-Fejer type inequalities via fractional integral of a function concerning another function

In this paper, we at first develop a generalized integral identity by associating Riemann-Liouville (RL) fractional integral of a function concerning another function. By using this identity estimates for various convexities are accomplish which are fractional integral inequalities. From our results, we obtained bounds of known fractional results which are discussed in detail. As applications of the derived results, we obtain the mid-point-type inequalities. These outcomes might be helpful in the investigation of the uniqueness of partial differential equations and fractional boundary value problems.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

Numerical solution of nonlinear differential boundary value problems using adaptive non-overlapping domain decomposition method

ABSTRACT This work aims is to study a nonlinear second-order boundary value differential elliptic problem in one dimension where the nonlinearity concerns the solution and its first derivative. We assume that the source term can be non-smooth and the nonlinearity can grow faster than quadratic. First, we show the existence of a non-negative weak solution if we assume the existence of a super-solution. Second, we present a numerical algorithm to compute an approximation of the non-negative weak solution. The proposed algorithm is decomposed in two steps, the first one is devoted to computing a super-solution, and in the second one, the algorithm computes a sequence of solutions of an intermediate problem obtained by using the Yosida approximation of the nonlinearity. This sequence converges to the non-negative weak solution of the nonlinear equation. The numerical method is an application of the Newton method to the discretized version of the problem, but at each iteration, the resulting system can be indefinite. To overcome this difficulty, we introduce an adaptive non-overlapping domain decomposition method.

Heat transfer enhancement due to nanoparticles, magnetic field, thermal and exponential space-dependent heat source aspects in nanoliquid flow past a stretchable spinning disk

This study explores the heat transfer characteristics of nanoliquid flowing over a rotating disk in the presence of the applied magnetic field and convective boundary condition. The nanoliquid is flowing due to the rotation of the disk with uniform stretching of a disk along the radial direction. Effects of ESHS (exponential space-related heat source) and THS (thermal-related heat source) are the focal concern of this article. The effective thermal conductivity of ethylene glycol (EG)-based graphene oxide (GO) nanoliquid is estimated by using Nan’s model whereas effective dynamic viscosity is calculated through Brinkman model. The partial differential system which governed the problem is transformed by using Von-Karman stretching transformations to the ordinary differential system. The subsequent two-point ODBVP (ordinary differential boundary value problem) is treated numerically. The consequence of effective parameters of the problem on different flow fields is illustrated graphically. The numerical values of shear stress and heat transfer rate (Nusselt number) are also calculated. Further, the slope of the data points is determined to quantify the outcome. Validation of the present results is made by direct comparison with the available results and an excellent agreement is found. It is found that the rate of heat transfer increased with nanoparticle volume fraction at the rate 0.4153 and the friction factor increased by increasing nanoparticle volume fraction at the rate 3.0681. The fluctuation rate of Nusselt number due to the variation of the ESHS parameter is almost three times more than that of THS parameter.

Melting heat transfer analysis of electrically conducting nanofluid flow over an exponentially shrinking/stretching porous sheet with radiative heat flux under a magnetic field

AbstractModern magnetic nanomaterial processing operations are progressing rapidly and require increasingly sophisticated mathematical models for their optimization. Stimulated by such developments, in this paper, a theoretical and computational study of a steady magnetohydrodynamic nanofluid over an exponentially stretching/shrinking permeable sheet with melting (phase change) and radiative heat transfer is presented. Besides, wall transpiration, that is, suction and blowing (injection), is included. This study deploys Buongiorno's nanofluid model, which simulates the effects of the Brownian motion and thermophoresis. The transport equations and boundary conditions are normalized via similarity transformations and appropriate variables, and the similarity solutions are shown to depend on the transpiration parameter. The emerging dimensionless nonlinear coupled ordinary differential boundary value problem is solved numerically with the Newton‐Fehlberg iteration technique. Validation with special cases from the literature is included. The increase in the magnetic field, that is, the Hartmann number, is observed to elevate nanoparticle concentration and temperature, whereas it dampens the velocity. Higher values of the melting parameter consistently decelerate the boundary layer flow and suppress temperature and nanoparticle concentration. A higher radiative parameter strongly increases temperature (and thermal boundary layer thickness) and weakly accelerates the flow. The increase in the Brownian motion reduces nanoparticle concentrations, whereas a greater thermophoretic body force strongly enhances them. The Nusselt number and Sherwood number are observed to be decreased with an increasing Hartmann number, whereas they are elevated with a stronger wall suction and melting parameter.

Computational Study of the Coupled Mechanism of Thermophoretic Transportation and Mixed Convection Flow around the Surface of a Sphere.

The main goal of the current work was to study the coupled mechanism of thermophoretic transportation and mixed convection flow around the surface of the sphere. To analyze the characteristics of heat and fluid flow in the presence of thermophoretic transportation, a mathematical model in terms of non-linear coupled partial differential equations obeying the laws of conservation was formulated. Moreover, the mathematical model of the proposed phenomena was approximated by implementing the finite difference scheme and boundary value problem of fourth order code BVP4C built-in scheme. The novelty point of this paper is that the primitive variable formulation is introduced to transform the system of partial differential equations into a primitive form to make the line of the algorithm smooth. Secondly, the term thermophoretic transportation in the mass equation is introduced in the mass equation and thus the effect of thermophoretic transportation can be calculated at different positions of the sphere. Basically, in this study, some favorite positions around the sphere were located, where the velocity field, temperature distribution, mass concentration, skin friction, and rate of heat transfer can be calculated simultaneously without any separation in flow around the surface of the sphere.

Existence of Positive Solutions of Boundary Value Problem For FractionalOrder Differential Equation

Recently boundary value problems for differential equations of non-integral order have studied in many papers ( see [1,2] ). Zaho etal [ 1 ] studied the following boundary value problem of fractional differential equations. Where denotes the Rimann-Liouville fractional derivative equation of order . By using the lower and upper solution method and fixed point theorem. Liang and Zhang [3] studied the non-linear fractional differential boundary value problem Where is a real number . is the Rimann-Liouville fractional differential operator of order . By means of fixed point theorems , they obtained results on the existence of positive solutions for boundary value problem of fractional differential equations. In this paper , we deal with some existence of positive solution of the following non-linear fractional differential equation. Where is a real number. denotes Rimann-Liouville fractional derivative of order . Our work based on Banach contraction mapping and Krasnoel'skii fixed point theorems to investigate the existence of positive solution. Finally , we suggest studing the existence solutions for the following Integrodifferential equation with boundary value conditions Where H is a nonlinear integral operator given as

Existence and monotone iteration of unique solution for tempered fractional differential equations Riemann\u2013Stieltjes integral boundary value problems

The primary objective of this research paper is to investigate two kinds of high-order tempered fractional differential equations integral boundary value problems. By means of the mixed monotone operators fixed point theorems with perturbation and the increasing varphi -(h,sigma )-concave operators fixed point theorems, we can not only guarantee the existence-uniqueness of solution, but also construct successively sequences for approximating the unique solution. In addition, we demonstrate the effectiveness of the main result by using two examples.

Biomathematical model for gyrotactic free-forced bioconvection with oxygen diffusion in near-wall transport within a porous medium fuel cell

Bioconvection has shown significant promise for environmentally friendly, sustainable “green” fuel cell technologies. The improved design of such systems requires continuous refinements in biomathematical modeling in conjunction with laboratory and field testing. Motivated by exploring deeper the near-wall transport phenomena involved in bio-inspired fuel cells, in the present paper, we examine analytically and numerically the combined free-forced convective steady boundary layer flow from a solid vertical flat plate embedded in a Darcian porous medium containing gyrotactic microorganisms. Gyrotaxis is one of the many taxes exhibited in biological microscale transport, and other examples include magneto-taxis, photo-taxis, chemotaxis and geo-taxis (reflecting the response of microorganisms to magnetic field, light, chemical concentration or gravity, respectively). The bioconvection fuel cell also contains diffusing oxygen species which mimics the cathodic behavior in a proton exchange membrane (PEM) system. The vertical wall is maintained at iso-solutal (constant oxygen volume fraction and motile microorganism density) and iso-thermal conditions. Wall values of these quantities are sustained at higher values than the ambient temperature and concentration of oxygen and biological microorganism species. Similarity transformations are applied to render the governing partial differential equations for mass, momentum, energy, oxygen species and microorganism species density into a system of ordinary differential equations. The emerging eight order nonlinear coupled, ordinary differential boundary value problem features several important dimensionless control parameters, namely Lewis number (Le), buoyancy ratio parameter i.e. ratio of oxygen species buoyancy force to thermal buoyancy force (Nr), bioconvection Rayleigh number (Rb), bioconvection Lewis number (Lb), bioconvection Péclet number (Pe) and the mixed convection parameter ([Formula: see text] spanning the entire range of free and forced convection. The transformed nonlinear system of equations with boundary conditions is solved numerically by a finite difference method with central differencing, tridiagonal matrix manipulation and an iterative procedure. Computations are validated with the symbolic Maple 14.0 software. The influence of buoyancy and bioconvection parameters on the dimensionless temperature, velocity, oxygen concentration and motile microorganism density distribution, Nusselt, Sherwood and gradient of motile microorganism density are studied. The work clearly shows the benefit of utilizing biological organisms in fuel cell design and presents a logical biomathematical modeling framework for simulating such systems. In particular, the deployment of gyrotactic microorganisms is shown to stimulate improved transport characteristics in heat and momentum at the fuel cell wall.

Existence and uniqueness of monotone positive solutions for fractional higher-order BVPs

We discuss the existence and uniqueness of monotone positive solutions for a class of higher-order nonlinear fractional differential equations infinite-point boundary value problems (for short, BVPs) on cones. The existence and uniqueness of solutions are obtained via applying the properties of the Green function and a fixed point theorem. Our analysis is based on the operator equation T ω + S ω = ω on an ordered Banach space. Finally, a example is given to illustrate our results.

Initialization of Electromagnetic Transient Simulation Using Boundary Value Solution Method

Initialization is an important preliminary work in electromagnetic transient simulation which is typically executed from the periodic steady-state. To this end, researchers have proposed so far a variety of initialization methods, among which the initialization based on fixed-point iteration or the so-called the EMTP-based approach has been widely used. However, this type of method can be very time consuming for a large power system due to its weak convergence. In this paper, the initialization is described as a two-point differential boundary value problem. On this basis, the boundary value solution technique is used to solve this problem, therefore, a new electromagnetic transient initialization method is derived. The proposed method is a strict Newton algorithm, so it has better convergence than the traditional fixed-point iteration method. Case studies conducted on two typical network have confirmed the effectiveness of the proposed method.

Initialization of Electromagnetic Transient Simulation Using Differential Quadrature Method

ABSTRACT The initialization problem of electromagnetic transient simulation is to calculate the steady-state solution of the differential equations describing the electromagnetic transients of power systems. Currently, the widely used initialization method in electromagnetic transient programs (EMTP) is the fixed-point iteration or the so-called EMTP-based approach where the steady-state periodic response is found for a given initial state by simply integrating the system equations until the response becomes periodic. In lightly damped systems, this type of method could extend over many periods making the computation costly. In this paper, the initialization is described as a two-point differential boundary value problem. On this basis, the differential quadrature method (DQM) is used to solve this problem, therefore, a new electromagnetic transient initialization method is derived. The proposed initialization method is a rigorous Newton algorithm and thus has better convergence than the fixed-point iterative method. In order to solve the problem that the Jacobian matrix involved in the DQM solution is very large, a block recursive solution method based on V -transformation is proposed. Case studies conducted on two typical networks have confirmed the effectiveness of the proposed method.

Impulsive quantum (p,q)-difference equations

In this paper we study quantum (p,q)-difference equations with impulse and initial or boundary conditions. We consider first order impulsive (p,q)-difference boundary value problems and second order impulsive (p,q)-difference initial value problems. Existence and uniqueness results are proved via Banach’s fixed point theorem.

Simulation of unsteady natural convection flow of a Casson viscoplastic fluid in a square enclosure utilizing a MAC algorithm

AbstractNon‐Newtonian fluids are increasingly being deployed in energy systems and materials processing. Motivated by these developments, in the current study, a numerical simulation is performed on two‐dimensional, unsteady buoyancy‐driven flow in a square cavity filled with non‐Newtonian fluid (Casson liquid). The enclosure geometry features vertical isothermal walls (with one at higher temperature than the other) and thermally insulated horizontal walls. The conservation equations for mass, momentum, and energy are normalized via appropriate transformations and the resulting dimensionless partial differential boundary value problem is solved computationally with a marker and cell algorithm, which features a finite difference scheme along with a staggered grid system. The projection method is employed to evaluate the pressure term. Extensive visualizations of the impact of emerging physical parameters (Rayleigh number and Casson viscoplastic parameter) on streamline and isotherm distributions in the cavity are presented for fixed Prandtl number. Nusselt number, that is, heat transfer rate, is increased with rising values of the Casson viscoplastic fluid parameter for any value of Rayleigh number. The density of streamlines increases with increasing values of Casson viscoplastic fluid parameter upto 1. Overall, the Casson fluid parameter plays a vital role in controlling the convective heat transfer within the enclosure. The computations are relevant to hybrid solar collectors, materials fabrication (polymer melts), etc.

Study of a generalized Riemann-Liouville fractional integral via convex functions

In this paper estimations in general form of sum of left and right sided Riemann-Liouville (RL) fractional integrals for convex functions are studied. Also some similar fractional inequalities for functions whose deriva tives in absolute value are convex, have been obtained. Associated fractional integral inequalities provide the bounds of different known fractional inequal ities. These results may be useful in in the study of uniqueness solutions of fractional differential equations and fractional boundary value problems.

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.