Differential Boundary Value Problems Research Articles

Well-posedness result for the flow of Maxwell viscoelastic fluid of White-Metzner type

In this paper, we investigate the flow characteristics of an unsteady quasistatic Maxwell viscoelastic fluid of White-Metzner type within a three-dimensional bounded domain. Our objective is to establish the well-posedness of the resulting partial differential equations initial boundary value problem, both locally and globally in time, under the condition of small initial data. To demonstrate the existence of a solution, we employ an iterative method that transforms our problem into a sequence of linear subproblems. At each iteration, we tackle both elliptic and hyperbolic problems. The convergence of this iterative process provides the solution to the original problem.

Solvability of a fractional differential equation multipoint boundary value problem at resonance in ℝn

Solvability of a fractional differential equation multipoint boundary value problem at resonance in ℝn

Numerical study of hybrid binary Al2O3-Cu-H2O nanofluid coating boundary layer flow from an exponentially stretching/shrinking perforated substrate with Cattaneo-Christov heat flux, heat source, suction and multiple slip effects

Modern coating systems are increasingly deploying nanomaterials which offer improved thermal performance. The optimization of these systems can benefit from more elegant fluid dynamic models of the coating process. Inspired by these advancements, the present investigation introduces a novel mathematical model to analyze the boundary layer transport in Al2O3-Cu-H2O hybrid binary nanofluid coating deposition on an exponentially stretching porous substrate (sheet). Lateral mass flux (suction) at the wall is also considered a heat source (generation) for hot spot manufacturing effects. To add further sophistication to the thermal conduction model, a non-Fourier approach is adopted which accurately incorporates thermal relaxation effects i.e. the Cattaneo-Christov heat flux (CCHF) model. The Tiwari-Das volume fraction nanoscale formulation is implemented for different combinations of Alumina (Al2O3) and Copper (Cu) metallic nanoparticles in an aqueous base fluid (H2O). Hydrodynamic wall and thermal slip are also included as they feature in coating processes. The fundamental equations governing mass, momentum, and energy conservation, along with the corresponding conditions at the wall (substrate) and free stream, are made dimensionless through suitable scaling similarity transformations. The resulting nonlinear coupled ordinary differential boundary value problem is subsequently addressed using the efficient MATLAB bvp4c routine. Special cases of the non-Fourier model are validated against published results, and the general model is further confirmed through validation using an Adams-Moulton 2-step predictor-corrector algorithm (AMPC). Furthermore, dual solutions of the generalized model are extracted and visualized. A comprehensive study of the effects of key thermophysical key parameters on momentum and thermal characteristics (velocity, temperature, skin friction and local Nusselt number) is conducted and solutions are visualized graphically. The computations show that higher temperatures are present with intensification in the thermal relaxation parameter (Cattaneo-Christov hyperbolic model parameter), which predicts therefore larger temperatures than the classical Parabolic heat conduction model using Fourier analysis. Effectively, thermal relaxation is responsible for finite heat wave propagation and produces more accurate estimations of heat transfer than the Fourier model. An increase in stretching parameter enhances the skin friction coefficient and elevates the local Nusselt number. As the thermal relaxation parameter increases, the temperature magnitudes corresponding to the first and second solutions are both enhanced as are thermal boundary layer thicknesses. An increase in wall suction induces an escalation in the first solution for velocity whereas it reduces the second solution velocity. Stronger wall suction suppresses temperatures. With increasing thermal slip, the boundary layer is cooled i.e. temperature plummets and thermal boundary layer thickness is decreased. Strong deceleration is induced with greater hydrodynamic (velocity) slip. Both first and second solutions for skin friction are elevated with shrinking whereas they are depressed with stretching. With greater velocity slip both the first and second solutions for skin friction are depleted with greater velocity slip for the shrinking case, whereas they are generally enhanced for the very strong stretching case. The computations offer some new insights into multi-physical thermal fluid dynamics of nanofluid-based coating systems.

Existence and Ulam-Hyers stability results for a class of fractional integro-differential equations involving nonlocal fractional integro-differential boundary conditions

In this paper, we investigate the existence and uniqueness of solutions for a class of fractional integro- differential boundary value problems involving both Riemann–Liouville and Caputo fractional derivatives, and supplemented with multi-point and nonlocal Riemann-Liouville fractional integral and Caputo fractional deriv- ative boundary conditions. Our results are based on some known tools of fixed point theory. We also study the Ulam–Hyers stability for the proposed fractional problems. Finally, some illustrative examples are included to verify the validity of our results.

Numerical solutions for magneto-convective boundary layer slip flow from a nonlinear stretching sheet with wall transpiration and thermal radiation effects

A mathematical model is presented for magnetohydrodynamic viscous incompressible flow of an electrically conducting Newtonian polymeric fluid from a moving permeable horizontal stretching sheet with variable magnetic field, momentum and thermal slip boundary conditions. The emerging nonlinear ordinary differential boundary value problem is solved numerically by the Runge-Kutta-Fehlberg fourth-fifth order numerical method in Maple symbolic software. The effects of the governing parameters on the dimensionless stream function, dimensionless flow velocity and the dimensionless temperature have been investigated, displayed graphically and discussed. It is found that greater momentum slip and magnetic field parameters reduce the dimensionless velocity. It is further observed that higher values of Prandtl number, thermal slip, and conductionradiation parameter (weaker radiation contribution) reduce the dimensionless temperature whilst stronger magnetic field increases the temperature due to the supplementary work expended in dragging the polymer against the action of the magnetic field which is dissipated as heat. Comparisons of numerical results with previous published results show an excellent agreement. The study finds applications in electro-conductive polymer (ECP) processing systems.

Triple diffusive free magneto-convection of electro-conductive nanofluid from a vertical stretching sheet

In the present investigation, an analysis is carried out to study the MHD triple diffusive free thermo-solutal convection boundary layer flow of an electro-conductive nanofluid flow over a vertical stretching sheet. This problem is relevant to magnetic nanomaterials fabrication operations in which multiple species in addition to nanoparticles are present. In addition to the nanoparticle diffusion, two different salts (species) having different properties are considered. A variable magnetic field is applied transverse to the vertical sheet. It is assumed that the surface is in contact with the hot magnetic nanofluid at a temperature which provides a variable heat transfer coefficient. Buongiorno’s model is employed for the nanofluid. It is also assumed that the Oberbeck-Boussinesq approximation is valid and the mixture of nanofluid and salts is homogenous and is in local thermal equilibrium. In addition, the thermal energy equation features cross-diffusion (Soret and Dufour) terms for both components of salts having different concentration. Appropriate similarity transformations are deployed to render the model non-dimensional. The emerging transformed dimensionless non-linear non-dimensional ordinary differential boundary value problem is solved with the robust bvp4c method in MATLAB. Validation with previous studies has been included for special cases of the general model. The simulations show that the addition of nanoparticles and salts, strongly modifies Temperature and nanoparticle and salt 1 and 2 concentrations. With stronger magnetic field the velocity is suppressed as is momentum boundary layer thickness whereas temperatures are boosted.

On Generalized Sehgal–Guseman-Like Contractions and Their Fixed-Point Results with Applications to Nonlinear Fractional Differential Equations and Boundary Value Problems for Homogeneous Transverse Bars

The goal of this study is to describe the class of modified Sehgal–Guseman-like contraction mappings and set up some fixed-point results in S-metric spaces. The class of generalized Sehgal–Guseman-like contraction mappings contains enhancements of Banach contractions, Kannan contractions, Chatterjee contractions, Chatterjee-type contractions, quasi-contractions, Ćirić–Reich–Rus-type contractions, Hardy–Rogers-type contractions, Reich-type contractions, interpolative Kannan contractions, interpolative Chatterjee contractions, among others, with their generalizations in S-metric spaces. We offer significant examples to substantiate the reliability of our results. This study also establishes consequential fixed-point results and applies them to nonlinear fractional differential equations and the boundary value problem for homogeneous transverse bars. At the end of the manuscript, we present an important open problem.

A new fixed point approach for solutions of a $ p $-Laplacian fractional $ q $-difference boundary value problem with an integral boundary condition

<p>We explored a class of quantum calculus boundary value problems that include fractional $ q $-difference integrals. Sufficient and necessary conditions for demonstrating the existence and uniqueness of positive solutions were stated using fixed point theorems in partially ordered spaces. Moreover, the existence of a positive solution for a boundary value problem with a Riemann-Liouville fractional derivative and an integral boundary condition was examined by utilizing a novel fixed point theorem that included a $ \mathfrak{a} $-$ \eta $-Geraghty contraction. Several examples were provided to demonstrate the efficacy of the outcomes.</p>

Electromagnetohydrodynamic (EMHD) convective transport of a reactive dissipative carreau fluid with thermal ignition in a non-Darcian vertical duct

A mathematical model is developed to simulate the steady, laminar exothermic reactive electromagneto-hydrodynamic combustible non-Newtonian natural convective transport in a vertical duct. Static uniform axial electrical field and transverse magnetic field are imposed. The Frank-Kamenetskii thermal explosion theory is utilized and also the Carreau fluid model, the latter due to its ability to simulate shear-thinning, Newtonian and shear-thickening behaviour. The duct contains a homogenous, isotropic porous medium and to accomodate Forchheimer inertial drag effects, a non-Darcian model is deployed. The duct walls are permeable enabling suction and injection effects to be studied. Viscous and Joule heating (Ohmic dissipation) are also featured in the model. Following a scaling transformation, the dimensionless emerging non-linear ordinary differential boundary value problem is solved with a robust numerical method (Mathematica shooting algorithm). Validation with an Adomian decomposition method (ADM) is included.

Cross-Diffusion and Higher-Order Chemical Reaction Effects on Hydromagnetic Copper–Water Nanofluid Flow Over a Rotating Cone in a Porous Medium

Spin coating of engineering components with advanced functional nanomaterials which respond to magnetic fields is growing. Motivated by exploring the fluid dynamics of such processes, a mathematical model is developed for chemically reactive Cu–H2O magnetohydrodynamic (MHD) nanofluid swirl coating flow on a revolving vertical electrically insulated cone adjacent to a porous medium under a radial static magnetic field. Heat and mass transfer is included and Dufour and Soret cross-diffusion effects are also incorporated in the model. Thermal and solutal buoyancy forces are additionally included. To simulate chemical reaction of the diffusing species encountered in manufacturing processes, a higher-order chemical reaction formulation is also featured. Via suitable scaling transformations, the governing nonlinear coupled partial differential conservation equations and associated boundary conditions are reformulated as a nonlinear ordinary differential boundary value problem. MATLAB-based shooting quadrature with a Runge–Kutta method is deployed to solve the emerging system. Concentration, temperature and velocity variations for various nondimensional flow parameters have been visualized and analyzed. In addition, key wall characteristics, i.e., radial and circumferential skin friction, Nusselt number and Sherwood number, have also been computed. Validation with earlier studies is also included. The simulations indicate that when compared to a lower-order chemical reaction, a higher-order chemical reaction allows a greater rate of heat and mass transfer at the cone surface. Increasing Dufour (diffuso-thermal) and Soret numbers generally reduces radial and circumferential skin friction and also Nusselt number, whereas it elevates the Sherwood number. Both skin friction components are also suppressed with increasing Richardson number. Strong deceleration in the tangential and circumferential velocity components is induced with greater magnetic field.

(ω,ρ)-BVP Solutions of Impulsive Differential Equations of Fractional Order on Banach Spaces

The paper focuses on exploring the existence and uniqueness of a specific solution to a class of Caputo impulsive fractional differential equations with boundary value conditions on Banach space, referred to as (ω,ρ)-BVP solution. The proof of the main results of this study involves the application of the Banach contraction mapping principle and Schaefer’s fixed point theorem. Furthermore, we provide the necessary conditions for the convexity of the set of solutions of the analyzed impulsive fractional differential boundary value problem. To enhance the comprehension and practical application of our findings, we conclude the paper by presenting two illustrative examples that demonstrate the applicability of the obtained results.

Some results on fractional Hahn difference boundary value problems

Abstract Fractional Hahn boundary value problems are significant tools to describe mathematical and physical phenomena depending on non-differentiable functions. In this work, we develop certain aspects of the theory of fractional Hahn boundary value problems involving fractional Hahn derivatives of the Caputo type. First, we construct the Green function for an α th \alpha {\rm{th}} -order fractional boundary value problem, with 1 < α < 2 1\lt \alpha \lt 2 , and discuss some important properties of the Green function. The solutions to the proposed problems are obtained in terms of the Green function. The uniqueness of the solutions is proved by various fixed point theorems. The Banach’s contraction mapping theorem, the Schauder’s theorem, and the Browder’s theorem are used.

Computation of EMHD ternary hybrid non-Newtonian nanofluid over a wedge embedded in a Darcy-Forchheimer porous medium with zeta potential and wall suction/injection effects

The field of engineering is witnessing an increasing number of applications for intelligent electromagnetic nano-coatings. These recent advancements serve as the impetus for an investigation into a theoretical and computational study of an unsteady electro-magneto-hydrodynamic (EMHD) incompressible two-dimensional tangent hyperbolic non-Newtonian ternary hybrid nanofluid boundary layer coating flow external to a two-dimensional porous wedge geometry adjacent to a Darcy-Forchheimer porous medium. Both aligned electrical and transverse magnetic field effects are included. The effects of zeta-potential and surface injection/suction are included. The ternary composite nanofluid comprises three nanoparticles ( A l 2 O 3 , T i O 2 and S i O 2 ) with aqueous base fluid ( H 2 O ) The governing conservation partial differential equations for continuity and momentum and associated boundary conditions at the wedge surface and free stream are transformed with appropriate similarity variables. The emerging ordinary differential nonlinear boundary value problem is numerically solved in MATLAB. Validation with earlier studies is included. A significant deceleration is induced in the flow of a ternary hybrid nanofluid with an increase in the electric field parameter. An increment in Falkner-Skan power-law constant (moving wedge angle parameter) boosts the local skin friction coefficient for both the ternary hybrid nanofluid (THNF) and unitary A l 2 O 3 nanofluid (UNF) cases. The flow is boosted with an increase in the power-law rheological characteristic, which ranges from shear-thinning n < 1 to shear-thickening n > 1.

Numerical study of dissipative SW/MWCNT-nanofluid coating flow from a stretching wall to a porous medium with shape factor effects

A mathematical model is developed for incompressible steady-state dissipative CNT-aqueous nanofluid boundary layer flow from a stretching sheet to a saturated isotropic porous medium. Three different CNT shapes (spheres, blades and platelets) are considered. Both single-walled (SWCNT) and multi-walled (MWCNT) carbon nanotubes are examined. A Darcy-Brinkman model is adopted for the porous medium and a modified viscous dissipation formulation is considered which features porous media influence in the energy conservation equation. Appropriate expressions are deployed for the CNT-modified nanofluid viscosity, density, specific heat capacity, thermal conductivity and CNT shape factor. Via similarity transformations, the governing conservation equations for mass, momentum and energy with associated boundary conditions are normalized to generate a coupled nonlinear ordinary differential boundary value problem. A numerical solution is presented with the robust MATLAB-based bvp4c method and 4th order Runge-Kutta shooting scheme. Validation with previous studies is included. Velocity, temperature, skin friction and Nusselt number are computed for a range of selected parameters. The computations show that elevation in CNT volume fraction parameter accelerates the boundary layer flow whereas increment in the Darcian (inverse permeability) parameter induces strong deceleration. MWCNTs produce higher velocities than SWCNTs. Temperature and thermal boundary layer thickness are found to be enhanced with increasing Eckert number, Darcian (inverse permeability) parameter, CNT volume fraction and CNT shape factor. Significantly greater temperatures are computed for MWCNTs.

Dual solutions of magnetite/cobalt/manganese-zinc-aqueous nano-ferrofluids from a stretching sheet with magnetic induction effects: MHD stagnation flow computation and analysis

A theoretical study is presented for the steady magnetohydrodynamic (MHD) boundary layer stagnation point flow of a nano-ferrofluid along a linearly moving stretching sheet, as a simulation of functional magnetic materials processing. Due to having imerging applications in heat transfer, the nano-ferrofluids draw the attention which comprise an aqueous base fluid doped with a variety of magnetic nanoparticles, i.e. magnetite (Fe3O4), cobalt ferrite (CoFe2O4) and Manganese-zinc (Mn-Zn) ferrite. A partial differential equation mathematical model is developed for mass, momentum, magnetic field continuity (induction), and energy with appropriate wall and free stream boundary conditions. Following similarity transformations, the dimensionless resultant nonlinear ordinary differential boundary value problem is solved numerically using the robust bvp4c function in MATLAB which features very efficient 4th order optimized Runge–Kutta quadrature. Dual solutions for the upper branch and lower branch separated by a critical point are identified. Visualization of velocity, temperature, and induced magnetic field function are presented graphically including validation of solutions with previous studies. Furthermore, skin-friction coefficient and the local Nusselt number are also computed. The impact of the controlling parameters, i.e. Prandtl number nanoparticle volume fraction parameter reciprocal of magnetic Prandtl number magnetic parameter and stretching rate ratio parameter have been illustrated through graphs and evaluated when a desire heat transfer can occur. Furthermore, resistance between fluid and the plate can be increased with the growing magnetic Prandtl number values. Increment in magnetic parameter () produces an elevation in the induced magnetic field magnitudes. Skin friction and Nusselt number are found to be greater for cobalt nanoparticles when compared to magnetite and Mn-Zn ferromagnetic nanoparticles when there is an increase in reciprocal magnetic Prandtl number. The simulations provide a deeper insight into the manufacturing flows of functional nano-ferromagnetic materials of relevance to deposition and coating systems.

Error estimates of Hermite‐Hadamard type inequalities with respect to a monotonically increasing function

Fractional calculus is used to examine and enhance the concept of calculus in diverse fields of science. In this paper, we establish Hermite‐Hadamard inequalities for composite ‐convex function. The generalized identities are established for Riemann‐type fractional integrals. The explored identities are used to examine error estimates of Hermite‐Hadamard inequalities for ‐convex function concerning a strictly monotone function. Some results that exist in literature are obtained as special cases to our general results. The conclusions of this article may be useful in determining the uniqueness of partial differential equations and fractional boundary value problems.

Adaptation of the Novel Cubic B-Spline Algorithm for Dealing with Conformable Systems of Differential Boundary Value Problems concerning Two Points and Two Fractional Parameters

Recently, conformable calculus has appeared in many abstract uses in mathematics and several practical applications in engineering and science. In addition, many methods and numerical algorithms have been adapted to it. In this paper, we will demonstrate, use, and construct the cubic B-spline algorithm to deal with conformable systems of differential boundary value problems concerning two points and two fractional parameters in both regular and singular types. Here, several linear and nonlinear examples will be presented, and a model for the Lane-Emden will be one of the applications presented. Indeed, we will show the complete construction of the used spline through the conformable derivative along with the convergence theory, and the error orders together with other results that we will present in detail in the form of tables and graphs using Mathematica software. Through the results we obtained, it became clear to us that the spline approach is effective and fast, and it requires little compulsive and mathematical burden in solving the problems presented. At the end of the article, we presented a summary that contains the most important findings, what we calculated, and some future suggestions.

Homotopy analysis of mixed convection flow of a magnetized viscoelastic nanofluid from a stretching surface in non-Darcy porous media with revised Fourier and Fickian approaches

This article addresses theoretically the mixed convection hydromagnetic flow of electrically conducting viscoelastic nanofluid from a vertical permeable stretching sheet in a non-Darcy porous medium with Cattaneo–Christov double-diffusion model to evaluate the heat transfer phenomena in steady boundary layer flow on a stretchable surface. In this regard, thermal and solutal hyperbolic wave relaxation effects are included for non-Fourier and non-Fickian models. Heat absorption is also included in the analysis. The Buongiorno two-component nanoscale model is adopted for simulating Brownian motion and thermophoretic body force effects. A non-Darcy drag force model is employed for the porous medium and the Reiner–Rivlin second-grade model for non-Newtonian characteristics. Via appropriate dimensionless similarity variables, the nonlinear dimensional partial differential conservation equations for momentum, energy and concentration with associated boundary conditions are rendered into a nonlinear dimensionless ordinary differential boundary value problem. The homotopy analysis method (HAM) is utilized to solve the boundary value problem and the impact of emerging parameters including thermal relaxation parameters, non-Newtonian material parameter, Darcy permeability parameter, porous inertial parameter and Hartmann magnetic number on the momentum, heat and mass transfer characteristics are visualized graphically and in tables. A 30th-order approximation for HAM is shown to produce sufficient accuracy for the velocity and temperature fields and a 40th-order estimates is adequate for the concentration field. It is observed that increasing the magnitude of thermal and solutal relaxation parameters has the opposite effect on the thermal and concentration distributions. The novelty of the work is the simultaneous inclusion of multiple effects (non-Fourier and non-Fickian thermal relaxation hyperbolic wave models, non-Darcy drag effects and heat generation/absorption) which are relevant to rheological nanomaterials processing and also the deployment of homotopy analysis as an alternative to conventional numerical methods such as finite differences, finite elements and MATLAB solvers. The study is relevant to the manufacture of electro-conductive polymers (ECPs) and smart (functional) magnetic nano-liquids.

Computation of swirling hydromagnetic nanofluid flow containing gyrotactic microorganisms from a spinning disk to a porous medium with hall current and anisotropic slip effects

AbstractPrompted by the advancements in hybrid bio‐nano‐swirling magnetic bioreactors, a mathematical model for the swirling flow from a rotating disk bioreactor to a magnetic fluid saturating a porous matrix and containing nanoparticles and gyrotactic micro‐organisms has been developed. An axial magnetic field is administered which is perpendicular to the disk and Hall currents are included. The disk is assumed to be impervious and stretches in the radial direction with a power‐law velocity. The Buongiorno nanoscale, Kuznetsov bioconvection and Darcy porous media models are deployed. Anisotropic momentum, thermal, nanoparticle concentration and motile micro‐organism slip effects are incorporated. Stefan blowing is also simulated. The governing conservation equations are transformed with appropriate variables to ordinary nonlinear differential equations. MATLAB bvp4c shooting quadrature is used to solve the emerging nonlinear, coupled ordinary differential boundary value problem under transformed boundary conditions. Verification with earlier solutions for the non‐magnetic Von Karman bioconvection nanofluid case is conducted. Further validation of the general magnetic model is conducted with the Adomian decomposition method (ADM). Extensive visualization of velocity, temperature, nanoparticle concentration and motile microorganism density number profiles is presented for the impact of various parameters including magnetic interaction parameter, Hall current parameter, Darcy number, momentum slip, thermal slip, nanoparticle slip and microorganism slip. Computations are also performed for skin friction, Nusselt number, Sherwood number and motile micro‐organism density number gradient. The simulations provide a useful benchmark for further studies.

$\alpha$-$(h,e)$-convex operators and applications for Riemann-Liouville fractional differential equations

In this paper, we consider a class of $\alpha$-$(h,e)$-convex operators defined in set $P_{h,e}$ and applications with $\alpha&gt; 1$. Without assuming the operator to be completely continuous or compact, by employing cone theory and monotone iterative technique, we not only obtain the existence and uniqueness of fixed point of $\alpha$-$(h,e)$-convex operators, but also construct two monotone iterative sequences to approximate the unique fixed point. At last, we investigate the existence-uniqueness of a nontrivial solution for Riemann-Liouville fractional differential equations integral boundary value problems by employing $\alpha$-$(h,e)$-convex operators fixed point theorem.