Showing papers on "Boundary value problem published in 2010"
TL;DR: In this paper, statistics obtained from seven different direct numerical simulations (DNSs) pertaining to a canonical turbulent boundary layer (TBL) under zero pressure gradient are compiled and compared, and the resulting comparison shows surprisingly large differences not only in both basic integral quantities such as the friction coefficient or the shape factor H12, but also in their predictions of mean and fluctuation profiles far into the sublayer.
Abstract: Statistics obtained from seven different direct numerical simulations (DNSs) pertaining to a canonical turbulent boundary layer (TBL) under zero pressure gradient are compiled and compared. The considered data sets include a recent DNS of a TBL with the extended range of Reynolds numbers Reθ = 500–4300. Although all the simulations relate to the same physical flow case, the approaches differ in the applied numerical method, grid resolution and distribution, inflow generation method, boundary conditions and box dimensions. The resulting comparison shows surprisingly large differences not only in both basic integral quantities such as the friction coefficient cf or the shape factor H12, but also in their predictions of mean and fluctuation profiles far into the sublayer. It is thus shown that the numerical simulation of TBLs is, mainly due to the spatial development of the flow, very sensitive to, e.g. proper inflow condition, sufficient settling length and appropriate box dimensions. Thus, a DNS has to be considered as a numerical experiment and should be the subject of the same scrutiny as experimental data. However, if a DNS is set up with the necessary care, it can provide a faithful tool to predict even such notoriously difficult flow cases with great accuracy.
752 citations
TL;DR: In this article, sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problems for fractional differential equations and inclusions involving the Caputo fractional derivative are established.
Abstract: In this survey paper, we shall establish sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problem for fractional differential equations and inclusions involving the Caputo fractional derivative. The both cases of convex and nonconvex valued right hand side are considered. The topological structure of the set of solutions is also considered.
742 citations
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12 Nov 2010TL;DR: In this paper, the authors transformed strongly elliptic boundary value problems of second order in domains \( \Omega \subset \mathbb{R}^3\) into boundary integral equations.
Abstract: In Chap. 3 we transformed strongly elliptic boundary value problems of second order in domains \( \Omega \subset \mathbb{R}^3\) into boundary integral equations. These integral equations were formulated as variational problems on a Hilbert space H:
713 citations
TL;DR: In this article, the authors considered nonnegative solutions of the Neumann boundary value problem for the chemotaxis system in a smooth bounded convex domain, where τ > 0, χ ∈ ℝ and f is a smooth function generalizing the logistic source.
Abstract: We consider nonnegative solutions of the Neumann boundary value problem for the chemotaxis system in a smooth bounded convex domain Ω ⊂ ℝ n , n ≥ 1, where τ > 0, χ ∈ ℝ and f is a smooth function generalizing the logistic source It is shown that if μ is sufficiently large then for all sufficiently smooth initial data the problem possesses a unique global-in-time classical solution that is bounded in Ω × (0, ∞). Known results, asserting boundedness under the additional restriction n ≤ 2, are thereby extended to arbitrary space dimensions.
607 citations
TL;DR: In this paper, a model-based description of the scaling and radial location of turbulent fluctuations in turbulent pipe flow is presented and used to illuminate the scaling behavior of the very large scale motions.
Abstract: A model-based description of the scaling and radial location of turbulent fluctuations in turbulent pipe flow is presented and used to illuminate the scaling behaviour of the very large scale motions. The model is derived by treating the nonlinearity in the perturbation equation (involving the Reynolds stress) as an unknown forcing, yielding a linear relationship between the velocity field response and this nonlinearity. We do not assume small perturbations. We examine propagating helical velocity response modes that are harmonic in the wall-parallel directions and in time, permitting comparison of our results to experimental data. The steady component of the velocity field that varies only in the wall-normal direction is identified as the turbulent mean profile. A singular value decomposition of the resolvent identifies the forcing shape that will lead to the largest velocity response at a given wavenumber–frequency combination. The hypothesis that these forcing shapes lead to response modes that will be dominant in turbulent pipe flow is tested by using physical arguments to constrain the range of wavenumbers and frequencies to those actually observed in experiments. An investigation of the most amplified velocity response at a given wavenumber–frequency combination reveals critical-layer-like behaviour reminiscent of the neutrally stable solutions of the Orr–Sommerfeld equation in linearly unstable flow. Two distinct regions in the flow where the influence of viscosity becomes important can be identified, namely wall layers that scale with R+1/2 and critical layers where the propagation velocity is equal to the local mean velocity, one of which scales with R+2/3 in pipe flow. This framework appears to be consistent with several scaling results in wall turbulence and reveals a mechanism by which the effects of viscosity can extend well beyond the immediate vicinity of the wall. The model reproduces inner scaling of the small scales near the wall and an approach to outer scaling in the flow interior. We use our analysis to make a first prediction that the appropriate scaling velocity for the very large scale motions is the centreline velocity, and show that this is in agreement with experimental results. Lastly, we interpret the wall modes as the motion required to meet the wall boundary condition, identifying the interaction between the critical and wall modes as a potential origin for an interaction between the large and small scales that has been observed in recent literature as an amplitude modulation of the near-wall turbulence by the very large scales.
594 citations
01 Jan 2010
TL;DR: The preprint version of this paper has different page and line numbers from the final version which appeared at Springer-Verlag as mentioned in this paper, which can be found on their personal web pages.
Abstract: Page and line numbers refer to the final version which appeared at Springer-Verlag. The preprint version, which can be found on our personal web pages, has different page and line numbers.
510 citations
Book•
24 Nov 2010
TL;DR: In this article, the authors introduce the concept of thermal expansion in pipes, and show that thermal expansion can be expressed as a combination of two-dimensional problems: 1.1 Steady State One-Dimensional Problems (Radial Flow) and 2.2 Steady-State Two-dimensional Problems 3.3 Transient Problems 3 Problems in Cylindrical coordinates 3.4 Transient problems 4 Problems in Spherical Coordinates 4.5 Bessel Functions and Fourier-Bessel series 2.6 Nonhomogeneous Differential Equations and Boundary Condition 2
Abstract: 1 Basic Laws of Thermoelasticity 1 Introduction 2 Stresses and Tractions 3 Equations of Motion 4 Coordinate Transformation. Principal Axes 5 Principal Stresses and Stress Invariants 6 Displacement and Strain Tensor 7 Compatibility Equations. Simply Connected Region 8 Compatibility Conditions. Multiply Connected Regions 9 Constitutive Laws of Linear Thermoelasticity 10 Displacement Formulation of Thermoelasticity 11 Stress Formulation of Thermoelasticity 12 Two-Dimensional Thermoelasticity 13 Michell conditions 2 Thermodynamics of Elastic Continuum 1 Introduction 2 Thermodynamics Definitions 3 First Law of Thermodynamics 4 Second Law of Thermodynamics 5 Variational Formulation of Thermodynamics 6 Thermodynamics of Elastic Continuum 7 General Theory of Thermoelasticity 8 Free Energy Function of Hookean Materials 9 Fourier's Law and Heat Conduction Equation 10 Generalized Thermoelasticity. Second Sound 11 Thermoelasticity without Energy Dissipation 12 Uniqueness Theorem 13 Variational Principle of Thermoelasticity 14 Reciprocity Theorem 15 Initial and Boundary Conditions 3 Basic Problems of Thermoelasticity 1 Introduction 2 Temperature Distribution for Zero Thermal Stress 3 Analogy of Thermal Gradient with Body Forces 4 General Solution of Thermoelastic Problems 5 General Solution in Cylindrical Coordinates 6 Solution of Two-Dimensional Navier Equations 7 Solution of Problems in Spherical Coordinates 4 Heat Conduction Problems 1 Introduction 2 Problems in Rectangular Cartesian Coordinates 2.1 Steady State One-Dimensional Problems 2.2 Steady Two-Dimensional Problems. Separation of Variables 2.3 Fourier Series 2.4 Double Fourier Series 2.5 Bessel Functions and Fourier-Bessel series 2.6 Nonhomogeneous Differential Equations and Boundary Condition 2.7 Lumped Formulation 2.8 Steady State Three-Dimensional Problems 2.9.Transient Problems 3 Problems in Cylindrical coordinates 3.1 Steady-State One-Dimensional Problems (Radial Flow) 3.2. Steady -State Two-Dimensional Problems 3.3 Steady-State Three-Dimensional Problems 3.4 Transient Problems 4 Problems in Spherical Coordinates 4.1 Steady-State One-Dimensional Problems 4.2 Steady-State Two- and Three-Dimensional Problems 4.3 Transient Problems 5 Thermal Stresses in Beams 1 Introduction 2 Elementary Theory of Thermal Stresses in Beams 3 Deflection Equation of Beams 4 Boundary Conditions 5 Shear Stress in a Beam 6 Beams of Rectangular Cross Section 7 Transient Thermal Stresses in Rectangular Beams 8 . Beam with Internal Heat Generation 9 Bimetallic Beam 10 Functionally Graded Beams 11 Transient Thermal Stresses in Functionally Graded Beams 12 Thermal Stresses in Thin Curved Beams and Rings 13 Deflection of Thin Curved Beams and Rings 6 Disks, Cylinders, and Spheres 1 Introduction 2 . Cylinders with Radial Temperature Variation 3 Thermal Stresses in Disks 4 Thick Spheres 5 Thermal Stresses in a Rotating Disk 6 Non-Axisymmetrically Heated cylinders 7 Method of Complex Variables 8 Functionally Graded Thick Cylinders 9 Axisymmetric Thermal Stresses in FGM Cylinders 10.Transient Thermal Stresses in Thick Spheres 11 Functionality Graded Spheres 7 Thermal Expansion in Piping Systems 1 Introduction 2 Definition of the Elastic Center 3 . Piping Systems in Two Dimensions 4 Piping Systems in Three-dimensions 5 Pipelines with Large Radius Elbows 8 Coupled and Generalized Thermoelasticity 1 Introduction 2 Governing Equations of Coupled Thermoelasticity 3 Coupled Thermoelasticity for Infinite Space 4 Variable Heat Source 5 One-Dimensional Coupled Problem 6 Propagation of Discontinuities 7 Half-Space Subjected to a Harmonic Temperature 8 Coupled Thermoelasticity of Thick Cylinders 9 Green-Naghdi Model of a Layer 10 Generalized Thermoelasticity of Layers 11 Generalized Thermoelasticity of Spheres and Cylinders
478 citations
TL;DR: In this paper, the existence and uniqueness of positive solutions for a nonlocal boundary value problem of fractional differential equation was investigated, and the uniqueness of the positive solution was obtained by the use of contraction map principle and some Lipschitz type conditions.
Abstract: In this paper, we investigate the existence and uniqueness of positive solutions for a nonlocal boundary value problem of fractional differential equation. Firstly, we give Green’s function and prove its positivity; secondly, the uniqueness of positive solution is obtained by the use of contraction map principle and some Lipschitz-type conditions; thirdly, by means of the fixed point index theory, we obtain some existence results of positive solution. The proofs are based upon the reduction of the problem considered to the equivalent Fredholm integral equation of second kind.
440 citations
TL;DR: In this article, a nonlinear size-dependent Timoshenko beam model based on modified couple stress theory is presented, a non-classical continuum theory capable of capturing the size effects.
381 citations
TL;DR: In this article, the fundamental frequency analysis of functionally graded (FG) beams having different boundary conditions is analyzed within the framework of the classical, the first-order and different higher-order shear deformation beam theories.
373 citations
TL;DR: It is shown that each of them, the total, the adiabatic, and the nonadiabatic trajectory entropy, separately satisfies a detailed fluctuation theorem.
Abstract: The total entropy production of a trajectory can be split into an adiabatic and a nonadiabatic contribution, deriving, respectively, from the breaking of detailed balance via nonequilibrium boundary conditions or by external driving. We show that each of them, the total, the adiabatic, and the nonadiabatic trajectory entropy, separately satisfies a detailed fluctuation theorem.
TL;DR: In this paper, the authors describe the methodology of continuum variational and diffusion quantum Monte Carlo calculations, which are based on many-body wavefunctions and are capable of achieving very high accuracy.
Abstract: This topical review describes the methodology of continuum variational and diffusion quantum Monte Carlo calculations. These stochastic methods are based on many-body wavefunctions and are capable of achieving very high accuracy. The algorithms are intrinsically parallel and well suited to implementation on petascale computers, and the computational cost scales as a polynomial in the number of particles. A guide to the systems and topics which have been investigated using these methods is given. The bulk of the article is devoted to an overview of the basic quantum Monte Carlo methods, the forms and optimization of wavefunctions, performing calculations under periodic boundary conditions, using pseudopotentials, excited-state calculations, sources of calculational inaccuracy, and calculating energy differences and forces.
01 Jan 2010
TL;DR: In this article, the authors proposed three standard variational iteration algorithms for solving differential equations, integro-differential equations, fractional differential equations and differential-difference equations.
Abstract: This paper proposes three standard variational iteration algorithms for solving differential equations, integro-differential equations, fractional differential equations, fractal differential equations, differential-difference equations and fractional/fractal differential-difference equations. The physical interpretations of the fractional calculus and the fractal derivative are given and an application to discrete lattice equations is discussed. The paper then examines the acceleration of some iteration formulae with particular emphasis being placed on the exponential Pade approximant that is suggested for solitary solutions and the sinusoidal Pade approximant that is usually used for periodic and compacton solutions. The paper points out that there may not be any physical meaning to the exact solutions of many nonlinear equations and stresses the importance of searching for approximate solutions that satisfy both the equations and the appropriate initial/boundary conditions. The variational iteration method is particularly suitable for solving this kind of problems. Approximate initial/boundary conditions and point boundary initial/conditions are also discussed, with the variational iteration method being capable of recovering the correct initial/boundary conditions and finding the solutions simultaneously.
01 Jan 2010
TL;DR: A unified approach to the method of Nehari manifold for functionals which have a local minimum at 0 is presented in this article, where several examples where this method is applied to the problem of finding ground states and multiple solutions for nonlinear elliptic boundary value problems.
Abstract: We present a unified approach to the method of Nehari manifold for functionals which have a local minimum at 0 and we give several examples where this method is applied to the problem of finding ground states and multiple solutions for nonlinear elliptic boundary value problems. We also consider a recent generalization of this method to problems where 0 is a saddle point of the functional.
TL;DR: In this article, a peridynamic model for transient heat and mass transfer in fracture-prone bodies is proposed, which is valid when the body undergoes damage or evolving cracks.
01 Jan 2010
TL;DR: The objective of this paper is to provide the theoretical background and illustrative examples for elastic buckling analysis of cold-formed steel members with general boundary conditions as implemented in the forthcoming update to CUFSM.
Abstract: The objective of this paper is to provide the theoretical background and illustrative examples for elastic buckling analysis of cold-formed steel members with general boundary conditions as implemented in the forthcoming update to CUFSM. CUFSM is an open source finite strip elastic stability analysis program freely distributed by the senior author. Although the finite strip method presents a general methodology, the conventional implementation (e.g. CUFSM v 3.13 or earlier) employs only simply-supported boundary conditions. In this paper, utilizing specially selected longitudinal shape functions, the conventional finite strip method is extended to general boundary conditions, including the conventional case: simply-simply supported, as well as: clamped-clamped, clamped-simply supported, clamped-free, and clamped-guided. The solution remains semi-analytical as the elastic and geometric stiffness matrices are derived in a general form with only specific integrals depending on the boundary conditions. An example of the stability solution is provided. The selection of longitudinal terms to be included in the analysis is discussed in terms of balancing accuracy with computational efficiency. Also herein, the constrained finite strip method is extended to general boundary conditions. Both modal decomposition and identification can be carried out based on the new bases developed for the constrained finite strip method, and illustrative examples are provided. This extension of CUFSM is intended to aid the implementation of the direct strength method to the case of general boundary conditions.
TL;DR: In this article, the authors discuss properties of a variational interface problem involving the fractional Laplacian, including optimal regularity, non-degeneracy, and smoothness of the free boundary.
Abstract: We discuss properties (optimal regularity, non-degeneracy, smoothness of the free boundary...) of a variational interface problem involving the fractional Laplacian; Due to the non-locality of the Dirichlet problem, the task is nontrivial. This difficulty is by-passed by an extension formula, discovered by the first author and Silvestre, which reduces the study to that of a co-dimension 2 (degenerate) free boundary.
TL;DR: An asymptotic analysis of the model equation with boundary rules for the Dirichlet and Neumann-type (specified flux) conditions is carried out to show that the model is first- and second-order accurate in time and space, respectively.
TL;DR: In this article, a hierarchical scheme is obtained by extending plates and shells Carrera's Unified Formulation (CUF) to beam structures and an N-order approximation via Mac Laurin's polynomials is assumed on the cross-section for the displacement unknown variables.
Abstract: This paper proposes several axiomatic refined theories for the linear static analysis of beams made of isotropic materials. A hierarchical scheme is obtained by extending plates and shells Carrera's Unified Formulation (CUF) to beam structures. An N-order approximation via Mac Laurin's polynomials is assumed on the cross-section for the displacement unknown variables. N is a free parameter of the formulation. Classical beam theories, such as Euler-Bernoulli's and Timoshenko's, are obtained as particular cases. According to CUF, the governing differential equations and the boundary conditions are derived in terms of a fundamental nucleo that does not depend upon the approximation order. The governing differential equations are solved via the Navier type, closed form solution. Rectangular and I-shaped cross-sections are accounted for. Beams undergo bending and torsional loadings. Several values of the span-to-height ratio are considered. Slender as well as deep beams are analysed. Comparisons with reference solutions and three-dimensional FEM models are given. The numerical investigation has shown that the proposed unified formulation yields the complete three-dimensional displacement and stress fields for each cross-section as long as the appropriate approximation order is considered. The accuracy of the solution depends upon the geometrical parameters of the beam and loading conditions.
TL;DR: In this article, three different schemes of adding force term to LBM with BGK method were evaluated and compared with results predicted by using finite volume method (FVM) for Ra = 10 6 and Pr = 0.71.
TL;DR: It is shown that the RBVMS formulation globally conserves angular momentum, a feature that is felt to be important for flows dominated by rotation, and that is not shared by standard stabilized formulations of fluid flow.
TL;DR: In this paper, the authors investigated the initial boundary value problems of the spacetime fractional diffusion equation and its numerical solutions and proposed a weak formulation of the weak solution, which is based on the Riemann-Liouville definition and the Caputo definition of the fractional derivative.
Abstract: . In this paper, we investigate initial boundary value problems of the spacetime fractional diffusion equation and its numerical solutions. Two definitions, i.e., Riemann-Liouville definition and Caputo one, of the fractional derivative are considered in parallel. In both cases, we establish the well-posedness of the weak solution. Moveover, based on the proposed weak formulation, we construct an efficient spectral method for numerical approximations of the weak solution. The main contribution of this work are threefold: First, a theoretical framework for the variational solutions of the space-time fractional diffusion equation is developed. We find suitable functional spaces and norms in which the space-time fractional diffusion problem can be formulated into an elliptic weak problem, and the existence and uniqueness of the weak solution are then proved by using existing theory for elliptic problems. Secondly, we show that in the case of Riemann-Liouville definition, the well-posedness of the space-time fractional diffusion equation does not require any initial conditions. This contrasts with the case of Caputo definition, in which the initial condition has to be integrated into the weak formulation in order to establish the well-posedness. Finally, thanks to the weak formulation, we are able to construct an efficient numerical method for solving the space-time fractional diffusion problem. AMS subject classifications: 35S10, 35A05, 65M70, 65M12
TL;DR: In this paper, a general mesh-free description of the magnetic field distribution in various electromagnetic machines, actuators, and devices is presented, based on transfer relations and Fourier theory.
Abstract: We present a general mesh-free description of the magnetic field distribution in various electromagnetic machines, actuators, and devices. Our method is based on transfer relations and Fourier theory, which gives the magnetic field solution for a wide class of two-dimensional (2-D) boundary value problems. This technique can be applied to rotary, linear, and tubular permanent-magnet actuators, either with a slotless or slotted armature. In addition to permanent-magnet machines, this technique can be applied to any 2-D geometry with the restriction that the geometry should consist of rectangular regions. The method obtains the electromagnetic field distribution by solving the Laplace and Poisson equations for every region, together with a set of boundary conditions. Here, we compare the method with finite-element analyses for various examples and show its applicability to a wide class of geometries.
TL;DR: In this article, the authors combine NURBS-based isogeometric analysis, residual-driven turbulence modeling and weak imposition of no-slip and no-penetration Dirichlet boundary conditions on unstretched meshes to compute wall-bounded turbulent flows.
TL;DR: An approach for prescribing lumped parameter outflow boundary conditions that accommodate transient phenomena is presented and applied to compute haemodynamic quantities in different physiologically relevant cardiovascular models to study non-periodic flow phenomena often observed in normal subjects and in patients with acquired or congenital cardiovascular disease.
Abstract: The simulation of blood flow and pressure in arteries requires outflow boundary conditions that incorporate models of downstream domains. We previously described a coupled multidomain method to couple analytical models of the downstream domains with 3D numerical models of the upstream vasculature. This prior work either included pure resistance boundary conditions or impedance boundary conditions based on assumed periodicity of the solution. However, flow and pressure in arteries are not necessarily periodic in time due to heart rate variability, respiration, complex transitional flow or acute physiological changes. We present herein an approach for prescribing lumped parameter outflow boundary conditions that accommodate transient phenomena. We have applied this method to compute haemodynamic quantities in different physiologically relevant cardiovascular models, including patient-specific examples, to study non-periodic flow phenomena often observed in normal subjects and in patients with acquired or congenital cardiovascular disease. The relevance of using boundary conditions that accommodate transient phenomena compared with boundary conditions that assume periodicity of the solution is discussed.
TL;DR: In this article, the authors studied the weak enforcement of Dirichlet boundary conditions for B-spline basis functions, with application to both second-and fourth-order problems.
Abstract: A key challenge while employing non-interpolatory basis functions in finite-element methods is the robust imposition of Dirichlet boundary conditions. The current work studies the weak enforcement of such conditions for B-spline basis functions, with application to both second- and fourth-order problems. This is achieved using concepts borrowed from Nitsche's method, which is a stabilized method for imposing constraints on surfaces. Conditions for the stability of the system of equations are derived for each class of problem. Stability parameters in the Nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the Dirichlet boundary. The approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not. Through several numerical examples, the approach is shown to yield optimal rates of convergence. Copyright © 2010 John Wiley & Sons, Ltd.
TL;DR: In this article, the nonlinear free vibration of single-walled carbon nanotubes (SWCNTs) is studied based on von Karman geometric nonlinearity and Eringen's nonlocal elasticity theory, where the effects of transverse shear deformation and rotary inertia are considered within the framework of Timoshenko beam theory.
Abstract: Nonlinear free vibration of single-walled carbon nanotubes (SWCNTs) is studied in this paper based on von Karman geometric nonlinearity and Eringen's nonlocal elasticity theory. The SWCNTs are modeled as nanobeams where the effects of transverse shear deformation and rotary inertia are considered within the framework of Timoshenko beam theory. The governing equations and boundary conditions are derived by using the Hamilton's principle. The differential quadrature (DQ) method is employed to discretize the nonlinear governing equations which are then solved by a direct iterative method to obtain the nonlinear vibration frequencies of SWCNTs with different boundary conditions. Zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs are considered in numerical calculations and the elastic modulus is obtained through molecular mechanics (MM) simulation. A detailed parametric study is conducted to study the influences of nonlocal parameter, length and radius of the SWCNTs and end supports on the nonlinear free vibration characteristics of SWCNTs.
Book•
02 Dec 2010
TL;DR: In this article, the authors propose models of higher order models of linear problems, eigenvalue problems, kernel estimates, positivity and lower order perturbations, and the dominance of positivity in linear equations.
Abstract: Models of Higher Order.- Linear Problems.- Eigenvalue Problems.- Kernel Estimates.- Positivity and Lower Order Perturbations.- Dominance of Positivity in Linear Equations.- Semilinear Problems.- Willmore Surfaces of Revolution.
TL;DR: A variational method EnVarA (energy variational analysis) is used that combines Hamilton's least action and Rayleigh's dissipation principles to create a variational field theory that includes flow, friction, and complex structure with physical boundary conditions of ions next to a charged wall.
Abstract: Ionic solutions are mixtures of interacting anions and cations. They hardly resemble dilute gases of uncharged noninteracting point particles described in elementary textbooks. Biological and electrochemical solutions have many components that interact strongly as they flow in concentrated environments near electrodes, ion channels, or active sites of enzymes. Interactions in concentrated environments help determine the characteristic properties of electrodes, enzymes, and ion channels. Flows are driven by a combination of electrical and chemical potentials that depend on the charges, concentrations, and sizes of all ions, not just the same type of ion. We use a variational method EnVarA (energy variational analysis) that combines Hamilton’s least action and Rayleigh’s dissipation principles to create a variational field theory that includes flow, friction, and complex structure with physical boundary conditions. EnVarA optimizes both the action integral functional of classical mechanics and the dissipation functional. These functionals can include entropy and dissipation as well as potential energy. The stationary point of the action is determined with respect to the trajectory of particles. The stationary point of the dissipation is determined with respect to rate functions (such as velocity). Both variations are written in one Eulerian (laboratory) framework. In variational analysis, an “extra layer” of mathematics is used to derive partial differential equations. Energies and dissipations of different components are combined in EnVarA and Euler–Lagrange equations are then derived. These partial differential equations are the unique consequence of the contributions of individual components. The form and parameters of the partial differential equations are determined by algebra without additional physical content or assumptions. The partial differential equations of mixtures automatically combine physical properties of individual (unmixed) components. If a new component is added to the energy or dissipation, the Euler–Lagrange equations change form and interaction terms appear without additional adjustable parameters. EnVarA has previously been used to compute properties of liquid crystals, polymer fluids, and electrorheological fluids containing solid balls and charged oil droplets that fission and fuse. Here we apply EnVarA to the primitive model of electrolytes in which ions are spheres in a frictional dielectric. The resulting Euler–Lagrange equations include electrostatics and diffusion and friction. They are a time dependent generalization of the Poisson–Nernst–Planck equations of semiconductors, electrochemistry, and molecular biophysics. They include the finite diameter of ions. The EnVarA treatment is applied to ions next to a charged wall, where layering is observed. Applied to an ion channel, EnVarA calculates a quick transient pile-up of electric charge, transient and steady flow through the channel, stationary “binding” in the channel, and the eventual accumulation of salts in “unstirred layers” near channels. EnVarA treats electrolytes in a unified way as complex rather than simple fluids. Ad hoc descriptions of interactions and flow have been used in many areas of science to deal with the nonideal properties of electrolytes. It seems likely that the variational treatment can simplify, unify, and perhaps derive and improve those descriptions.
TL;DR: This paper obtains the existence and multiplicity results of positive solutions of fractional order derivative of D 0 + α by using some fixed point theorems.
Abstract: In this paper, we are concerned with the nonlinear differential equation of fractional order D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 t 1 , 1 α ≤ 2 , where D 0 + α is the standard Riemann–Liouville fractional order derivative, subject to the boundary conditions u ( 0 ) = 0 , D 0 + β u ( 1 ) = a D 0 + β u ( ξ ) . We obtain the existence and multiplicity results of positive solutions by using some fixed point theorems.