Homogeneous Boundary Value Problem Research Articles

Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer

We consider boundary value problems for a nonlinear mass transfer model, which generalizes the classical Boussinesq approximation, under inhomogeneous Dirichlet boundary conditions for the velocity and the substance’s concentration. It is assumed that the leading coefficients of viscosity and diffusion and the buoyancy force in the model equations depend on concentration. We develop a mathematical apparatus for studying the inhomogeneous boundary value problems under consideration. It is based on using a weak solution of the boundary value problem and on the construction of liftings of the inhomogeneous boundary data. They remove the inhomogeneity of the data and reduce initial problems to equivalent homogeneous boundary value problems. Based on this apparatus we will prove the theorem of the global existence of a weak solution to the boundary value problem under study and establish important properties of the solution. In particular, we will prove the validity of the maximum principle for the substance’s concentration. We will also establish sufficient conditions for the problem data, ensuring the local uniqueness of weak solutions.

LYAPUNOV-TYPE INEQUALITY FOR CERTAIN HALF-LINEAR LOCAL FRACTIONAL ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we establish a Lyapunov-type inequality for the half-linear local fractional ordinary differential equation based on the formulation of the local fractional derivative. In addition, we apply the inequality to investigate the non-existence and uniqueness of solutions for related homogeneous and non-homogeneous local fractional boundary value problems.

EXACT SOLUTIONS TO INHOMOGENEOUS BOUNDARY VALUE PROBLEMS OF THE THEORY OF ELASTICITY IN A RECTANGLE

A method is proposed for constructing exact solutions to boundary value problems of the theory of elasticity in a rectangle with stiffeners located inside the region (an inhomogeneous problem). The solutions are represented by series in Papkovich–Fadle eigenfunctions, the coefficients of which are determined explicitly. The method is based on the Papkovich orthogonality relation and the theory of expansions in Papkovich–Fadle eigenfunctions developed by the authors in homogeneous boundary value problems of the theory of elasticity in a rectangle (the biharmonic problem). The solution sequence is demonstrated with the example of an even-symmetric problem for a rectangle the sides of which are free, and an external load acts along the stiffener located on the axis of symmetry of the rectangle.

Chemotaxis driven formation of time periodic pattern and steady state pattern

This paper is concerned with a chemotaxis-fluid model with realistic boundary conditions matching the experiments of Kessler et al. Different from the initial and boundary value problem, the existence of time periodic solution is proved under no any smallness assumptions on chemotactic coefficient. Subsequently, we investigate the reason of pattern formation. It is generally known that there is no pattern formation for the homogeneous boundary value problem. Therefore, the pattern formation is partly caused by non-homogeneous boundary conditions, but chemotaxis is also an indispensable condition. In fact, it is proved that the density of bacteria will tend to be evenly distributed when the chemotactic intensity tends to 0. At last, the idea of finding time periodic pattern is also applied to prove the steady-state pattern formation of chemotaxis-Navier-Stokes model.

Comparative Study of Guided Electromagnetic Wave Propagation for Two Models of an Open Tape Helix

Rigorous methods to solve the homogeneous boundary value problems and the techniques to resolve illposed mathematical issues arising in the problem formulation of electromagnetic wave propagation of an Open tape helix slow wave structure with anisotropically conducting and completely conducting model is presented. The dispersion equation of both the models require convergent coeefficients of the infinte linear homogeneous equations whose determinant of the coefficent matrix is zero. The coefficient matrix of the anisotropically conducting model is rapidly convergent, while in the case of completely conducting model, the matrix entries which are infinite summation series representations turns out to be divergent, the not so well-posed boundary value problem is regularised with the method of mollification functions. The dispersion characteristics are plotted after truncating the infinite series expansions to adequate number of terms and summing the terms of converging series expansions(after regularization in the case of the completely conducting tape-helix model) in dispersion equations. The tape-current distribution for both models are estimated from the null-space vector of the truncated coefficient matrix corresponding to a particular (β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> – <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ) root of the dispersion equation. A comparison of the numerical results for both models show that the neglect of the perpendicular tape-current density component entails a substantial modification of the dispersion characteristics of guided waves supported by an open tape helix.

The enhanced boundary knot method with fictitious sources for solving Helmholtz-type equations

The boundary knot method (BKM) is a boundary meshless method for solving homogeneous boundary value problems. The fact that the BKM uses non-singular general solution without the need of source points outside the domain is at the expense of the accuracy when compared to the well-known method of fundamental solutions. In this paper, a new approach called the enhanced BKM (EBKM) is proposed to greatly improve the accuracy of the BKM by introducing fictitious sources which can be placed inside and/or outside the domain quite freely. We also adopt various techniques to place these fictitious sources in an optimal way.

On solvability of homogeneous Riemann boundary value problems in Hardy-Orlicz classes

This work deals with the Orlicz space and the Hardy-Orlicz classes generated by this space, which consist of analytic functions inside and outside the unit disk. The homogeneous Riemann boundary value problems with piecewise continuous coefficients are considered in these classes. New characteristic of Orlicz space is defined which depends on whether the power function belongs to this space or not. Relationship between this characteristic and Boyd indices of Orlicz space is established. The concept of canonical solution of homogeneous problem is defined, which depends on the argument of the coefficient. In terms of the above characteristic, a condition on the jumps of the argument is found which is sufficient for solvability of these problems, and, in case of solvability, a general solution is constructed. It is established the basicity of the parts of exponential system in Hardy-Orlicz classes.

On a Dirichlet boundary value problem for an Ermakov–Painlevé I equation. A Hamiltonian EPI system

Here, a proto-type Ermakov–Painlevé I equation is introduced and a homogeneous Dirichlet-type boundary value problem analysed. In addition, a novel Ermakov–Painlevé I system is set down which is reducible by an involutory transformation to the autonomous Ermakov–Ray–Reid system augmented by a single component Ermakov–Painlevé I equation. Hamiltonian such systems are delimited

On the non-uniqueness of the solution to a boundary value problem of heat conduction with a load in the form of a fractional derivative

The paper deals with the second boundary value problem for the loaded heat equation in the first quadrant. The loaded term contains a fractional derivative in the Caputo sense of an order α, 2&lt;α&lt;3. The boundary value problem is reduced to an integro-differential equation with a difference kernel by inverting the differential part. It is proved that a homogeneous integro-differential equation has at least one non-zero solution. It is shown that the solution of the homogeneous boundary value problem corresponding to the original boundary value problem is not unique, and the load acts as a strong perturbation of the boundary value problem.

On Poncelet problem, Abel equation and a Goursat-type boundary value problem

This paper explains a connection between the famous Poncelet problem, the Abel equation, a new moment problem, problems of solution existence and uniqueness of boundary value problems for the string equation and some others in short presentation. We prove a connection of above problem with one problem of Goursat-type for a hyperbolic-like equation of fourth order. We consider a fourth-order differential equation with two multiple characteristics and parameters in the lower terms and a Goursat-type homogeneous boundary value problem with two real boundary conditions in some space of real entire functions. It is proven that this problem has a nontrivial solution in some space if and only if a parameter belongs denumerable dense set of real number.

Influence of internal stresses on stability of multilayer micropolar tubes

Within the nonlinear Cosserat continuum, the stability of a compressed multilayer tube under internal and external hydrostatic pressures is studied. The tube is formed by attaching to each other N hollow cylinders that have been subjected to preliminary deformations of axial extension-compression and contains internal stresses. For the model of a physically linear micropolar material, the system of linearized equilibrium equations is derived, which describes the behavior of a multilayer cylindrical tube in a perturbed state. Using a special substitution, the stability analysis is reduced to solving a linear homogeneous boundary-value problem for a system of 6N ordinary differential equations. In the case of a three-layer tube made of dense polyurethane foam, the stability regions were constructed in the planes of loading parameters (the relative axial compression and the relative external or internal pressure) for different preliminary deformations of inner and outer layers (coatings). According to the results obtained, the preliminary extension of both the outer and inner coatings generally stabilizes the considered deformations of the three-layer tube, while the effect of their preliminary compression is negative. It should be noted here that the pre-stressed outer coating has a more significant influence on the tube stability than the pre-stressed inner coating. In addition, the described effects of preliminary deformations are more pronounced for tubes with thicker coatings.

Modelling and investigation of the stress state in a finite cylinder with a circular crack under sudden torsional loading

This paper is devoted to the analysis of the dynamic stress state of a finite cylinder with an internal circular crack. One of the cylinder’s bases is fixed, and a time-dependent torque is applied to the other base. For solving this problem, a modified finite difference method was applied. This method consists in the difference approximation of only the time derivative. As a result, the original problem is reduced to a set of sequentially solved homogeneous boundary value problems. Their solutions contain an unknown displacement jump on the crack. The integral equation for the unknown jump is reduced to the Fredholm integral equation of the second kind, which was solved numerically. This numerical solution made it possible to obtain an approximate formula for calculating the stress intensity factor (SIF).

Sonic-supersonic solutions for the two dimensional steady relativistic Euler equations

In this paper, we study the existence of the sonic-supersonic solution for the 2D steady isentropic relativistic Euler equations. Employing the characteristic decomposition of two dependent variables, 2D relativistic Euler equations are transformed into the first-order hyperbolic equations. Based on the first-order hyperbolic equations and the transformation (q2−a2,Φ) introduced by Zhang and Zheng (2014), we transform the problem into a homogeneous boundary value problem. Utilizing the method of iteration, the existence of the classical solution is proved in (t,Φ) plane. Moreover, the sonic-supersonic solution is verified in (x,y) plane. Finally, we give the C1,16 regularity of sonic-supersonic solution near the sonic curve.

Ağırlıklı iç çarpım ile zaman kesirli problem

In this research, we discuss the construction of analytic solution of homogenous initial boundary value problem including PDEs of fractional order. Since homogenous initial boundary value problem involves Caputo fractional order derivative, it has classical initial and boundary conditions. By means of separation of variables method and the inner product defined on L^2 [0,l], the solution is constructed in the form of a Fourier series with respect to the eigenfunctions of a corresponding Sturm-Liouville eigenvalue problem including fractional derivative in Caputo sense used in this study. We defined a new inner product with a weighted function to get coefficients in the Fourier series. Illustrative example presents the applicability and influence of separation of variables method on fractional mathematical problems.

Non-local problem with an integral condition for a parabolic equation with a Bessel operator

For the parabolic equation with the Bessel operator ∂u/∂t=(∂^2 u)/(∂x^2 )+k/x ∂u/∂x in the rectangular domain 0&lt;x&lt;l, 0&lt;t≤T, we consider a boundary value problem with the non-local integral condition of the first kind ∫_0^l▒〖u(x,t)xdx=0, 0≤t≤T.〗 This problem is reduced to an equivalent boundary value problem with mixed boundary conditions of the first and third kind. It is shown that the homogeneous equivalent boundary value problem has only a trivial zero solution, and hence the original inhomogeneous problem cannot have more than one solution. This proof uses Gronwall’s lemma. Then, by the method of spectral analysis, the existence theorem for a solution to an equivalent problem is proved. This solution is defined explicitly in the form of a Dini series. Sufficient conditions with respect to the initial condition are obtained. These conditions guarantee the convergence of the constructed series in the class of regular solutions.

The nonstationary thermoelectric elasticity problem for a long piezoceramic cylinder

А new closed-loop solution for the coupled nonstationary problem of thermoelectric elasticity is designed for a long piezoceramic radially polarized cylinder. The case of the nonstationary load acting on its inner cylindrical surface is considered as a function of temperature change at a given law of the convection heat exchange on the outer face wall (boundary conditions of heat conductivity of the 1st and 3rd types). Electrodynamic cylinder surfaces are connected to a measuring device with a high input resistance (electric idling). We investigate the problem where the rate of the temperature load changes does not affect the inertial characteristics of the elastic system. It makes it possible to expand the initial linear computational relations with the equilibrium, electrostatics and heat conductivity equations with respect to the radial component of the displacement vector, electric potential as well as the function of temperature field changes. Hyperbolic LS-theory of the thermal conductivity is used in the computations. The problem is solved with a generalized method of biorthogonal finite integral transformation based on a multicomponent ratio of eigen functions of two homogeneous boundary value problems. The structural algorithm of this approach allows identifying a conjugated operator, without which it is impossible to solve non-self-conjugated linear problems in mathematical physics. The resulted computational relations make it possible to determine the stress-strain state, temperature and electric fields induced in the piezoceramic element under an arbitrary external temperature effect. By connecting the electroelastic system to the measuring tool, we can find voltage. Firstly, the analysis of the numerical results allows identifying the rate of the temperature load changes, at which it is necessary to use the hyperbolic theory of thermal conductivity. Secondly, it allows determining the physical characteristics of the piezoceramic material for the case when the rate of changing the body volume leads to a redistribution of the temperature field. The developed computational algorithm can be used to design non-resonant piezoelectric temperature sensors.

The Homogeneous Riemann Boundary Value Problem with General Continuous Coefficient on Rectifiable Open Curves

The Homogeneous Riemann Boundary Value Problem with General Continuous Coefficient on Rectifiable Open Curves

Diffusion coefficient calculated by time-lag using the film-roll method

A method for determining the diffusion coefficient by time-lag using the film-roll method for the sublimation diffusion of disperse dye was proposed. A polyethylene terephthalate film-roll coated with dye paste was treated at 170–190°C for various times. A solution consisting of the sum of a steady-state solution and a transient solution was obtained by the homogeneous boundary value problem from a trigonometrical series. The boundary conditions of the steady-state first layer and the steady-state first layer amount of dye were determined from the steady-state concentration distribution. For various diffusion times, the steady-state first layer-passed total amounts of dye that passed through the first layer in the steady-state condition were obtained by subtracting the steady-state first layer amounts from the total amounts. The time-lag was calculated from the linear regression line for the plot of the steady-state first layer-passed total number(X) of positive values against time. The diffusion coefficient was calculated by the boundary conditions of the steady-state first layer and the time-lag. For diffusion at 170°C, 180°C, and 190°C, the correlations of the steady-state first layer-passed total amounts with respect to time were very linear and the reliability of the diffusion coefficients obtained by the time-lag was proved by the good linearity of the Arrhenius plot. The activation energy obtained was 36.8 kcal[Formula: see text]mol−1.

The dynamical problem on acting concentrated load on the elastic quarter space

The wave field of an elastic quarter space is constructed when one face is rigidly fixed and a dynamic normal compressive load acts on the other along a rectangular section at the initial moment of time. Integral Laplace and Fourier transforms are applied sequentially to the equations of motion and boundary conditions in contrast to traditional approaches when integral transforms are applied to solutions' representations through harmonic functions. This leads to a one-dimensional vector homogeneous boundary value problem with respect to unknown displacement's transformants. The problem was solved using matrix differential calculus. The original displacement field was found after applying inverse integral transforms. For the case of stationary vibrations a method of calculating integrals in the solution in the near loading zone was indicated. For the analysis of oscillations in a remote zone the asymptotic formulas were constructed. The amplitude of vertical vibrations was investigated depending on the shape of the load section, natural frequencies of vibrations and the material of the medium.

Note on the conformable boundary value problems: Sturm’s theorems and Green’s function

Recently, the conformable derivative and its properties have been introduced. In this paper, we propose and prove some new results on conformable Boundary Value Problems. First, we introduce a conformable version of classical Sturm´s separation, and comparison theorems. For a conformable Sturm-Liouville problem, Green's function is constructed, and its properties are also studied. In addition, we propose the applicability of the Green´s Function in solving conformable inhomogeneous linear differential equations with homogeneous boundary conditions, whose associated homogeneous boundary value problem has only trivial solution. Finally, we prove the generalized Hyers-Ulam stability of the conformable inhomogeneous boundary value problem.