Point Boundary Conditions Research Articles

Lyapunov matrices for a class of neutral type time delay systems

A class of neutral type time delay systems is presented for which computation of Lyapunov matrices is reduced to a solution of an auxiliary delay free system of linear differential matrix equations with a special set of two point boundary conditions.

Multilayer differential discrete ordinate method for inhomogeneous participating media

This paper presents a multilayer differential discrete ordinate method to solve the radiative transfer equation for an absorbing, emitting and scattering inhomogeneous plane parallel medium. This method reduces the integro-differential equation into a set of coupled first order ordinary differential equations with two point boundary conditions on using a suitable quadrature scheme. These equations are then solved numerically. Numerical validation of the method for gray medium is done by comparing the results obtained with benchmark cases available in the literature. Validation for a non-gray medium is done by considering a problem concerning radiative transfer from the atmosphere. The brightness temperature at the top of the atmosphere is calculated at various frequencies and validated with those obtained by several other numerical methods.

Numerical solution of special 12th-order boundary value problems using differential transform method

In this paper, a differential transform method (DTM) is used to find the numerical solution of a special 12th-order boundary value problems with two point boundary conditions. The analysis is accompanied by testing differential transform method both on linear and nonlinear problems from the literature [Wazwaz AM. Approximate solutions to boundary value problems of higher-order by the modified decomposition method. Comput Math Appl 2000:40;679–91; Siddiqi SS, Ghazala Akram. Solutions of 12th order boundary value problems using non-polynomial spline technique. Appl Math Comput 2007. doi:10.1016/j.amc.2007.10.015; Siddiqi SS, Twizell EH. Spline solutions of linear 12th-order boundary value problems. J Comput Appl Math 1997;78:371–90]. Numerical experiments and comparison with existing methods are performed to demonstrate reliability and efficiency of the proposed method.

A Two-Region Hydraulic Averaging Model for Cuttings Transport During Horizontal Well Drilling

Abstract The two-region hydraulic averaging model was used to analyze the problem of cuttings transport during horizontal well drilling. This model considers a two-phase two region system composed of a moving bed (ω-region) and a stationary bed of drill cuttings modelled as a porous medium (η-region). The ω-region is made up of a solid phase (σ-phase) dispersed in a continuous fluid phase (β-phase), while the η-region consists of a stationary solid phase (σ-phase) and a fluid phase (β-phase). The volume averaging method was applied to obtain the volume-averaged transport equations for both the moving bed and the porous medium regions. These equations are based on the non-local form of the volume-averaged momentum transport equation that is valid within the interface region. The three main flow patterns of the horizontal cuttings transport process analyzed were: Case 1 - fully suspended flow, Case 2 - flow with a stationary bed and Case 3 - flow with a moving bed. The one-dimensional models for all cases were solved numerically using the finite difference technique with an implicit scheme. The numerical results were compared with experimental data and theoretical results reported in the literature and a good agreement was found. Introduction Due to the presence of two phases (solid and liquid) where the solid particles tend to settle at the bottom of the pipe(1), the hydraulic transport of solid particles in horizontal pipes is a very complex physical phenomenon. Such a phenomenon is relevant in several areas, such as the chemical, geothermal, mining and oil industries. In the oil industry, horizontal drilling is used to exploit reservoirs exhibiting thin pay zones to solve the problems related to water and gas conning, to obtain greater drainage area and to maximize the productive potential in naturally fractured reservoirs. However, a major deterrent in horizontal drilling is the reduction in performance of the transport of solid rock fragments called cuttings transport(2). A detailed review of published experimental data reveals that the cuttings transport characteristics change with an increase in wellbore angle. Tomren et al.(3) and Ford et al.(4) carried out experimental work on cuttings transport in inclined wellbores and observed the existence of different layers that might occur during mud and cuttings flow in a wellbore: a stationary bed, a sliding bed and a heterogeneous suspension or clear mud. Leising and Walton(5), Sifferman and Becker(6) and Peden et al.(7) reported that under a certain range of well deviation, the cuttings bed in annuli is unstable. Doron and Barnea(8) carried out experimental work on solid-liquid flow in pipes and observed the existence of three mainflow patterns: fully suspended flow, flow with a stationary bed and flow with a moving bed. On the other hand, numerous mathematical and empirical models for the prediction of cuttings transport in horizontal and directional wells have been developed by several researchers(9–18). However, many of the previous models (two- and three-layer approaches) have been constructed on an intuitive basis rather than on a rigorous analysis of the governing point equations and boundary conditions. Intuitive analysis often leads to hidden assumptions and unsupported simplifications.

Non-isothermal film casting: Determination of draw resonance

The important industrial process of casting polymeric films suffers from the “draw resonance” instability that appears as sudden oscillations in the product dimensions. This instability influences the quality of the end-product and negatively limits productivity and efficiency of the process. The draw resonance originates when a material is being processed beyond the limits of its intrinsic properties. Research is conducted with the intention to find those process and material properties that allow to optimize the production process while keeping it stable. This paper concentrates on a non-isothermal analysis of the stability of the film casting. The mathematical model of the process is given by a quasi-linear system of first order PDEs with two point boundary conditions. The constitutive polymer behavior is approximated by the modified Giesekus model. Linear stability analysis combined with the Laplace transformation of the resulting linear system is applied to find parameters that determine mathematical and thus process instability. It all comes down to determining the spectrum of a compact operator; corresponding eigenfunctions can be regarded as the characteristic modes of the system. For implementation, the modification of Galerkin approach is used. The major advantage of the mathematical and numerical method is that the full spectrum is calculated in a matter of seconds. Our results agree perfectly with the ones from literature for isothermal case, and with the experimental data for the non-isothermal case. The results also indicate that non-isothermality is highly important and cannot be excluded from modeling.

Quartic spline solution of linear fifth order boundary value problems

Some techniques are available to solve numerically fifth order boundary value problems with two point boundary condition. This paper describes quartic spline method for the numerical solution of linear special case fifth order boundary value problem. Three examples are considered for the numerical illustration of the method developed. The method is also compared with that developed by Caglar et al. [H.N. Caglar, S.H. Caglar, E.H. Twizell, The numerical solution of fifth order boundary value problems with sixth degree B-Spline functions, Appl. Math. Lett. 12 (1999) 25–30] and Khan [M.S. Khan, Finite difference solutions of fifth order boundary value problems, Ph.D. thesis, Brunal University, England, 1994]. This comparison provides that developed method is more efficient that effective tool and yields better results.

Quintic spline solution of linear sixth-order boundary value problems

There are few techniques available to numerically solve sixth-order boundary value problems with two point boundary conditions. In this paper, quintic spline method is developed for the numerical solutions of linear special case sixth-order boundary value problems. Two examples are considered for the numerical illustration of the method developed. The method is also compared with those developed by EL-Gamel et al. [Mohamed EL-Gamel, John R. Cannon, Ahmed I. Zayed, Sinc–Galerkin method for solving sixth-order BVPs, Math. Comput. 73 (247) (2003)] and Wazwaz [A. Wazwaz, The numerical solution of sixth-order boundary value problems by the modified decomposition method, Appl. Math. Comput. 118 (2001) 311–325]. The comparison shows that the quintic spline method is more efficient and effective tool and yields better results.

Uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations

For the nth order differential equation, y ( n ) = f ( x , y , y ′ , … , y ( n − 1 ) ) , we consider uniqueness implies existence results for solutions satisfying certain nonlocal ( k + 2 ) -point boundary conditions, 1 ⩽ k ⩽ n − 1 . Uniqueness of solutions when k = n − 1 is intimately related to uniqueness of solutions when 1 ⩽ k ⩽ n − 2 . These relationships are investigated as well.

Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type

New criteria are established for the existence of multiple positive solutions of a Hammerstein integral equation of the form $$ u(t)= \int_{0}^1 k(t,s)g(s)f(s,u(s))ds \equiv Au(t) $$ where $k$ can have discontinuities in its second variable and $g \in L^{1}$. These criteria are determined by the relationship between the behaviour of $f(t,u)/u$ as $u$ tends to $0^+$ or $\infty$ and the principal (positive) eigenvalue of the linear Hammerstein integral operator $$ Lu(t)=\int_{0}^1 k(t,s)g(s)u(s)ds. $$ We obtain new results on the existence of multiple positive solutions of a second order differential equation of the form $$ u''(t)+g(t)f(t,u(t))=0 \quad\text{a.e. on } [0,1], $$ subject to general separated boundary conditions and also to nonlocal $m$-point boundary conditions. Our results are optimal in some cases. This work contains several new ideas, and gives a {\it unified} approach applicable to many BVPs.

Point Constriction, Interface, and Boundary Conditions for Kinematic-Wave Model

The entropy solutions of the Riemann problem for the kinematic-wave model are described for (entrant and exiting) boundary conditions, interfaces (e.g., lane drops), and point constrictions (e.g., bottlenecks or signalized intersections) for kinematic-wave models. These results are used to obtain analytic solutions and computational solutions, by the method of Godunov, for the three test scenarios earlier suggested by Ross. Results show that the kinematic-wave model, with a correct computational solution (i.e., converging to the entropy solution), correctly produces the analytic (entropy) solutions of the kinematic-wave model and that analytic and computational solutions both satisfy intuitive expectations at least as well as the novel model suggested by Ross and even avoid some acknowledged deficiencies of the latter. They also emphasize the necessity of using discrete approximations that satisfy the entropy condition and of exercising care in ensuring that the discretization parameters are sufficiently small.

Effect of Constriction Oscillation on Flow for Potential Application to Vocal Fold Mechanics: Numerical Analysis and Experiment

Oscillation of the vocal folds makes a sound source of the human voiced sound. Understanding of the oscillation mechanism, which is a complex flow-structure interaction problem in the airway, is crucial for considering clinical diagnosis of voice disorders. However, details of the oscillation mechanism are still unclear partly because, from a fluid mechanical viewpoint, the effect of oscillation of the vocal fold wall during the phonation on airflow behaviors remains elusive. In the present study, flow characteristics in a sinusoidally-oscillating constriction mimicking the vocal fold were investigated by numerical and experimental approaches. The numerical analyses focused in particular on the effect of constriction oscillation on flow separation demonstrated that the flow separation point moves continuously and periodically in a frequency-dependent manner. In the experimental study, an apparatus was newly designed, with a view to detect the oscillation-induced movement of the flow separation point, to enable detailed measurement of pressure distribution along the constriction with an interval of 2 mm that is synchronized with measurement of constriction displacement. Although movement of the separation point as seen in the numerical analyses was not detected by this limiting resolution of the apparatus, we obtained pressure-width relations that is partly contrary to the numerical results but is presumably dependent on the inlet boundary condition. These findings indicate that appropriate evaluations of separation point and inlet boundary conditions are key factors to characterize the flow in oscillating constriction, which is crucial for better understanding of the vocal fold mechanics.

Point Constriction, Interface, and Boundary Conditions for Kinematic-Wave Model

The entropy solutions of the Riemann problem for the kinematic-wave model are described for (entrant and exiting) boundary conditions, interfaces (e.g., lane drops), and point constrictions (e.g., bottlenecks or signalized intersections) for kinematic-wave models. These results are used to obtain analytic solutions and computational solutions, by the method of Godunov, for the three test scenarios earlier suggested by Ross. Results show that the kinematic-wave model, with a correct computational solution (i.e., converging to the entropy solution), correctly produces the analytic (entropy) solutions of the kinematic-wave model and that analytic and computational solutions both satisfy intuitive expectations at least as well as the novel model suggested by Ross and even avoid some acknowledged deficiencies of the latter. They also emphasize the necessity of using discrete approximations that satisfy the entropy condition and of exercising care in ensuring that the discretization parameters are sufficiently small.

Precise time integration for linear two-point boundary value problems

The precise time integration (PTI) method is proposed to solve the linear two-point boundary value problem (TPBVP). By employing the method of dimensional expanding, the non-homogeneous ordinary differential equations (ODEs) can be transformed into homogeneous ones and then the original PTI algorithms can be applied directly to the TPBVP. The PTI consists of two methods: the method of matrix exponential and the method of Riccati equations, both of which utilize the merit of 2 N algorithm and guarantee high precise numerical results. The method of matrix exponential follows the similar scheme proposed for initial value problem of ODEs, and uses the matrix exponential to link boundary conditions between the two points. The method of Riccati equations employs the Riccati equations to express the relationships of two point boundary conditions. And then in terms of the relationships of boundary conditions at two points, the full conditions at initial point can be obtained and then the TPBVP can be transformed into initial value problem and then solved by direct time marching scheme. With some modifications, the above algorithms can be directly extended to the infinite-interval problem and the variant coefficient ODEs problem. In the program implementation, the object-oriented (OO) design of PTI is proposed to demonstrate the applicability and easy maintenance of OO techniques in numerical computation. Finally, four selected numerical examples are given to show the high precision characteristics of PTI.

Some disconjugacy criteria for differential equations with oscillatory coefficients

AbstractWe determine conditions on the coefficients of both second and fourth order differential operators which give disconjugacy. The conditions exploit the change of sign of the coefficients to show how the change of sign enhances disconjugacy. An application is given to the stability of solutions of equations with periodic coefficients. Focal point and other boundary conditions are considered as well, and sufficient conditions are given which imply the nonexistence of nontrivial solutions of the differential equations satisfying the boundary conditions. Some open problems are stated for the minimization of certain nonlinear functionals. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The generalized quasilinearization method and three point boundary value problems on time scales

In this paper, we study the convergence of monotone sequences of iterates for nonlinear second order dynamic equations with three point boundary conditions on time scales. We prove that it is possible to construct two sequences converging to the unique solution of the three point boundary value problem from above and below with high rate of convergence.

Modeling of counter current moving bed gas-solid reactor used in direct reduction of iron ore

In this work, the shaft furnace reactor of the MIDREX ® process is simulated. This is a counter current gas-solid reactor, which transforms iron ore pellets into sponge iron. Simultaneous mass and energy balance along the reactor leads to a set of ordinary differential equation with two points boundary conditions. The iron ore reduction kinetics was modelated with the unreacted shrinking core model. Solving the ODE system allows to know the concentration and temperature profiles of all species within the reactor. The model was able to satisfactorily reproduce the data of two MIDREX ® plants: Siderca (ARGENTINA) and Gilmore Steel Corporation (U.S.A.). Also, it was used to explore the performance of the reactor under different operating conditions. This capacity could be used for design and control purpose.

Three Point Boundary Value Problems on Time Scales

This work formulates existence theorems for solutions to three-point boundary value problems on time scales. The ideas are based on a relationship between the three point boundary conditions, lower and upper solutions and topological degree theory.

Existence results for singular three point boundary value problems on time scales

We prove the existence of a positive solution for the three point boundary value problem on time scale T given by y ΔΔ +f(x,y)=0, x∈(0,1]∩ T, y(0)=0, y(p)=y σ 2(1) , where p∈(0,1)∩ T is fixed and f( x, y) is singular at y=0 and possibly at x=0, y=∞. We do so by applying a fixed point theorem due to Gatica, Oliker, and Waltman [J. Differential Equations 79 (1989) 62] for mappings that are decreasing with respect to a cone. We also prove the analogous existence results for the related dynamic equations y ∇∇+ f( x, y)=0, y Δ∇+ f( x, y)=0, and y ∇Δ+ f( x, y)=0 satisfying similar three point boundary conditions.

On the solvability of some discontinuous third order nonlinear differential equations with two point boundary conditions

We prove the existence of extremal solutions for the third order discontinuous functional nonlinear problem − φ u″(t) ′=f t,u,u′(t),u″(t) for a.e. t∈[a,b], u(a)=A(A∈ R), L 1 u,u′,u′(a),u′(b),u″(a) =0, L 2 u,u′(a),u′(b),u″(b) =0, by using a fixed point theorem after having established some existence results for some auxiliary second order nonlinear problems. We observe that, together with the discontinuities allowed on the spacial variable u, with adequate modifications of technical type, analogous results can be obtained when the equation has a second member not necessarily continuous in u″.

On the Displacement Field for Unidirectional Off-Axis Composites in 3-Point Flexure — Part 1: Analytical Approach

The displacement field of a unidirectional off-axis specimen in 3-point flexure test is obtained in a new way, by applying the Second Castigliano's theorem and the unitarial load method. The displacement field is calculated for two cases: in the first one, lift-off between specimen and fixture support occurs caused by bending-twisting coupling, and specimen is supposed to contact with the test fixture support at two diagonally opposite points. In the second case, lift-off does not occur and specimen is supposed to contact with the fixture support at four corners. The limit between two cases is established by the calculation of the critical value of the span-to-width ratio for lift-off to occur. The displacement field calculated includes shear effects and satisfies displacements compatibility and equilibrium conditions, except in the vicinity of point reactions and applied load. Neither displacement field solutions for composite plates with point boundary conditions, nor solutions based on the Second Castigliano's theorem concerning composite plates have been encountered in the literature survey.