New enclosure algorithms for the verified solutions of nonlinear Volterra integral equations

This paper is concerned with new algorithms which provide the sharp bounds that are guaranteed to contain the exact solutions of nonlinear Volterra integral equations. We develop new enclosure algorithms based on the interval methods which was first introduced by Moore in [24] together with the Taylor polynomials to improve the accuracy of the scheme by reducing the width of interval solutions. The modified methods calculate a priori bound automatically in parallel with the computation of solutions of integral equations. We will show that the accuracy of the proposed algorithms is dependent on the number of interval subdivisions. Some numerical experiments are also included to demonstrate the validity and applicability of the scheme and showing a marked improvement in comparison with the recent existing numerical results.     

On the Uniqueness Theorem for Generalized Solutions of Initial-Boundary Problems for the Marguerre—Vlasov Vibrations of Shallow Shells with Clamped Boundary Conditions

The uniqueness theorem for generalized solutions of initial-boundary problems for the Marguerre—Vlasov vibrations of shallow shells with clamped boundary conditions is proved. A unique method developed by the author, based upon a nonstandard treatment of smoothing operators, is applied instead of using an enclosure theorem at the critical values of indices. Accepted 10 November 1997     

Numerical and analytical modelling of the MHD buoyancy-driven flow in a Bridgman crystal growth configuration

We present analytical and numerical models of magnetohydrodynamic(MHD) buoyancy-driven flow within the liquid pool of a horizontal Bridgman crystal growth furnace, under the influence of a uniform vertical magnetic field B₀. A horizontal differentially heated cylinder, whose aspect ratio (radius to length) is small enough for a fully developed regime to be established in the central core, is considered. With Hartmann layers remaining electrically inactive, a modified Rayleigh number Ra_G, which is the ration of the ordinary Rayleigh number to the square of the Hartmann number, is found to control the MHD reorganisation of the flow. This modified Rayleigh number is a measure of the importance of thermal convection relative to diffusion if velocity is estimated from the balance between the torques of buoyancy and the Laplace force. When Ra_G is much smaller than unity (quasi-diffusive regime), an analytical modelling of the flow, based on a power series of Ra_G, demonstrates that this balance requires secondary vortices within vertical mid-planes of the cylinder, both within the core flow and near the end walls. A 3-D numerical calculation of the flow provides evidence of the transition from a convective MHD flow (when Ra_G is still of the order of unity) to the quasi-diffusive flow, analytically studied. Indeed, this transition takes the form of a rather complex 3-D MHD organisation of the flow which is due to the nonuniformity of the axial temperature gradient along the cylinder.     

Enclosure results for quasilinear systems of variational inequalities

In this paper we consider systems of quasilinear elliptic variational inequalities, and prove the existence of minimal and maximal (in the set theoretical sense) solutions within some ordered interval of an appropriately defined pair of sub- and supersolutions. We show that the notion of sub- and supersolutions of variational inequalities introduced here is consistent with the usual notion of sub-supersolutions for (variational) equations. For weakly coupled quasimonotone systems of variational inequalities the existence of smallest and greatest solutions is proved.     

Distinguishing between the interior pressures induced from two independent sources within a room using the probabilistic approach

In this paper, a probabilistic approach is introduced and used to distinguish between the interior pressures that are induced from two independent sound sources within a rectangular room. One source is a vibrating wall of the room and the other is an interior point source. The model is set up using the modal analysis method. In the probabilistic method, the probability densities of the uncertain acoustic model parameters are computed. The values of highest probability density are identified and assigned to the uncertain parameters in the acoustic model. The contributions of the sound sources to the total pressure are then obtained from the acoustic model input with the identified parameters. The effects of the modal truncations, the modeling errors, and the measurement locations on the accuracies of the identification process are studied.     

Numerical Verification Methods for Solutions of the Free Boundary Problem

We propose two methods to enclose the solution of an ordinary free boundary problem. The problem is reformulated as a nonlinear boundary value problem on a fixed interval including an unknown parameter. By appropriately setting a functional space that depends on the finite element approximation, the solution is represented as a fixed point of a compact map. Then, by using the finite element projection with constructive error estimates, a Newton-type verification procedure is derived. In addition, numerical examples confirming the effectiveness of current methods are given.     

MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure

This study looks at MHD natural convection flow and heat transfer in a laterally heated enclosure with an off-centred partition. Governing equations in the form of vorticity–stream function formulation are solved using the polynomial differential quadrature (PDQ) method. Numerical results are obtained for various values of the partition location, Rayleigh, Prandtl and Hartmann numbers. The results indicate that magnetic field significantly suppresses flow, and thus heat transfer, especially for high Rayleigh number values. The results also show that the x-directional magnetic field is more effective in damping convection than the y-directional magnetic field, and the average heat transfer rate decreases with an increase in the distance of the partition from the hot wall. The average heat transfer rate decreases up to 80% if the partition is placed at the midpoint and an x-directional magnetic field is applied. The results also show that flow and heat transfer have little dependence on the Prandtl number.     

封闭腔内纳米流体强化自然对流换热的数值模拟

采用F luen软件对封闭腔内Cu-H2O纳米流体强化自然对流换热进行了数值模拟,重点分析Cu纳米粒子添加量和Gr数对换热性能的影响,并解释其换热机理。研究结果表明:在水基液中加入Cu纳米粒子可以显著提高基液的自然对流换热特性。对于一给定的Gr数,随着纳米粒子质量分数的增加,纳米流体的速度组成部分增加,纳米流体质量分数越大,x方向和y方向的速度峰越大,因此加速了流体中能量传输。另一方面,随着Gr数的增加,流线图中旋涡逐渐变大,流线间强度增加,说明换热效果逐渐增强。当Gr数较小时,传热主要是由热壁和冷壁之间的热传导引起的,随着Gr数的增大,换热逐渐变为由对流换热占主导地位。     

Computational analysis of non-isothermal temperature distribution on natural convection in nanofluid filled enclosures

In this study, the problem of steady state natural convection in an enclosure filled with a nanofluid has been analyzed numerically by using heating and cooling by sinusoidal temperature profiles on one side. The governing partial differential equations, in terms of the dimensionless stream function–vorticity and temperature, are solved numerically using the finite volume method for various inclination angles 0^°≤?≤90^°$0^{°}\le ?\le 90^{°}$, different types of nanoparticles (TiO₂ and Al₂O₃) and fractions of nanoparticles 0≤φ≤0.1$0\le \phi \le 0.1$, whereas the range of the Rayleigh number Ra is 10³–10⁵. It is found that the addition of nanoparticles into water affects the fluid flow and temperature distribution especially for higher Rayleigh numbers. An enhancement in heat transfer rate was registered for the whole range of Rayleigh numbers. However, low Rayleigh numbers show more enhancement compared to high Rayleigh numbers.