Search

The presence of a particulate phase defines the effective response of a suspension to changes of the average heat flux. Using the random-point approximation we show that within the first order in the concentration, one needs to solve the problem for the temperature field created by a single inclusion in a matrix subject to an unsteady temperature gradient at infinity. We solve this problem by means of Laplace transform and use the solution as the first-order kernel in the functional expansion. From this kernel, we find the statistical average for the heat flux, which turns out to be a memory integral of the spatially averaged time-dependent temperature gradient. Thus, we discover that the constructive relationship between the average flux and averaged temperature gradient is not local in time, but rather involves a convolution integral that represents the memory due to the heterogeneity of the system. This is a novel result, which inter alia gives a rigorous justification to the usage of generalizations of the heat conduction law involving fractional time derivatives. The decay of the kernel is very close to t −1/2 for dimensionless times lesser than one and abruptly changes to t −3/2 for larger times.

The temperature distribution around two spheres is considered when the main field has a constant gradient at infinity. Bispherical coordinates are used, together with a transformation of the dependent variable that leads to separation of variables. Then the solution can be sought in Legendre series with respect to one of the bispherical coordinates. An important element of the proposed work is the effective way to reduce an essentially 3D problem to a set of three 2D problems. The Legendre spectral method is shown to have an exponential convergence which is confirmed by the computations. The efficiency is so high that even for the hard cases of two closely situated spheres, an accuracy of 10−10 is achieved with as few as 20 terms in the expansion. Solutions with both longitudinal and transverse gradients at infinity are obtained, and the contour lines of the temperature field are presented graphically. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

We consider the nonlinear system of equations built up from a generalized Boussinesq equation coupled with a wave equation which is a model for the one-dimensional dynamics of phases in martensitic alloys. The strongly implicit scheme employing Newton's quasilinearisation allows us to track the long time evolution of the localized solutions of the system. Two distinct classes of solutions are encountered for the pure Boussinesq equation. The first class consists of oscillatory pulses whose envelopes are localized waves. The second class consists of smoother solutions whose shapes are either heteroclinic (kinks) or homoclinic (bumps). The homoclinics decrease in amplitude with time while their support increases. An appropriate self-similar scaling is found analytically and confirmed by the direct numerical simulations to high accuracy. The rich phenomenology resulting from the coupling with the wave equation is also investigated.

The Boussinesq model of shallow water flow is considered, which contains nonlinearity and fourth‐order dispersion. Boussinesq equation has been one of the main soliton models in 1D. To find its 2D solutions, a perturbation series with respect to the small parameter e : = c2 is developed in the present work, where c is the phase speed of the localized wave. Within the order O(e2) = O(c4), a hierarchy is derived consisting of fourth‐order ordinary differential equations (ODEs). The Bessel operators involved are reformulated to facilitate the creation of difference schemes for the ODEs from the hierarchy. The numerical scheme uses a special approximation for the behavioral condition in the singularity point (the origin). The results of this work show that at infinity the 2D wave shape decays algebraically, rather than exponentially as in the 1D cases. The new result can be instrumental for understanding the interaction of 2D Boussinesq solitons, and for creating more efficient numerical algorithms explicitly...

In this paper we prove that the global existence and the blow up of the weak solutions to Boussinesq Paradigm Equation (BPE) depend not only on the initial energy but also on the profiles of the initial data.The constant d of the critical initial energy which guaranties the above properties of the solution is found explicitly by means of the exact constant of the Sobolev embedding theorem. We demonstrate numerically in the one dimensional case that this constant d is the best possible one for the global existence and a lack of global existence.In this way we can find the intervals for the velocities c of the solitary wave solutions to BPE in which the solution to BPE with initial data close to the solitons exists globally in time. Thus for different parameters of BPE we give numerically some ranges of the stability of the solitary waves.

We consider the 2D stationary propagating solitary waves of the so‐called Boussinesq Paradigm equation. The fourth‐ order elliptic boundary value problem on infinite interval is solved by a Galerkin spectral method. An iterative procedure based on artificial time (‘false transients’) and operator splitting is used. Results are obtained for the shapes of the solitary waves for different values of the dispersion parameters for both subcritical and supercritical phase speeds.

In this paper we develop further the Galerkin technique based on the so-called beam functions with application to nonUnear problems. We make use of the formidas expressing a product of two beam functions into a series with respect to the system. First we prove that the overall convergence rate for a fourth-order linear b.v.p is algebraic fifth order, provided that the derivatives of the sought function up to fifth order exist. It is then shown that the inclusion of a quadratic nonlinear term in the equation does not degrade the fifth-order convergence. We validate our findings on a model problem which possesses analytical solution in the linear case. The agreement between the beam-Galerkin solution and the analytical solution for the linear problem is better than 10^'^ for 200 terms. We also show that the error introduced by the expansion of the nonlinear term is lesser than 10^'. The foeam-Galerkin method outperforms finite differences due to its superior accuracy whilst its advantage over the Chebyshev-tau method is attributed to the smaller condition number of the matrices involved in the former

This paper deals with a possible generalization of Fourier's law that incorporates spatial memory into the constitutive relation. The integral and differential versions of the memory terms in the constitutive relation are discussed. It is shown that the asymptotically correct model contains the biharmonic operator as the vehicle for the higher‐order heat diffusion that also accounts for the spatial memory of the processes. Different solutions in 1D, 2D and 3D are presented to show the applicability of the new model.

We consider the problem of acceleration of the Alternative Directions Implicit (ADI) scheme for Dirichlet problem for biharmonic equation. The second Douglas scheme is used as the main vehicle and two full time steps are organised in a single iteration unit in which the explicit operators are arranged differently for the second step. Using an a priori estimate for the spectral radius of the operator, we show that there exists an optimal value for the acceleration parameter. An algorithm is devised implementing the scheme and the optimal range of the parameter is verified through numerical experiments. One iteration unit speeds up the convergence from two to three times in comparison with the standard ADI scheme. To obtain more significant acceleration for the cases when the standard ADI scheme has extremely slow convergence, a generalised multiunit scheme is constructed by introducing another acceleration parameter and treating two consecutive iteration units as one basic element.

A new complete orthonormal system of functions in the $L^2 ( - \infty ,\infty )$ space is introduced. The system consists of two sequences composed of odd and even functions respectively. Unlike Hermite and Laguerre sets of functions which behave exponentially at infinity, the new system exhibits asymptoticbehavior $x^{ - 1} $ for the odd sequence and $x^{ - 2} $ for the even one.Formulae representing products, derivatives, etc. in series in the system are developed. A nonlinear differential equation with a requirement for summability of the square of its solutions, instead of boundary conditions, is solved. The example displays most of the features of the new method important for essentially nonlinear problems.

For the pressure-driven viscous flow between two flat plates and in tubes (Poiseuille flows) the principle of minimal dissipation is applied. Minimum is sought in the class of point random functions. For kernels of stochastic integrals involved a boundary value problem is derived and appropriate numerical method is proposed. Calculations show that beyond certain critical magnitude of Reynolds number a bifurcation takes place and non-trivial stochastic solution emerges. Various characteristics of this solution are calculated and most of them are in good quantitative or qualitative agreement with the experimental data concerning turbulent Poiseuille flows. The kernels of stochastic integrals are interpreted as large eddies (coherent structures).