Integral results for nonlinear diffusion equations

We show how to construct integral results for the multi-dimensional nonlinear diffusion equation c/t=·(D(c)c) and for some generalisations of this. For appropriate boundary conditions these become integral invariants. An application of these results to determining the large-time behaviour of some radially symmetric problems is indicated.     

On the strain-displacement equations of a geometrically nonlinear curved beam

The paper is concerned with the strain-displacement equations of a curved beam undergoing arbitrarily large deflections/rotations. Two alternative procedures are used for this derivation: a mathematically consistent and an engineering-type approach. It is demonstrated that upon making certain simplifying assumptions regarding higher-order terms both approaches lead to identical results. VDI     

Comparison of two iteration procedures for a class of nonlinear jerk equations

A modified Michens iteration procedure and a direct Michens iteration procedure are applied to determine approximate periods and analytical approximate periodic solutions of a class of nonlinear jerk equations. When we use the modified Michens iteration procedure to deal with the nonlinear jerk equations, we need to solve the nonlinear algebra equation for determining the approximate angular frequency. The higher the number of iterations, the more difficult it is to deal with the nonlinear algebra equation. This is because the kth-order approximate solution obtained by the modified Michens iteration procedure is a function of the angular frequency. However, since the kth-order approximate solution obtained by the direct Michens iteration procedure is independent on the kth-order approximate angular frequency, the form of nonlinear algebra equation that determines the kth-order approximate angular frequency of nonlinear jerk equation is similar. The second-order approximate period obtained by the direct Michens iteration procedure provides very accurate results for the large initial velocity amplitudes. But the second-order approximate period obtained by the modified Michens iteration procedure is invalid for all amplitudes B of the initial velocity. A comparison of the first- and second-order approximate periodic solutions obtained by the direct Michens iteration procedure with the numerically exact solutions shows that the second-order approximate periodic solution is much more accurate than the first one. Thus, the direct Michens iteration procedure is very effective for the class of nonlinear jerk equations.     

Application of meshfree collocation method to a class of nonlinear partial differential equations

In this work, an algorithm for the numerical solution of the generalized Hirota–Satsuma equations and Jaulent–Miodek equations based on meshless radial basis functions (RBFs) method using collocation points, called Kansa's method, is presented. Four model problems with six different initial conditions are considered for the computation. A fairly explicit scheme is used to approximate the solution. The comparison is made with the exact solutions of each problem of the generalized Hirota–Satsuma coupled Korteweg–de Vries equations. A system consisting highly nonlinear partial differential equations known as Jaulent–Miodek equations and generalized Hirota–Satsuma coupled modified-Korteweg–de Vries equations are considered for comparison with the work already published. The multiquadric RBF results are compared with homotopy perturbation method (HPM) and variational iteration method (VIM) to highlight the excellent performance of the method.     

Non-stationary statistical solutions of a class of random diffusion equations: Analytical and numerical considerations

Non-stationary, behaviour of statistical moments up to the second order, of solutions of linear one-dimensional diffusion equations having random initial conditions and random external excitations the latter of which is represented by non-stationary Gaussian white noises, is considered. Three approaches are presented: the first is concerned with the analytical solution procedure based on separation of variables together with the superposition principle; the second deals with a semi-analytical approach by the use of finite element and finite difference approximations in space; and the third is related to numerical analysis using the simplest explicit and implicit finite differences. Comparison is made for the results obtained by these three solution procedures. Convergence behaviour of the analytical solutions is investigated, and the consideration of stability of the finite difference solutions is also given.     

On approximate constitutive equations in nonlinear viscoelasticity

Summary In this paper a simplified constitutive equation for nonlinear slightly viscoelastic materials is derived by a method similar to that ofColeman andNoll. This equation is also valid for short times, slow motions, or small deformations superposed on a large steady deformation for all viscoelastic materials. An experimental method for determining the material parameters contained in this constitutive equation for dynamic conditions is also presented.

---

Über genäherte Materialgleichungen in der nichtlinearen Viskoelastizität

Zusammenfassung Mittels einer Methode ähnlich der vonColeman undNoll wird eine vereinfachte Materialgleichung für nichtlineare, schwach viskoelastische Stoffe hergeleitet. Diese Gleichung gilt auch für das Kurzzeitverhalten, für langsame Bewegungen, oder für die Überlagerung kleiner Deformationen auf große stationäre Verformungen bei allen viskoelastischen Materialien. Eine experimentelle Methode zur Bestimmung der in der Materialgleichung auftretenden Stoffkonstanten für dynamische Bedingungen wird gleichfalls angegeben.

     

On a bounded solution to a pair of coupled nonlinear ordinary differential equations

We present a bounded, decaying solution to a pair of coupled, nonlinear second-order ordinary differential equations arising in the theory of natural convection. The solution is found by transforming the problem into a non-autonomous system in the phase-plane. A uniqueness proof is given for the bounded solution.     

On a bounded solution to a pair of coupled nonlinear ordinary differential equations

We present a bounded, decaying solution to a pair of coupled, nonlinear second-order ordinary differential equations arising in the theory of natural convection. The solution is found by transforming the problem into a non-autonomous system in the phase-plane. A uniqueness proof is given for the bounded solution.     

On the two-scales method for a class of integro-differential equations appearing in viscoelasticity

In this paper, we consider vibrations of a linear viscoelastic material, with long memory and short relaxation time (introduction of the small parameter ?). Related to this problem, an integro-differential equation is to be resolved. An approximate solution is constructed by using the two-scales method. It is shown that this formal approximation is valid on time intervals of order ?^?1. The method of averaging leads to the same first approximation.     

A class of particular solutions of one-dimensional nonlinear equations of heat conduction

A class of solutions is obtained for the heat-conduction equations in the case of a power relation between the coefficient of thermal conductivity and the temperature.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 32, No. 3, pp. 508–511, March, 1977.     

On a class of exact solutions to the equations of motion of a second grade fluid

A class of exact solutions to the equations of motion of a second grade fluid is exhibited wherein the non-linearities which occur in the equations of motion are self-cancelling though individually nonvanishing. These flows are those in which the vorticity and the Laplacian of the vorticity remain constant along stream lines.     

Approximate solutions to a nonlinear diffusion equation

Approximate similarity solutions to the porous-medium equation, c ₁ = · (c ^m c), are obtained in one and two dimensions. The problems considered arise in the modelling of dopant diffusion in semiconductors, the two-dimentional problems corresponding to diffusion under a mask edge.     

On a class of constitutive equations in viscoplasticity: Formulation and computational issues

The viscoplastic constitutive model is formulated based on the existence of the dissipation potential which embodies the notion of the gauge (Minkowski) function of the convex set. A perturbation method is used for a solution of stiff differential equations characterizing the associated problem of evolution. It relies on a discrete formulation of viscoplasticity which results from the regularized version of the principle of maximum plastic dissipation. The operator split methodology and the Newton-Raphson method are used to obtain the numerical solution of the discretized equations of evolution. The consistent tangent modulus is expressed in a closed form as a result of the exact linearization of the discretized evolution equations. For several variants of the flow potential function, including some representative stiff functional forms, numerical tests of the integration algorithm based on iso-error maps are provided. Finally, a numerical example is presented to illustrate the robustness and the effectiveness of the proposed approach.     

Approximate analytical solutions of a nonlinear diffusion equation

Approximate analytical solutions of a nonlinear diffusion equation $\frac{{\partial c}}{{\partial t}} = \frac{\partial }{{\partial x}}\left( {D(c)\frac{{\partial c}}{{\partial x}}} \right)$ are obtained in the practically important case of constant boundary conditions corresponding to the diffusion in a homogeneously doped half-space at a zero surface concentration for D(c) = ac, ac ², and a√c (a > 0). The error of approximation for these D(c) dependences in the concentration interval (1–2) × 10^?3 < c < (0.92–0.99) does not exceed 1–2%.     

General solutions of the coupled diffusion equations

New finite integral transform and the corresponding infinite series are introduced, which brings the solution of the coupled diffusion equations within the realm of integral transform theory. The formulation of this transform leads to an eigenvalue problem which is not of the conventional Sturm-Liuoville type and therefore a special integral condition was derived which serves as an orthogonality relation. The solution obtained can be applied when studying diffusion in a tubular reactor, heat transfer by a turbulenty flowing fluid-solids mixture in a pipe.     

Physicomechanical state of the plate in diffusion saturation with nonlinear diffusion

Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 25, No. 2, pp. 42–45, March–April, 1989.