H.A. Erbay

Higher-order split-step Fourier schemes for the generalized nonlinear Schrödinger equation

The generalized nonlinear Schrodinger (GNLS) equation is solved numerically by a split-step Fourier method. The first, second and fourth-order versions of the method are presented. A classical problem concerning the motion of a single solitary wave is used to compare the first, second and fourth-order schemes in terms of the accuracy and the computational cost. This numerical experiment shows that the split-step Fourier method provides highly accurate solutions for the GNLS equation and that the fourth-order scheme is computationally more efficient than the first-order and second-order schemes. Furthermore, two test problems concerning the interaction of two solitary waves and an exact solution that blows up in finite time, respectively, are investigated by using the fourth-order split-step scheme and particular attention is paid to the conserved quantities as an indicator of the accuracy. The question how the present numerical results are related to those obtained in the literature is discussed.

Wave propagation in fluid filled nonlinear viscoelastic tubes

SummaryThe present work considers one dimensional wave propagation in an infinitely long, straight and homogeneous nonlinear viscoelastic tube filled with an incompressible, inviscid fluid. In order to include the geometric dispersion in the analysis, the tube wall inertia effects are added to the pressure-area relation. Using the reductive perturbation technique, the propagation of weakly nonlinear waves in the long-wave approximation is examined. In the long-wave approximation, a general equation is obtained, and it is shown that by a proper scaling this equation reduces to the well-known nonlinear evolution equations. Intensifying the effect of nonlinearity in the perturbation process, the modified forms of these evolution equations are also obtained. In the absence of nonlinear viscoelastic effects all the equations reduce to those of the linear viscoelastic tube.

A split-step Fourier method for the complex modified Korteweg-de Vries equation☆

Abstract In this study, the complex modified Korteweg-de Vries (CMKdV) equation is solved numerically by three different split-step Fourier schemes. The main difference among the three schemes is in the order of the splitting approximation used to factorize the exponential operator. The space variable is discretized by means of a Fourier method for both linear and nonlinear subproblems. A fourth-order Runge-Kutta scheme is used for the time integration of the nonlinear subproblem. Classical problems concerning the motion of a single solitary wave with a constant polarization angle are used to compare the schemes in terms of the accuracy and the computational cost. Furthermore, the interaction of two solitary waves with orthogonal polarizations is investigated and particular attention is paid to the conserved quantities as an indicator of the accuracy. Numerical tests show that the split-step Fourier method provides highly accurate solutions for the CMKdV equation.

Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity

We study the initial-value problem for a general class of nonlinear nonlocal wave equations arising in one-dimensional nonlocal elasticity. The model involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. We show that some well-known examples of nonlinear wave equations, such as Boussinesq-type equations, follow from the present model for suitable choices of the kernel function. We establish global existence of solutions of the model assuming enough smoothness on the initial data together with some positivity conditions on the nonlinear term. Furthermore, conditions for finite time blow-up are provided.

A continuum model for liquid phase electroepitaxy

Abstract This paper presents a macroscopic continuum model for liquid phase electroepitaxial growth of single crystal semiconductors. The governing equations and associated boundary and interface conditions of the model are obtained from the fundamental principles of electrodynamics and thermomechanics of continua. The constitutive equations of the substrate/source and the liquid phase are derived from an irreversible rational thermodynamic theory. By means of systematic simplifications, special forms of the governing equations and associated interface conditions are presented in order to obtain tractable equations and gain physical insight for various thermoelectric effects involved in the process. The formulation presented here is valid for general LPEE growth processes with any configuration, and takes into account electromigration and the well-known thermoelectric effects such as Joule, Peltier, Thomson, Dufour, and Sorel. The fundamental equations derived here can also be used to model the growth process of ternary compound semiconductors, either directly or with modifications depending on the type of compositions considered.

An asymptotic theory of thin micropolar plates

The asymptotic expansion technique is used to obtain the two-dimensional dynamic equations of thin micropolar elastic plates from the three-dimensional dynamic equations of micropolar elasticity theory. To this end, all the field variables are scaled via an appropriate thickness parameter such that it reflects the expected behavior of the plate. A formal power series expansion of the three-dimensional solution is used by considering the thickness parameter as a small parameter. Without any a priori assumption on the form of the field variables, it is shown that the zeroth-order approximation simultaneously includes both the plate equations previously presented in the literature and the standard assumptions on the specific forms of the field variables. Some aspects of the present asymptotic plate equations are also discussed.

Thermally induced vibrations in a generalized thermoelastic solid with a cavity

The present work deals with thermally induced vibrations in an infinite solid with a cavity. The medium is assumed to be linear, isotropic, temperature-rate-dependent thermoelastic. The problem is solved for the cases of cylindrical and spherical cavities. The surface of the cavity is assumed to be subjected to a temperature varying harmonically with time, and free of stress. For the cases considered, the coupled field equations admit exact solutions in terms of Hankel and the spherical Hankel functions, respectively. Numerical results are compared with those of classical ther-moelaslicity. The contribution of the second sound parameters in these problems becomes more significant as the frequency of applied termperature increases.

The dynamic response of an incompressible non-linearly elastic membrane tube subjected to a dynamic extension

Abstract The dynamic response of an isotropic hyperelastic membrane tube, subjected to a dynamic extension at its one end, is studied. In the first part of the paper, an asymptotic expansion technique is used to derive a non-linear membrane theory for finite axially symmetric dynamic deformations of incompressible non-linearly elastic circular cylindrical tubes by starting from the three-dimensional elasticity theory. The equations governing dynamic axially symmetric deformations of the membrane tube are obtained for an arbitrary form of the strain-energy function. In the second part of the paper, finite amplitude wave propagation in an incompressible hyperelastic membrane tube is considered when one end is fixed and the other is subjected to a suddenly applied dynamic extension. A Godunov-type finite volume method is used to solve numerically the corresponding problem. Numerical results are given for the Mooney–Rivlin incompressible material. The question how the present numerical results are related to those obtained in the literature is discussed.

Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations

Abstract We study the initial-value problem for a general class of nonlinear nonlocal coupled wave equations. The problem involves convolution operators with kernel functions whose Fourier transforms are nonnegative. Some well-known examples of nonlinear wave equations, such as coupled Boussinesq-type equations arising in elasticity and in quasi-continuum approximation of dense lattices, follow from the present model for suitable choices of the kernel functions. We establish local existence and sufficient conditions for finite-time blow-up and as well as global existence of solutions of the problem.

On the asymptotic membrane theory of thin hyperelastic plates

Applying the asymptotic expansion technique to the three-dimensional equations of non-linear elasticity, a non-linear asymptotic membrane theory considering large deflections and strains is obtained for thin hyperelastic plates. To this end, the displacement vector and stress tensor components are scaled via an appropriate thickness parameter such that the present approximation takes into account larger deflections compared with those of the von Karman plate theory. Later, for an arbitrary form of the strain energy function, the hierarchy of the field equations is obtained by expanding the displacement vector and the stress tensor in terms of powers of the square root of the thickness parameter. The equations belonging to the first three orders of this hierarchy are studied in detail. It is shown that the zeroth order approximation corresponds to the well-known Foppl membrane theory, the first order approximation includes bending effects, and the effect of material non-linearity appears in the second order approximation. Solving the problem of an infinitely long strip under uniform load for clamped edge conditions, the effect of material non-linearity is discussed numerically for both compressible and incompressible hyperelastic solids. The results are also compared with the solutions of the asymptotic approximation which gives the von Karman plate equations in the zeroth order approximation.