Search

In previous works, it was shown that the propagation of magnetic spin waves in thin films can be approximated by a nonlinear Schrodinger-type equation. The formulation begins with the magnetostatic equations (Gauss and Ampere’s laws of magnetism) and the Landau–Lifshitz equation. The solution of this system is a potential function whose dimensionless amplitude is the solution of a nonlinear Schrodinger. In the current work, we are demonstrating an efficient infinite series solution using the Christov functions. This is the first time the functions are used in problems involving complex arithmetics. The solutions of the time-independent and time-dependent problems are given in complex series form.

A robust two-dimensional (2D) formulation for the electrical characterization of nematic liquid crystals (N-LCs) under low-frequency (LF) AC biasing conditions is proposed. The finite-difference (FD) method is first implemented to solve Poisson's equation in the domain of interest in order to obtain the governing LF electric field, which affects the local dielectric properties of the anisotropic material. Then, the nonlinear Euler–Lagrange partial differential equation (PDE), governing the orientation of the directors, is solved using one of three FD schemes with relaxation proposed in this paper. Once the N-LC layer is characterized, the average refractive index as a function of the x -coordinate is calculated assuming a normally incident monochromatic laser beam. The results are compared with published data in the literature obtained using a finite element method (FEM). Solution of the PDE governing the orientation of the directors in a non-uniform 2D electric field is obtained using either strong anchoring or soft anchoring. An investigation of the effects of boundary conditions on the average refractive index is presented.

We apply the Christov-Galerkin spectral method for the numerical investigation of the interaction of solitons in the Cubic Nonlinear Schrodinger Equation. The issues of convergence are addressed and an algorithm is devised for the application of the method. Results are obtained for the interaction of solitons with different phase velocities and different carrier frequencies. The interactions are shown to be elastic, save for the phase shifts. The latter are extracted from the numerical solution and discussed.

We develop a robust Christov-Galerkin spectral technique for computing interacting localized wave solutions of and fourth and sixth-order generalized wave equations. To this end, a special complete orthonormal system of functions in L^2(-~,~) is used whose rate of convergence is shown to be exponential for the cases under consideration. For the time-stepping, an implicit algorithm is chosen which makes use of the banded structure of the matrices representing the different spatial derivatives. As featuring examples, the head-on collision of solitary waves is investigated for a sixth-order generalized Boussinesq equation and a fourth-order Boussinesq type equation with a linear term. Its solutions comprise monotone shapes (sech-es) and damped oscillatory shapes (Kawahara solitons). The numerical results are validated against published data in the literature using the method of variational imbedding.

A frequency-agile microstrip patch antenna printed on a biased nematic liquid crystal (N-LC) substrate is accurately modelled. The dielectric properties of the N-LC are controlled through the strength of the bias voltage. The model is based on the solution of Poisson's equation coupled to a nonlinear partial differential equation (PDE) describing the orientation of the directors in a non-homogeneous electric field. The coupled problem was solved using an iterative finite difference (FD) scheme. The obtained numerical results are compared and verified against measurements already published in the literature.

This paper presents a robust numerical method for the analysis of wave propagation in nematic liquid crystals. The structure is excited by a plane wave incident at an oblique angle with respect to the normal to the liquid-crystal cell. The underlined formulation is based on an eigenvalue problem which is solved analytically in order to obtain the governing field expressions inside a homogeneous, thin crystal layer. The liquid-crystal cell is comprised of N such layers. Enforcing the continuity of the tangential electric and magnetic fields at the interfaces formed by the various layers, a matrix system is generated. Solution of the linear system of equations results in the light intensity inside the liquid crystal, which is coupled to a non-linear differential equation for the director tilt angle. This equation is solved using either an explicit or implicit finite-difference scheme. An iteration process continues until convergence is reached for the coupled problem. The proposed numerical method was validated against published results that were generated by approximate analytical methods. Further simulations and studies were conducted emphasizing on the physics of the problem and related interesting phenomena.

This work reports on the implementation of an an iterative procedure to solve the non-linear problem of wave propagation in homeotropic Nematic Liquid Crystals (N-LC). The nematic structure of the crystal molecules is strongly dependent on the applied external electromagnetic field. In our case, a monochromatic plane wave is normally incident on the liquid crystal, which is sandwiched between two glass layers. The orientation of these molecules, called the directors, determine the dielectric tensor properties of the medium. A Mode-Matching Technique (MMT) was used to accurately solve for the governing fields in each of the subdivided layers composing the crystal, whereas an explicit finite-difference scheme with relaxation was implemented to solve for the directors' orientation. The non-linear problem was also solved using a more efficient implicit finite difference scheme characterized by a faster convergence rate. Obtained computational results were compared to published data indicating good agreement.

In the current work, we develop a numerical method suitable for treating the problem of nonlinear two‐dimensional flows in rectangular domains. For the spatial approximation we employ the Fourier‐Galerkin approach. More specifically, our basis functions are products of trigonometric and Beam functions. This choice means that the solutions automatically satisfy the boundary and periodic conditions in the x and y directions respectively.The accuracy of the method is assessed by applying it to model problems which admit exact analytical solutions. The numerical and analytic solutions are found to be in good agreement. The convergence rate of the spectral coefficients is found to be fifth‐order algebraic in the x‐direction and y‐direction, confirming the efficiency and speed of our technique.

In this work, we investigate the 2D flow in a rectangular cavity subject to both vertical and horizontal temperature gradients. The linearized model is studied and the effect of thermal relaxation, as described by the Maxwell‐Cattaneo law of heat conduction is examined. To this end, a spectral numerical model is created based on a Galerkin expansion. The basis is the Cartesian product of systems of beam functions and trigonometric functions.The natural modes of the system are derived for both the Fourier and non‐Fourier models. The results are compared to earlier works for the plain Fourier law. Our computations show that for the same set of parameters, the Maxwell‐Cattaneo law yields modes which are quantitatively different from the Fourier. It is found that the real parts of the eigenvalues increase with the Straughan number Sg, which quantifies the non‐Fourier effects. This confirms the destabilizing effect of the MC‐law on the convective flow.

We study the effect of the Maxwell‐Cattaneo law of heat conduction (MCHC) on the 1D flow in a vertical slot subject to both vertical and horizontal temperature gradients. The gravitational acceleration is allowed to oscillate, which provides an opportunity to investigate the quantitative contribution of thermal inertia as epitomized by MCHC. The addition of the time derivative in MCHC increases the order of the system. We use a spectral expansion with Rayleigh’s beam functions as the basis set, which is especially suited to fourth order boundary value problems (BVP). We show that the time derivative (relaxation of the thermal flux) has a dissipative nature and leads to the appearance of purely real negative eigenvalues. Yet it also increases the absolute value of the imaginary part and decreases the absolute value of the real part of the complex eigenvalues. Thus, the system has a somewhat more oscillatory behavior than the one based on Fourier’s heat conduction law (FHC).

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