Search results for: Coupled Schrödinger equation
1522 Numerical Study of Some Coupled PDEs by using Differential Transformation Method
Authors: Reza Abazari, Rasool Abazari
In this paper, the two-dimension differential transformation method (DTM) is employed to obtain the closed form solutions of the three famous coupled partial differential equation with physical interest namely, the coupled Korteweg-de Vries(KdV) equations, the coupled Burgers equations and coupled nonlinear Schrödinger equation. We begin by showing that how the differential transformation method applies to a linear and non-linear part of any PDEs and apply on these coupled PDEs to illustrate the sufficiency of the method for this kind of nonlinear differential equations. The results obtained are in good agreement with the exact solution. These results show that the technique introduced here is accurate and easy to apply.
Keywords: Coupled Korteweg-de Vries(KdV) equation, Coupled Burgers equation, Coupled Schrödinger equation, differential transformation method.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 2849
1521 Approximate Solution to Non-Linear Schrödinger Equation with Harmonic Oscillator by Elzaki Decomposition Method
Authors: Emad K. Jaradat, Ala’a Al-Faqih
Nonlinear Schrödinger equations are regularly experienced in numerous parts of science and designing. Varieties of analytical methods have been proposed for solving these equations. In this work, we construct an approximate solution for the nonlinear Schrodinger equations, with harmonic oscillator potential, by Elzaki Decomposition Method (EDM). To illustrate the effects of harmonic oscillator on the behavior wave function, nonlinear Schrodinger equation in one and two dimensions is provided. The results show that, it is more perfectly convenient and easy to apply the EDM in one- and two-dimensional Schrodinger equation.
Keywords: Non-linear Schrodinger equation, Elzaki decomposition method, harmonic oscillator, one and two- dimensional Schrodinger equation.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 753
1520 An Expansion Method for Schrödinger Equation of Quantum Billiards with Arbitrary Shapes
Authors: İnci M. Erhan
Abstract:A numerical method for solving the time-independent Schrödinger equation of a particle moving freely in a three-dimensional axisymmetric region is developed. The boundary of the region is defined by an arbitrary analytic function. The method uses a coordinate transformation and an expansion in eigenfunctions. The effectiveness is checked and confirmed by applying the method to a particular example, which is a prolate spheroid.
Keywords: Bessel functions, Eigenfunction expansion, Quantum billiard, Schrödinger equation, Spherical harmonicsProcedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 5074
1519 Constructing Distinct Kinds of Solutions for the Time-Dependent Coefficients Coupled Klein-Gordon-Schrödinger Equation
Authors: Anupma Bansal
We seek exact solutions of the coupled Klein-Gordon-Schrödinger equation with variable coefficients with the aid of Lie classical approach. By using the Lie classical method, we are able to derive symmetries that are used for reducing the coupled system of partial differential equations into ordinary differential equations. From reduced differential equations we have derived some new exact solutions of coupled Klein-Gordon-Schrödinger equations involving some special functions such as Airy wave functions, Bessel functions, Mathieu functions etc.
Keywords: Klein-Gordon-Schödinger Equation, Lie Classical Method, Exact SolutionsProcedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 4407
1518 Bound State Solutions of the Schrödinger Equation for Hulthen-Yukawa Potential in D-Dimensions
Authors: I. Otete, A. I. Ejere, I. S. Okunzuwa
In this work, we used the Hulthen-Yukawa potential to obtain the bound state energy eigenvalues of the Schrödinger equation in D-dimensions within the frame work of the Nikiforov-Uvarov (NU) method. We demonstrated the graphical behaviour of the Hulthen and the Yukawa potential and investigated how the screening parameter and the potential depth affected the structure and the nature of the bound state eigenvalues. The results we obtained showed that increasing the screening parameter lowers the energy eigenvalues. Also, the eigenvalues acted as an inverse function of the potential depth. That is, increasing the potential depth reduces the energy eigenvalues.
Keywords: Schrödinger's equation, bound state, Hulthen-Yukawa potential, Nikiforov-Uvarov, D-dimensionsProcedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 272
1517 An H1-Galerkin Mixed Method for the Coupled Burgers Equation
Authors: Xianbiao Jia, Hong Li, Yang Liu, Zhichao Fang
In this paper, an H1-Galerkin mixed finite element method is discussed for the coupled Burgers equations. The optimal error estimates of the semi-discrete and fully discrete schemes of the coupled Burgers equation are derived.
Keywords: The coupled Burgers equation, H1-Galerkin mixed finite element method, Backward Euler's method, Optimal error estimates.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1416
1516 Photon Localization inside a Waveguide Modeled by Uncertainty Principle
Authors: Shilpa N. Kulkarni, Sujata R. Patrikar
Abstract:In the present work, an attempt is made to understand electromagnetic field confinement in a subwavelength waveguide structure using concepts of quantum mechanics. Evanescent field in the waveguide is looked as inability of the photon to get confined in the waveguide core and uncertainty of position is assigned to it. The momentum uncertainty is calculated from position uncertainty. Schrödinger wave equation for the photon is written by incorporating position-momentum uncertainty. The equation is solved and field distribution in the waveguide is obtained. The field distribution and power confinement is compared with conventional waveguide theory. They were found in good agreement with each other.
Keywords: photon localization in waveguide, photon tunneling, quantum confinement of light, Schrödinger wave equation, uncertainty principle.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 2707
1515 Calculation of Wave Function at the Origin (WFO) for Heavy Mesons by Numerical Solving of the Schrodinger Equation
Authors: M. Momeni Feyli
Abstract:Many recent high energy physics calculations involving charm and beauty invoke wave function at the origin (WFO) for the meson bound state. Uncertainties of charm and beauty quark masses and different models for potentials governing these bound states require a simple numerical algorithm for evaluation of the WFO's for these bound states. We present a simple algorithm for this propose which provides WFO's with high precision compared with similar ones already obtained in the literature.
Keywords: Mesons, Bound states, Schrodinger equation, Nonrelativistic quark model.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1395
1514 FWM Wavelength Conversion Analysis in a 3-Integrated Portion SOA and DFB Laser using Coupled Wave Approach and FD-BPM Method
Authors: M. K. Moazzam, A. Salmanpour, M. Nirouei
Abstract:In this paper we have numerically analyzed terahertzrange wavelength conversion using nondegenerate four wave mixing (NDFWM) in a SOA integrated DFB laser (experiments reported both in MIT electronics and Fujitsu research laboratories). For analyzing semiconductor optical amplifier (SOA), we use finitedifference beam propagation method (FDBPM) based on modified nonlinear SchrÖdinger equation and for distributed feedback (DFB) laser we use coupled wave approach. We investigated wavelength conversion up to 4THz probe-pump detuning with conversion efficiency -5dB in 1THz probe-pump detuning for a SOA integrated quantum-well
Keywords: distributed feedback laser, nondegenerate fourwave mixing, semiconductor optical amplifier, wavelengthconversionProcedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1393
1513 Mathematical Approach for Large Deformation Analysis of the Stiffened Coupled Shear Walls
Authors: M. J. Fadaee, H. Saffari, H. Khosravi
Abstract:Shear walls are used in most of the tall buildings for carrying the lateral load. When openings for doors or windows are necessary to be existed in the shear walls, a special type of the shear walls is used called "coupled shear walls" which in some cases is stiffened by specific beams and so, called "stiffened coupled shear walls". In this paper, a mathematical method for geometrically nonlinear analysis of the stiffened coupled shear walls has been presented. Then, a suitable formulation for determining the critical load of the stiffened coupled shear walls under gravity force has been proposed. The governing differential equations for equilibrium and deformation of the stiffened coupled shear walls have been obtained by setting up the equilibrium equations and the moment-curvature relationships for each wall. Because of the complexity of the differential equation, the energy method has been adopted for approximate solution of the equations.
Keywords: Buckling load, differential equation, energy method, geometrically nonlinear analysis, mathematical method, Stiffened coupled shear walls.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1503
1512 CO-OFDM DSP Channel Estimation
Authors: Pranav Ravikumar, Arunabha Bera, Vijay K. Mehra, Anand Kumar
Abstract:This paper solves the Non Linear Schrodinger Equation using the Split Step Fourier method for modeling an optical fiber. The model generates a complex wave of optical pulses and using the results obtained two graphs namely Loss versus Wavelength and Dispersion versus Wavelength are generated. Taking Chromatic Dispersion and Polarization Mode Dispersion losses into account, the graphs generated are compared with the graphs formulated by JDS Uniphase Corporation which uses standard values of dispersion for optical fibers. The graphs generated when compared with the JDS Uniphase Corporation plots were found to be more or less similar thus verifying that the model proposed is right. MATLAB software was used for doing the modeling.
Keywords: Modulation, Non Linear Schrodinger Equation, Optical fiber, Split Step Fourier Method.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 2643
1511 Instability of Soliton Solutions to the Schamel-nonlinear Schrödinger Equation
Authors: Sarun Phibanchon, Michael A. Allen
A variational method is used to obtain the growth rate of a transverse long-wavelength perturbation applied to the soliton solution of a nonlinear Schr┬¿odinger equation with a three-half order potential. We demonstrate numerically that this unstable perturbed soliton will eventually transform into a cylindrical soliton.
Keywords: Soliton, instability, variational method, spectral method.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 3529
1510 Impact of the Existence of One-Way Functionson the Conceptual Difficulties of Quantum Measurements
Authors: Arkady Bolotin
Abstract:One-way functions are functions that are easy to compute but hard to invert. Their existence is an open conjecture; it would imply the existence of intractable problems (i.e. NP-problems which are not in the P complexity class). If true, the existence of one-way functions would have an impact on the theoretical framework of physics, in particularly, quantum mechanics. Such aspect of one-way functions has never been shown before. In the present work, we put forward the following. We can calculate the microscopic state (say, the particle spin in the z direction) of a macroscopic system (a measuring apparatus registering the particle z-spin) by the system macroscopic state (the apparatus output); let us call this association the function F. The question is: can we compute the function F in the inverse direction? In other words, can we compute the macroscopic state of the system through its microscopic state (the preimage F -1)? In the paper, we assume that the function F is a one-way function. The assumption implies that at the macroscopic level the Schrödinger equation becomes unfeasible to compute. This unfeasibility plays a role of limit of the validity of the linear Schrödinger equation.
Keywords: One-way functions, P versus NP problem, quantummeasurements.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1196
1509 Stability of Stochastic Model Predictive Control for Schrödinger Equation with Finite Approximation
Authors: Tomoaki Hashimoto
Abstract:Recent technological advance has prompted significant interest in developing the control theory of quantum systems. Following the increasing interest in the control of quantum dynamics, this paper examines the control problem of Schrödinger equation because quantum dynamics is basically governed by Schrödinger equation. From the practical point of view, stochastic disturbances cannot be avoided in the implementation of control method for quantum systems. Thus, we consider here the robust stabilization problem of Schrödinger equation against stochastic disturbances. In this paper, we adopt model predictive control method in which control performance over a finite future is optimized with a performance index that has a moving initial and terminal time. The objective of this study is to derive the stability criterion for model predictive control of Schrödinger equation under stochastic disturbances.
Keywords: Optimal control, stochastic systems, quantum systems, stabilization.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 2058
1508 Schrödinger Equation with Position-Dependent Mass: Staggered Mass Distributions
Authors: J. J. Peña, J. Morales, J. García-Ravelo, L. Arcos-Díaz
The Point canonical transformation method is applied for solving the Schrödinger equation with position-dependent mass. This class of problem has been solved for continuous mass distributions. In this work, a staggered mass distribution for the case of a free particle in an infinite square well potential has been proposed. The continuity conditions as well as normalization for the wave function are also considered. The proposal can be used for dealing with other kind of staggered mass distributions in the Schrödinger equation with different quantum potentials.
Keywords: Free particle, point canonical transformation method, position-dependent mass, staggered mass distribution.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1135
1507 A Novel System of Two Coupled Equations for the Longitudinal Components of the Electromagnetic Field in a Waveguide
Authors: Arti Vaish, Harish Parthasarathy
Abstract:In this paper, a novel wave equation for electromagnetic waves in a medium having anisotropic permittivity has been derived with the help of Maxwell-s curl equations. The x and y components of the Maxwell-s equations are written with the permittivity () being a 3 × 3 symmetric matrix. These equations are solved for Ex , Ey, Hx, Hy in terms of Ez, Hz, and the partial derivatives. The Z components of the Maxwell-s curl are then used to arrive to the generalized Helmholtz equations for Ez and Hz.
Keywords: Electromagnetism, Maxwell's Equations, Anisotropic permittivity, Wave equation, Matrix Equation, Permittivity tensor.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1569
1506 Numerical Solution of Linear Ordinary Differential Equations in Quantum Chemistry by Clenshaw Method
Authors: M. Saravi, F. Ashrafi, S.R. Mirrajei
Abstract:As we know, most differential equations concerning physical phenomenon could not be solved by analytical method. Even if we use Series Method, some times we need an appropriate change of variable, and even when we can, their closed form solution may be so complicated that using it to obtain an image or to examine the structure of the system is impossible. For example, if we consider Schrodinger equation, i.e., We come to a three-term recursion relations, which work with it takes, at least, a little bit time to get a series solution. For this reason we use a change of variable such as or when we consider the orbital angular momentum, it will be necessary to solve. As we can observe, working with this equation is tedious. In this paper, after introducing Clenshaw method, which is a kind of Spectral method, we try to solve some of such equations.
Keywords: Chebyshev polynomials, Clenshaw method, ODEs, Spectral methodsProcedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1301
1505 Soliton Interaction in Multi-Core Optical Fiber: Application to WDM System
Authors: S. Arun Prakash, V. Malathi, M. S. Mani Rajan
Abstract:The analytical bright two soliton solution of the 3- coupled nonlinear Schrödinger equations with variable coefficients in birefringent optical fiber is obtained by Darboux transformation method. To the design of ultra-speed optical devices, Soliton interaction and control in birefringence fiber is investigated. Lax pair is constructed for N coupled NLS system through AKNS method. Using two-soliton solution, we demonstrate different interaction behaviors of solitons in birefringent fiber depending on the choice of control parameters. Our results shows that interactions of optical solitons have some specific applications such as construction of logic gates, optical computing, soliton switching, and soliton amplification in wavelength division multiplexing (WDM) system.
Keywords: Optical soliton, soliton interaction, soliton switching, WDM.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1860
1504 On the Integer Solutions of the Pell Equation x2 - dy2 = 2t
Authors: Ahmet Tekcan, Betül Gezer, Osman Bizim
Let k ≥ 1 and t ≥ 0 be two integers and let d = k2 + k be a positive non-square integer. In this paper, we consider the integer solutions of Pell equation x2 - dy2 = 2t. Further we derive a recurrence relation on the solutions of this equation.
Keywords: Pell equation, Diophantine equation.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 2194
1503 Mathematical Modelling of Transport Phenomena in Radioactive Waste-Cement-Bentonite Matrix
Authors: Ilija Plecas, Uranija Kozmidis-Luburic, Radojica Pesic
The leaching rate of 137Cs from spent mix bead (anion and cation) exchange resins in a cement-bentonite matrix has been studied. Transport phenomena involved in the leaching of a radioactive material from a cement-bentonite matrix are investigated using three methods based on theoretical equations. These are: the diffusion equation for a plane source an equation for diffusion coupled to a firstorder equation and an empirical method employing a polynomial equation. The results presented in this paper are from a 25-year mortar and concrete testing project that will influence the design choices for radioactive waste packaging for a future Serbian radioactive waste disposal center.
Keywords: bentonite, cement , radioactive waste, composite, disposal, diffusionProcedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 2178
1502 Dust Acoustic Shock Waves in Coupled Dusty Plasmas with Kappa-Distributed Ions
Authors: Hamid Reza Pakzad
We have considered an unmagnetized dusty plasma system consisting of ions obeying superthermal distribution and strongly coupled negatively charged dust. We have used reductive perturbation method and derived the Kordeweg-de Vries-Burgers (KdV-Burgers) equation. The behavior of the shock waves in the plasma has been investigated.
Keywords: Shock, Soliton, Coupling, Superthermal ions.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1786
1501 The Proof of Two Conjectures Related to Pell-s Equation x2 −Dy2 = ± 4
Authors: Armend Sh. Shabani
Abstract:Let D ≠ 1 be a positive non-square integer. In this paper are given the proofs for two conjectures related to Pell-s equation x2 -Dy2 = ± 4, proposed by A. Tekcan.
Keywords: Pell's equation, solutions of Pell's equation.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1120
1500 DQ Analysis of 3D Natural Convection in an Inclined Cavity Using an Velocity-Vorticity Formulation
In this paper, the differential quadrature method is applied to simulate natural convection in an inclined cubic cavity using velocity-vorticity formulation. The numerical capability of the present algorithm is demonstrated by application to natural convection in an inclined cubic cavity. The velocity Poisson equations, the vorticity transport equations and the energy equation are all solved as a coupled system of equations for the seven field variables consisting of three velocities, three vorticities and temperature. The coupled equations are simultaneously solved by imposing the vorticity definition at boundary without requiring the explicit specification of the vorticity boundary conditions. Test results obtained for an inclined cubic cavity with different angle of inclinations for Rayleigh number equal to 103, 104, 105 and 106 indicate that the present coupled solution algorithm could predict the benchmark results for temperature and flow fields. Thus, it is convinced that the present formulation is capable of solving coupled Navier-Stokes equations effectively and accurately.
Keywords: Natural convection, velocity-vorticity formulation, differential quadrature (DQ).Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1469
1499 An Analytical Method for Solving General Riccati Equation
Authors: Y. Pala, M. O. Ertas
In this paper, the general Riccati equation is analytically solved by a new transformation. By the method developed, looking at the transformed equation, whether or not an explicit solution can be obtained is readily determined. Since the present method does not require a proper solution for the general solution, it is especially suitable for equations whose proper solutions cannot be seen at first glance. Since the transformed second order linear equation obtained by the present transformation has the simplest form that it can have, it is immediately seen whether or not the original equation can be solved analytically. The present method is exemplified by several examples.
Keywords: Riccati Equation, ordinary differential equation, nonlinear differential equation, analytical solution, proper solution.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1872
1498 The Pell Equation x2 − Py2 = Q
Authors: Ahmet Tekcan, Arzu Özkoç, Canan Kocapınar, Hatice Alkan
Abstract:Let p be a prime number such that p ≡ 1(mod 4), say p = 1+4k for a positive integer k. Let P = 2k + 1 and Q = k2. In this paper, we consider the integer solutions of the Pell equation x2-Py2 = Q over Z and also over finite fields Fp. Also we deduce some relations on the integer solutions (xn, yn) of it.
Keywords: Pell equation, solutions of Pell equation.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1961
1497 The Diophantine Equation y2 − 2yx − 3 = 0 and Corresponding Curves over Fp
Authors: Ahmet Tekcan, Arzu Özkoç, Hatice Alkan
Abstract:In this work, we consider the number of integer solutions of Diophantine equation D : y2 - 2yx - 3 = 0 over Z and also over finite fields Fp for primes p ≥ 5. Later we determine the number of rational points on curves Ep : y2 = Pp(x) = yp 1 + yp 2 over Fp, where y1 and y2 are the roots of D. Also we give a formula for the sum of x- and y-coordinates of all rational points (x, y) on Ep over Fp.
Keywords: Diophantine equation, Pell equation, quadratic form.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1163
1496 Equations of Pulse Propagation in Three-Layer Structure of As2S3 Chalcogenide Plasmonic Nano-Waveguides
Authors: Leila Motamed-Jahromi, Mohsen Hatami, Alireza Keshavarz
This research aims at obtaining the equations of pulse propagation in nonlinear plasmonic waveguides created with As2S3 chalcogenide materials. Via utilizing Helmholtz equation and first-order perturbation theory, two components of electric field are determined within frequency domain. Afterwards, the equations are formulated in time domain. The obtained equations include two coupled differential equations that considers nonlinear dispersion.
Keywords: Nonlinear optics, propagation equation, plasmonic waveguide.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1020
1495 Solution of The KdV Equation with Asymptotic Degeneracy
Authors: Tapas Kumar Sinha, Joseph Mathew
Recently T. C. Au-Yeung, C.Au, and P. C. W. Fung  have given the solution of the KdV equation  to the boundary condition , where b is a constant. We have further extended the method of  to find the solution of the KdV equation with asymptotic degeneracy. Via simulations we find both bright and dark Solitons (i.e. Solitons with opposite phases).
Keywords: KdV equation, Asymptotic Degeneracy, Solitons, Inverse ScatteringProcedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1462
1494 Exact Solutions of the Helmholtz equation via the Nikiforov-Uvarov Method
Authors: Said Laachir, Aziz Laaribi
The Helmholtz equation often arises in the study of physical problems involving partial differential equation. Many researchers have proposed numerous methods to find the analytic or approximate solutions for the proposed problems. In this work, the exact analytical solutions of the Helmholtz equation in spherical polar coordinates are presented using the Nikiforov-Uvarov (NU) method. It is found that the solution of the angular eigenfunction can be expressed by the associated-Legendre polynomial and radial eigenfunctions are obtained in terms of the Laguerre polynomials. The special case for k=0, which corresponds to the Laplace equation is also presented.
Keywords: Helmholtz equation, Nikiforov-Uvarov method, exact solutions, eigenfunctions.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 2869
1493 Study of Cahn-Hilliard Equation to Simulate Phase Separation
Authors: Nara Guimarães, Marcelo Aquino Martorano, Douglas Gouvêa
An investigation into Cahn-Hilliard equation was carried out through numerical simulation to identify a possible phase separation for one and two dimensional domains. It was observed that this equation can reproduce important mass fluxes necessary for phase separation within the miscibility gap and for coalescence of particles.
Keywords: Cahn-Hilliard equation, miscibility gap, phase separation.Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1727