A Combined MPI-CUDA Parallel Solution of Linear and Nonlinear Poisson-Boltzmann Equation
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José Colmenares
2014-01-01
Full Text Available The Poisson-Boltzmann equation models the electrostatic potential generated by fixed charges on a polarizable solute immersed in an ionic solution. This approach is often used in computational structural biology to estimate the electrostatic energetic component of the assembly of molecular biological systems. In the last decades, the amount of data concerning proteins and other biological macromolecules has remarkably increased. To fruitfully exploit these data, a huge computational power is needed as well as software tools capable of exploiting it. It is therefore necessary to move towards high performance computing and to develop proper parallel implementations of already existing and of novel algorithms. Nowadays, workstations can provide an amazing computational power: up to 10 TFLOPS on a single machine equipped with multiple CPUs and accelerators such as Intel Xeon Phi or GPU devices. The actual obstacle to the full exploitation of modern heterogeneous resources is efficient parallel coding and porting of software on such architectures. In this paper, we propose the implementation of a full Poisson-Boltzmann solver based on a finite-difference scheme using different and combined parallel schemes and in particular a mixed MPI-CUDA implementation. Results show great speedups when using the two schemes, achieving an 18.9x speedup using three GPUs.
International Nuclear Information System (INIS)
Rasulova, M.Yu
1998-01-01
A study has been made of a system of charged particles and inhomogeneities randomly distributed in accordance with the same law in the neighborhoods of corresponding sites of a planar crystal lattice. The existence and uniqueness of the solution of the generalized Poisson-Boltzmann's equation for the average self-consistent potential and average density of surface charges are proved. (author)
A modified Poisson-Boltzmann equation applied to protein adsorption.
Gama, Marlon de Souza; Santos, Mirella Simões; Lima, Eduardo Rocha de Almeida; Tavares, Frederico Wanderley; Barreto, Amaro Gomes Barreto
2018-01-05
Ion-exchange chromatography has been widely used as a standard process in purification and analysis of protein, based on the electrostatic interaction between the protein and the stationary phase. Through the years, several approaches are used to improve the thermodynamic description of colloidal particle-surface interaction systems, however there are still a lot of gaps specifically when describing the behavior of protein adsorption. Here, we present an improved methodology for predicting the adsorption equilibrium constant by solving the modified Poisson-Boltzmann (PB) equation in bispherical coordinates. By including dispersion interactions between ions and protein, and between ions and surface, the modified PB equation used can describe the Hofmeister effects. We solve the modified Poisson-Boltzmann equation to calculate the protein-surface potential of mean force, treated as spherical colloid-plate system, as a function of process variables. From the potential of mean force, the Henry constants of adsorption, for different proteins and surfaces, are calculated as a function of pH, salt concentration, salt type, and temperature. The obtained Henry constants are compared with experimental data for several isotherms showing excellent agreement. We have also performed a sensitivity analysis to verify the behavior of different kind of salts and the Hofmeister effects. Copyright © 2017 Elsevier B.V. All rights reserved.
Fellner, Klemens; Kovtunenko, Victor A
2016-01-01
A nonlinear Poisson-Boltzmann equation with inhomogeneous Robin type boundary conditions at the interface between two materials is investigated. The model describes the electrostatic potential generated by a vector of ion concentrations in a periodic multiphase medium with dilute solid particles. The key issue stems from interfacial jumps, which necessitate discontinuous solutions to the problem. Based on variational techniques, we derive the homogenisation of the discontinuous problem and establish a rigorous residual error estimate up to the first-order correction.
Structural interactions in ionic liquids linked to higher-order Poisson-Boltzmann equations
Blossey, R.; Maggs, A. C.; Podgornik, R.
2017-06-01
We present a derivation of generalized Poisson-Boltzmann equations starting from classical theories of binary fluid mixtures, employing an approach based on the Legendre transform as recently applied to the case of local descriptions of the fluid free energy. Under specific symmetry assumptions, and in the linearized regime, the Poisson-Boltzmann equation reduces to a phenomenological equation introduced by Bazant et al. [Phys. Rev. Lett. 106, 046102 (2011)], 10.1103/PhysRevLett.106.046102, whereby the structuring near the surface is determined by bulk coefficients.
Advanced diffusion model in compacted bentonite based on modified Poisson-Boltzmann equations
International Nuclear Information System (INIS)
Yotsuji, K.; Tachi, Y.; Nishimaki, Y.
2012-01-01
Document available in extended abstract form only. Diffusion and sorption of radionuclides in compacted bentonite are the key processes in the safe geological disposal of radioactive waste. JAEA has developed the integrated sorption and diffusion (ISD) model for compacted bentonite by coupling the pore water chemistry, sorption and diffusion processes in consistent way. The diffusion model accounts consistently for cation excess and anion exclusion in narrow pores in compacted bentonite by the electric double layer (EDL) theory. The firstly developed ISD model could predict the diffusivity of the monovalent cation/anion in compacted bentonite as a function of dry density. This ISD model was modified by considering the visco-electric effect, and applied for diffusion data for various radionuclides measured under wide range of conditions (salinity, density, etc.). This modified ISD model can give better quantitative agreement with diffusion data for monovalent cation/anion, however, the model predictions still disagree with experimental data for multivalent cation and complex species. In this study we extract the additional key factors influencing diffusion model in narrow charged pores, and the effects of these factors were investigated to reach a better understanding of diffusion processes in compacted bentonite. We investigated here the dielectric saturation effect and the excluded volume effect into the present ISD model and numerically solved these modified Poisson-Boltzmann equations. In the vicinity of the negatively charged clay surfaces, it is necessary to evaluate concentration distribution of electrolytes considering the dielectric saturation effects. The Poisson-Boltzmann (P-B) equation coupled with the dielectric saturation effects was solved numerically by using Runge-Kutta and Shooting methods. Figure 1(a) shows the concentration distributions of Na + as numerical solutions of the modified and original P-B equations for 0.01 M pore water, 800 kg m -3
Altman, Michael D.; Bardhan, Jaydeep P.; White, Jacob K.; Tidor, Bruce
2009-01-01
We present a boundary-element method (BEM) implementation for accurately solving problems in biomolecular electrostatics using the linearized Poisson–Boltzmann equation. Motivating this implementation is the desire to create a solver capable of precisely describing the geometries and topologies prevalent in continuum models of biological molecules. This implementation is enabled by the synthesis of four technologies developed or implemented specifically for this work. First, molecular and accessible surfaces used to describe dielectric and ion-exclusion boundaries were discretized with curved boundary elements that faithfully reproduce molecular geometries. Second, we avoided explicitly forming the dense BEM matrices and instead solved the linear systems with a preconditioned iterative method (GMRES), using a matrix compression algorithm (FFTSVD) to accelerate matrix-vector multiplication. Third, robust numerical integration methods were employed to accurately evaluate singular and near-singular integrals over the curved boundary elements. Finally, we present a general boundary-integral approach capable of modeling an arbitrary number of embedded homogeneous dielectric regions with differing dielectric constants, possible salt treatment, and point charges. A comparison of the presented BEM implementation and standard finite-difference techniques demonstrates that for certain classes of electrostatic calculations, such as determining absolute electrostatic solvation and rigid-binding free energies, the improved convergence properties of the BEM approach can have a significant impact on computed energetics. We also demonstrate that the improved accuracy offered by the curved-element BEM is important when more sophisticated techniques, such as non-rigid-binding models, are used to compute the relative electrostatic effects of molecular modifications. In addition, we show that electrostatic calculations requiring multiple solves using the same molecular geometry
AQUASOL: An efficient solver for the dipolar Poisson-Boltzmann-Langevin equation.
Koehl, Patrice; Delarue, Marc
2010-02-14
The Poisson-Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE
Xie, Yang; Ying, Jinyong; Xie, Dexuan
2017-03-30
SMPBS (Size Modified Poisson-Boltzmann Solvers) is a web server for computing biomolecular electrostatics using finite element solvers of the size modified Poisson-Boltzmann equation (SMPBE). SMPBE not only reflects ionic size effects but also includes the classic Poisson-Boltzmann equation (PBE) as a special case. Thus, its web server is expected to have a broader range of applications than a PBE web server. SMPBS is designed with a dynamic, mobile-friendly user interface, and features easily accessible help text, asynchronous data submission, and an interactive, hardware-accelerated molecular visualization viewer based on the 3Dmol.js library. In particular, the viewer allows computed electrostatics to be directly mapped onto an irregular triangular mesh of a molecular surface. Due to this functionality and the fast SMPBE finite element solvers, the web server is very efficient in the calculation and visualization of electrostatics. In addition, SMPBE is reconstructed using a new objective electrostatic free energy, clearly showing that the electrostatics and ionic concentrations predicted by SMPBE are optimal in the sense of minimizing the objective electrostatic free energy. SMPBS is available at the URL: smpbs.math.uwm.edu © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.
Womack, James C; Anton, Lucian; Dziedzic, Jacek; Hasnip, Phil J; Probert, Matt I J; Skylaris, Chris-Kriton
2018-03-13
The solution of the Poisson equation is a crucial step in electronic structure calculations, yielding the electrostatic potential-a key component of the quantum mechanical Hamiltonian. In recent decades, theoretical advances and increases in computer performance have made it possible to simulate the electronic structure of extended systems in complex environments. This requires the solution of more complicated variants of the Poisson equation, featuring nonhomogeneous dielectric permittivities, ionic concentrations with nonlinear dependencies, and diverse boundary conditions. The analytic solutions generally used to solve the Poisson equation in vacuum (or with homogeneous permittivity) are not applicable in these circumstances, and numerical methods must be used. In this work, we present DL_MG, a flexible, scalable, and accurate solver library, developed specifically to tackle the challenges of solving the Poisson equation in modern large-scale electronic structure calculations on parallel computers. Our solver is based on the multigrid approach and uses an iterative high-order defect correction method to improve the accuracy of solutions. Using two chemically relevant model systems, we tested the accuracy and computational performance of DL_MG when solving the generalized Poisson and Poisson-Boltzmann equations, demonstrating excellent agreement with analytic solutions and efficient scaling to ∼10 9 unknowns and 100s of CPU cores. We also applied DL_MG in actual large-scale electronic structure calculations, using the ONETEP linear-scaling electronic structure package to study a 2615 atom protein-ligand complex with routinely available computational resources. In these calculations, the overall execution time with DL_MG was not significantly greater than the time required for calculations using a conventional FFT-based solver.
Fogolari, Federico; Corazza, Alessandra; Esposito, Gennaro
2015-04-05
The generalized Born model in the Onufriev, Bashford, and Case (Onufriev et al., Proteins: Struct Funct Genet 2004, 55, 383) implementation has emerged as one of the best compromises between accuracy and speed of computation. For simulations of nucleic acids, however, a number of issues should be addressed: (1) the generalized Born model is based on a linear model and the linearization of the reference Poisson-Boltmann equation may be questioned for highly charged systems as nucleic acids; (2) although much attention has been given to potentials, solvation forces could be much less sensitive to linearization than the potentials; and (3) the accuracy of the Onufriev-Bashford-Case (OBC) model for nucleic acids depends on fine tuning of parameters. Here, we show that the linearization of the Poisson Boltzmann equation has mild effects on computed forces, and that with optimal choice of the OBC model parameters, solvation forces, essential for molecular dynamics simulations, agree well with those computed using the reference Poisson-Boltzmann model. © 2015 Wiley Periodicals, Inc.
Large Time Behavior of the Vlasov-Poisson-Boltzmann System
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Li Li
2013-01-01
Full Text Available The motion of dilute charged particles can be modeled by Vlasov-Poisson-Boltzmann system. We study the large time stability of the VPB system. To be precise, we prove that when time goes to infinity, the solution of VPB system tends to global Maxwellian state in a rate Ot−∞, by using a method developed for Boltzmann equation without force in the work of Desvillettes and Villani (2005. The improvement of the present paper is the removal of condition on parameter λ as in the work of Li (2008.
Poisson-Boltzmann versus Size-Modified Poisson-Boltzmann Electrostatics Applied to Lipid Bilayers.
Wang, Nuo; Zhou, Shenggao; Kekenes-Huskey, Peter M; Li, Bo; McCammon, J Andrew
2014-12-26
Mean-field methods, such as the Poisson-Boltzmann equation (PBE), are often used to calculate the electrostatic properties of molecular systems. In the past two decades, an enhancement of the PBE, the size-modified Poisson-Boltzmann equation (SMPBE), has been reported. Here, the PBE and the SMPBE are reevaluated for realistic molecular systems, namely, lipid bilayers, under eight different sets of input parameters. The SMPBE appears to reproduce the molecular dynamics simulation results better than the PBE only under specific parameter sets, but in general, it performs no better than the Stern layer correction of the PBE. These results emphasize the need for careful discussions of the accuracy of mean-field calculations on realistic systems with respect to the choice of parameters and call for reconsideration of the cost-efficiency and the significance of the current SMPBE formulation.
Poisson-Boltzmann-Nernst-Planck model
International Nuclear Information System (INIS)
Zheng Qiong; Wei Guowei
2011-01-01
The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external
Poisson-Boltzmann-Nernst-Planck model.
Zheng, Qiong; Wei, Guo-Wei
2011-05-21
The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external
A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments
International Nuclear Information System (INIS)
Fisicaro, G.; Goedecker, S.; Genovese, L.; Andreussi, O.; Marzari, N.
2016-01-01
The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into account the non-trivial electrostatic screening coming from the solvent and the electrolytes. As a consequence, the electrostatic potential has to be found by solving the generalized Poisson and the Poisson-Boltzmann equations for neutral and ionic solutions, respectively. In the present work, solvers for both problems have been developed. A preconditioned conjugate gradient method has been implemented for the solution of the generalized Poisson equation and the linear regime of the Poisson-Boltzmann, allowing to solve iteratively the minimization problem with some ten iterations of the ordinary Poisson equation solver. In addition, a self-consistent procedure enables us to solve the non-linear Poisson-Boltzmann problem. Both solvers exhibit very high accuracy and parallel efficiency and allow for the treatment of periodic, free, and slab boundary conditions. The solver has been integrated into the BigDFT and Quantum-ESPRESSO electronic-structure packages and will be released as an independent program, suitable for integration in other codes
A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments.
Fisicaro, G; Genovese, L; Andreussi, O; Marzari, N; Goedecker, S
2016-01-07
The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into account the non-trivial electrostatic screening coming from the solvent and the electrolytes. As a consequence, the electrostatic potential has to be found by solving the generalized Poisson and the Poisson-Boltzmann equations for neutral and ionic solutions, respectively. In the present work, solvers for both problems have been developed. A preconditioned conjugate gradient method has been implemented for the solution of the generalized Poisson equation and the linear regime of the Poisson-Boltzmann, allowing to solve iteratively the minimization problem with some ten iterations of the ordinary Poisson equation solver. In addition, a self-consistent procedure enables us to solve the non-linear Poisson-Boltzmann problem. Both solvers exhibit very high accuracy and parallel efficiency and allow for the treatment of periodic, free, and slab boundary conditions. The solver has been integrated into the BigDFT and Quantum-ESPRESSO electronic-structure packages and will be released as an independent program, suitable for integration in other codes.
A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments
Energy Technology Data Exchange (ETDEWEB)
Fisicaro, G., E-mail: giuseppe.fisicaro@unibas.ch; Goedecker, S. [Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel (Switzerland); Genovese, L. [University of Grenoble Alpes, CEA, INAC-SP2M, L-Sim, F-38000 Grenoble (France); Andreussi, O. [Institute of Computational Science, Università della Svizzera Italiana, Via Giuseppe Buffi 13, CH-6904 Lugano (Switzerland); Theory and Simulations of Materials (THEOS) and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, Station 12, CH-1015 Lausanne (Switzerland); Marzari, N. [Theory and Simulations of Materials (THEOS) and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, Station 12, CH-1015 Lausanne (Switzerland)
2016-01-07
The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into account the non-trivial electrostatic screening coming from the solvent and the electrolytes. As a consequence, the electrostatic potential has to be found by solving the generalized Poisson and the Poisson-Boltzmann equations for neutral and ionic solutions, respectively. In the present work, solvers for both problems have been developed. A preconditioned conjugate gradient method has been implemented for the solution of the generalized Poisson equation and the linear regime of the Poisson-Boltzmann, allowing to solve iteratively the minimization problem with some ten iterations of the ordinary Poisson equation solver. In addition, a self-consistent procedure enables us to solve the non-linear Poisson-Boltzmann problem. Both solvers exhibit very high accuracy and parallel efficiency and allow for the treatment of periodic, free, and slab boundary conditions. The solver has been integrated into the BigDFT and Quantum-ESPRESSO electronic-structure packages and will be released as an independent program, suitable for integration in other codes.
PB-AM: An open-source, fully analytical linear poisson-boltzmann solver.
Felberg, Lisa E; Brookes, David H; Yap, Eng-Hui; Jurrus, Elizabeth; Baker, Nathan A; Head-Gordon, Teresa
2017-06-05
We present the open source distributed software package Poisson-Boltzmann Analytical Method (PB-AM), a fully analytical solution to the linearized PB equation, for molecules represented as non-overlapping spherical cavities. The PB-AM software package includes the generation of outputs files appropriate for visualization using visual molecular dynamics, a Brownian dynamics scheme that uses periodic boundary conditions to simulate dynamics, the ability to specify docking criteria, and offers two different kinetics schemes to evaluate biomolecular association rate constants. Given that PB-AM defines mutual polarization completely and accurately, it can be refactored as a many-body expansion to explore 2- and 3-body polarization. Additionally, the software has been integrated into the Adaptive Poisson-Boltzmann Solver (APBS) software package to make it more accessible to a larger group of scientists, educators, and students that are more familiar with the APBS framework. © 2016 Wiley Periodicals, Inc. © 2016 Wiley Periodicals, Inc.
Directory of Open Access Journals (Sweden)
Thereza A. Soares
2004-08-01
Full Text Available The ability of biomolecules to catalyze chemical reactions is due chiefly to their sensitivity to variations of the pH in the surrounding environment. The reason for this is that they are made up of chemical groups whose ionization states are modulated by pH changes that are of the order of 0.4 units. The determination of the protonation states of such chemical groups as a function of conformation of the biomolecule and the pH of the environment can be useful in the elucidation of important biological processes from enzymatic catalysis to protein folding and molecular recognition. In the past 15 years, the theory of Poisson-Boltzmann has been successfully used to estimate the pKa of ionizable sites in proteins yielding results, which may differ by 0.1 unit from the experimental values. In this study, we review the theory of Poisson-Boltzmann under the perspective of its application to the calculation of pKa in proteins.
Poisson-Boltzmann theory of the charge-induced adsorption of semi-flexible polyelectrolytes.
Ubbink, Job; Khokhlov, Alexei R
2004-03-15
A model is suggested for the structure of an adsorbed layer of a highly charged semi-flexible polyelectrolyte on a weakly charged surface of opposite charge sign. The adsorbed phase is thin, owing to the effective reversal of the charge sign of the surface upon adsorption, and ordered, owing to the high surface density of polyelectrolyte strands caused by the generally strong binding between polyelectrolyte and surface. The Poisson-Boltzmann equation for the electrostatic interaction between the array of adsorbed polyelectrolytes and the charged surface is solved for a cylindrical geometry, both numerically, using a finite element method, and analytically within the weak curvature limit under the assumption of excess monovalent salt. For small separations, repulsive surface polarization and counterion osmotic pressure effects dominate over the electrostatic attraction and the resulting electrostatic interaction curve shows a minimum at nonzero separations on the Angstrom scale. The equilibrium density of the adsorbed phase is obtained by minimizing the total free energy under the condition of equality of chemical potential and osmotic pressure of the polyelectrolyte in solution and in the adsorbed phase. For a wide range of ionic conditions and charge densities of the charged surface, the interstrand separation as predicted by the Poisson-Boltzmann model and the analytical theory closely agree. For low to moderate charge densities of the adsorbing surface, the interstrand spacing decreases as a function of the charge density of the charged surface. Above about 0.1 M excess monovalent salt, it is only weakly dependent on the ionic strength. At high charge densities of the adsorbing surface, the interstrand spacing increases with increasing ionic strength, in line with the experiments by Fang and Yang [J. Phys. Chem. B 101, 441 (1997)]. (c) 2004 American Institute of Physics.
Poisson-Boltzmann theory of charged colloids: limits of the cell model for salty suspensions
International Nuclear Information System (INIS)
Denton, A R
2010-01-01
Thermodynamic properties of charge-stabilized colloidal suspensions and polyelectrolyte solutions are commonly modelled by implementing the mean-field Poisson-Boltzmann (PB) theory within a cell model. This approach models a bulk system by a single macroion, together with counterions and salt ions, confined to a symmetrically shaped, electroneutral cell. While easing numerical solution of the nonlinear PB equation, the cell model neglects microion-induced interactions and correlations between macroions, precluding modelling of macroion ordering phenomena. An alternative approach, which avoids the artificial constraints of cell geometry, exploits the mapping of a macroion-microion mixture onto a one-component model of pseudo-macroions governed by effective interparticle interactions. In practice, effective-interaction models are usually based on linear-screening approximations, which can accurately describe strong nonlinear screening only by incorporating an effective (renormalized) macroion charge. Combining charge renormalization and linearized PB theories, in both the cell model and an effective-interaction (cell-free) model, we compute osmotic pressures of highly charged colloids and monovalent microions, in Donnan equilibrium with a salt reservoir, over a range of concentrations. By comparing predictions with primitive model simulation data for salt-free suspensions, and with predictions from nonlinear PB theory for salty suspensions, we chart the limits of both the cell model and linear-screening approximations in modelling bulk thermodynamic properties. Up to moderately strong electrostatic couplings, the cell model proves accurate for predicting osmotic pressures of deionized (counterion-dominated) suspensions. With increasing salt concentration, however, the relative contribution of macroion interactions to the osmotic pressure grows, leading predictions from the cell and effective-interaction models to deviate. No evidence is found for a liquid
Bijma, K; Engberts, J B F N
This paper describes how the theory of the ''dressed micelle'', which is based on the nonlinear Poisson-Boltzmann equation, can be used to calculate a number of thermodynamic quantities for micellization of sodium p-alkylbenzenesulphonates. From the Gibbs energy of micellization, the enthalpy of
Ca/Na selectivity coefficients from the Poisson-Boltzmann theory
International Nuclear Information System (INIS)
Hedstroem, Magnus; Karnland, Ola
2010-01-01
Document available in extended abstract form only. A possible scenario in the post-glacial evolution of the bentonite buffer used in a KBS-3 repository for spent nuclear fuel is that parts of the buffer may erode due to sol formation caused by the extensive swelling of, in particular, Na-montmorillonite in water of low ionic strength. Presence of calcium in the interlayer has been shown to promote gel formation even in electrolytes with ionic strengths in the vicinity of those in glacial melt waters. In order to estimate the amount of calcium in the clay at the onset of glaciation one needs information of the selectivity coefficient for Ca/Na exchange. Hitherto, most experimental data for evaluating the Gaines-Thomas selectivity coefficient, K GT have been obtained in batch experiments, i.e. at high water-to-solid ratios. The conditions in highly compacted bentonite are, however, radically different in many respects, e.g. the interlayer space is on the nanometre scale and the concentration of counterions is in molar range. Therefore we would like to theoretically investigate the transferability of the selectivity coefficients, determined in batch experiments, to compacted conditions. We solve the Poisson-Boltzmann (PB) equation for two parallel charged surfaces in equilibrium with an external NaCl/CaCl 2 mixed solution. Integration of the ion concentration profiles obtained from the PB equation gives the occupancy of Na + and Ca 2+ in the clay. That information together with the composition of the external electrolyte is all that is needed for the calculation of K GT . With a surface layer-charge density of one charge per 145 A 2 , which is close to the value for Wyoming montmorillonite, we find a variation of the selectivity coefficient from about 4 M in batch to 8 M for compacted montmorillonite with dry density 1700 kg/m 3 . The significance as well as the physics behind these results will be presented in detail. The predictions, based on the PB theory, will
Acceleration of Linear Finite-Difference Poisson-Boltzmann Methods on Graphics Processing Units.
Qi, Ruxi; Botello-Smith, Wesley M; Luo, Ray
2017-07-11
Electrostatic interactions play crucial roles in biophysical processes such as protein folding and molecular recognition. Poisson-Boltzmann equation (PBE)-based models have emerged as widely used in modeling these important processes. Though great efforts have been put into developing efficient PBE numerical models, challenges still remain due to the high dimensionality of typical biomolecular systems. In this study, we implemented and analyzed commonly used linear PBE solvers for the ever-improving graphics processing units (GPU) for biomolecular simulations, including both standard and preconditioned conjugate gradient (CG) solvers with several alternative preconditioners. Our implementation utilizes the standard Nvidia CUDA libraries cuSPARSE, cuBLAS, and CUSP. Extensive tests show that good numerical accuracy can be achieved given that the single precision is often used for numerical applications on GPU platforms. The optimal GPU performance was observed with the Jacobi-preconditioned CG solver, with a significant speedup over standard CG solver on CPU in our diversified test cases. Our analysis further shows that different matrix storage formats also considerably affect the efficiency of different linear PBE solvers on GPU, with the diagonal format best suited for our standard finite-difference linear systems. Further efficiency may be possible with matrix-free operations and integrated grid stencil setup specifically tailored for the banded matrices in PBE-specific linear systems.
Nonextensive statistical mechanics of ionic solutions
International Nuclear Information System (INIS)
Varela, L.M.; Carrete, J.; Munoz-Sola, R.; Rodriguez, J.R.; Gallego, J.
2007-01-01
Classical mean-field Poisson-Boltzmann theory of ionic solutions is revisited in the theoretical framework of nonextensive Tsallis statistics. The nonextensive equivalent of Poisson-Boltzmann equation is formulated revisiting the statistical mechanics of liquids and the Debye-Hueckel framework is shown to be valid for highly diluted solutions even under circumstances where nonextensive thermostatistics must be applied. The lowest order corrections associated to nonadditive effects are identified for both symmetric and asymmetric electrolytes and the behavior of the average electrostatic potential in a homogeneous system is analytically and numerically analyzed for various values of the complexity measurement nonextensive parameter q
Stability of Nonlinear Wave Patterns to the Bipolar Vlasov-Poisson-Boltzmann System
Li, Hailiang; Wang, Yi; Yang, Tong; Zhong, Mingying
2018-04-01
The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction waves for the bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, motivated by the micro-macro decomposition to the Boltzmann equation in Liu and Yu (Commun Math Phys 246:133-179, 2004) and Liu et al. (Physica D 188:178-192, 2004), we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave patterns, viscous shock waves and rarefaction waves are proved for the 1D bipolar VPB system. More precisely, it is first proved that the linear superposition of two Boltzmann shock profiles in the first and third characteristic fields is nonlinearly stable to the 1D bipolar VPB system up to some suitable shifts without the zero macroscopic mass conditions on the initial perturbations. Then the time-asymptotic stability of the rarefaction wave fan to compressible Euler equations is proved for the 1D bipolar VPB system. These two results are concerned with the nonlinear stability of wave patterns for Boltzmann equation coupled with additional (electric) forces, which together with spectral analysis made in Li et al. (Indiana Univ Math J 65(2):665-725, 2016) sheds light on understanding the complicated dynamic behaviors around the wave patterns in the transportation of charged particles under the binary collisions, mutual interactions, and the effect of the electrostatic potential forces.
Directory of Open Access Journals (Sweden)
L.B.Bhuiyan
2005-01-01
Full Text Available The density functional and modified Poisson-Boltzmann descriptions of a spherical (electric double layer are compared and contrasted vis-a-vis existing Monte Carlo simulation data (for small ion diameter 4.25·10-10 m from the literature for a range of physical parameters such as macroion surface charge, macroion radius, valencies of the small ions, and electrolyte concentration. Overall, the theoretical predictions are seen to be remarkably consistent between themselves, being also in very good agreement with the simulations. Some modified Poisson-Boltzmann results for the zeta potential at small ion diameters of 3 and 2·10-10 m are also reported.
Sun, Hui; Wen, Jiayi; Zhao, Yanxiang; Li, Bo; McCammon, J Andrew
2015-12-28
Dielectric boundary based implicit-solvent models provide efficient descriptions of coarse-grained effects, particularly the electrostatic effect, of aqueous solvent. Recent years have seen the initial success of a new such model, variational implicit-solvent model (VISM) [Dzubiella, Swanson, and McCammon Phys. Rev. Lett. 96, 087802 (2006) and J. Chem. Phys. 124, 084905 (2006)], in capturing multiple dry and wet hydration states, describing the subtle electrostatic effect in hydrophobic interactions, and providing qualitatively good estimates of solvation free energies. Here, we develop a phase-field VISM to the solvation of charged molecules in aqueous solvent to include more flexibility. In this approach, a stable equilibrium molecular system is described by a phase field that takes one constant value in the solute region and a different constant value in the solvent region, and smoothly changes its value on a thin transition layer representing a smeared solute-solvent interface or dielectric boundary. Such a phase field minimizes an effective solvation free-energy functional that consists of the solute-solvent interfacial energy, solute-solvent van der Waals interaction energy, and electrostatic free energy described by the Poisson-Boltzmann theory. We apply our model and methods to the solvation of single ions, two parallel plates, and protein complexes BphC and p53/MDM2 to demonstrate the capability and efficiency of our approach at different levels. With a diffuse dielectric boundary, our new approach can describe the dielectric asymmetry in the solute-solvent interfacial region. Our theory is developed based on rigorous mathematical studies and is also connected to the Lum-Chandler-Weeks theory (1999). We discuss these connections and possible extensions of our theory and methods.
Analytical estimation of effective charges at saturation in Poisson-Boltzmann cell models
International Nuclear Information System (INIS)
Trizac, Emmanuel; Aubouy, Miguel; Bocquet, Lyderic
2003-01-01
We propose a simple approximation scheme for computing the effective charges of highly charged colloids (spherical or cylindrical with infinite length). Within non-linear Poisson-Boltzmann theory, we start from an expression for the effective charge in the infinite-dilution limit which is asymptotically valid for large salt concentrations; this result is then extended to finite colloidal concentration, approximating the salt partitioning effect which relates the salt content in the suspension to that of a dialysing reservoir. This leads to an analytical expression for the effective charge as a function of colloid volume fraction and salt concentration. These results compare favourably with the effective charges at saturation (i.e. in the limit of large bare charge) computed numerically following the standard prescription proposed by Alexander et al within the cell model
Electro-osmosis of non-Newtonian fluids in porous media using lattice Poisson-Boltzmann method.
Chen, Simeng; He, Xinting; Bertola, Volfango; Wang, Moran
2014-12-15
Electro-osmosis in porous media has many important applications in various areas such as oil and gas exploitation and biomedical detection. Very often, fluids relevant to these applications are non-Newtonian because of the shear-rate dependent viscosity. The purpose of this study was to investigate the behaviors and physical mechanism of electro-osmosis of non-Newtonian fluids in porous media. Model porous microstructures (granular, fibrous, and network) were created by a random generation-growth method. The nonlinear governing equations of electro-kinetic transport for a power-law fluid were solved by the lattice Poisson-Boltzmann method (LPBM). The model results indicate that: (i) the electro-osmosis of non-Newtonian fluids exhibits distinct nonlinear behaviors compared to that of Newtonian fluids; (ii) when the bulk ion concentration or zeta potential is high enough, shear-thinning fluids exhibit higher electro-osmotic permeability, while shear-thickening fluids lead to the higher electro-osmotic permeability for very low bulk ion concentration or zeta potential; (iii) the effect of the porous medium structure depends significantly on the constitutive parameters: for fluids with large constitutive coefficients strongly dependent on the power-law index, the network structure shows the highest electro-osmotic permeability while the granular structure exhibits the lowest permeability on the entire range of power law indices considered; when the dependence of the constitutive coefficient on the power law index is weaker, different behaviors can be observed especially in case of strong shear thinning. Copyright © 2014 Elsevier Inc. All rights reserved.
Duval, J.F.L.
2005-01-01
In a previous study (Langmuir 2004, 20, 10324), the electrokinetic properties of diffuse soft layers were theoretically investigated within the framework of the Debye-H¿ckel approximation valid in the limit of sufficiently low values for the Donnan potential. In the current paper, the
Directory of Open Access Journals (Sweden)
L.B. Bhuiyan
2017-12-01
Full Text Available The modified Poisson-Boltzmann theory of the restricted primitive model double layer is revisited and recast in a fresh, slightly broader perspective. Derivation of relevant equations follow the techniques utilized in the earlier MPB4 and MPB5 formulations and clarifies the relationship between these. The MPB4, MPB5, and a new formulation of the theory are employed in an analysis of the structure and charge reversal phenomenon in asymmetric 2:1/1:2 valence electrolytes. Furthermore, polarization induced surface charge amplification is studied in 3:1/1:3 systems. The results are compared to the corresponding Monte Carlo simulations. The theories are seen to predict the "exact" simulation data to varying degrees of accuracy ranging from qualitative to almost quantitative. The results from a new version of the theory are found to be of comparable accuracy as the MPB5 results in many situations. However, in some cases involving low electrolyte concentrations, theoretical artifacts in the form of un-physical "shoulders" in the singlet ionic distribution functions are observed.
Directory of Open Access Journals (Sweden)
V.Dorvilien
2013-01-01
Full Text Available The structure of cylindrical double layers is studied using a modified Poisson Boltzmann theory and the density functional approach. In the model double layer the electrode is a cylindrical polyion that is infinitely long, impenetrable, and uniformly charged. The polyion is immersed in a sea of equi-sized rigid ions embedded in a dielectric continuum. An in-depth comparison of the theoretically predicted zeta potentials, the mean electrostatic potentials, and the electrode-ion singlet density distributions is made with the corresponding Monte Carlo simulation data. The theories are seen to be consistent in their predictions that include variations in ionic diameters, electrolyte concentrations, and electrode surface charge densities, and are also able to reproduce well some new and existing Monte Carlo results.
Mean electrostatic and Poisson-Boltzmann models for multicomponent transport through compacted clay
International Nuclear Information System (INIS)
Steefel, C.I.; Galindez, J.M.
2012-01-01
Document available in extended abstract form only. Electrical double layer effects in the pore space of clays become increasingly important as the level of compaction increases and intergrain and interlayer spacings shift towards the range of nano-meters. At such scales, solute transport can no longer be explained by concentration gradients alone and it becomes necessary to include the electrostatic effects on chemical potentials. In fact, the electrical double layer (EDL) that develops in the neighborhood of the negatively charged clay surfaces can extend well into the aqueous phase, effectively constraining the space available to anions (known as anion exclusion), thus distorting the spatial distribution of ionic species in solution. In this study, we make use of two approaches for addressing the accumulation and transport of charged ionic species in the electrical double layers of compacted bentonite: 1) a mean electrostatic approach based on the assumption of Donnan equilibrium, and 2) a 2D numerical approach based on the multicomponent Poisson-Nernst-Planck (NPP) set of equations. For the mean electrostatic or Donnan approach to the electrical double layer [1], two options are considered: 1) a model in which surface complexation in the Stern layer may partly balance the fixed charge of the montmorillonite making up the bentonite buffer, and 2) a model in which the fixed mineral charge is balanced completely by the diffuse layer. In the mean electrostatic approach, one additional equation that balances the charge between the Stern layer and the diffuse layer is added to the multicomponent reactive transport code CrunchFlow. The only additional unknown that is required is the mean electrostatic potential, although it may be necessary in certain cases to consider the volume (or width) of the electrical double layer as an additional implicit unknown. Both ions and neutral species may diffuse within the diffuse layer according to their gradients and species
The time-dependent development of electric double-layers in saline solutions
International Nuclear Information System (INIS)
Morrow, R; McKenzie, D R; Bilek, M M M
2006-01-01
We have studied the time-dependent development of electric double-layers (ionic sheaths) in saline solutions by simultaneously solving the sodium and chlorine ion continuity equations coupled with Poisson's equation in one dimension. The study of the effects of time-varying electric fields in solution is relevant to the possible health effect of radio-frequency electric fields on cells in the human body and to assessing the potential of using external electric fields to orient proteins for attachment to surfaces for biosensing applications. Our calculations, for applied voltages of 10-175 mV between the electrode and the solution, predict time scales of ∼0.1-110 μs for the formation of double-layers in solutions of concentration between 0.001 and 1.0 M. We develop an empirical equation that can predict the double-layer formation time to within 10% over this wide parameter range. The method has been validated by comparing the solutions obtained, once the program has run to a steady state, with the standard non-linear Poisson-Boltzmann equations. Excellent agreement is found with the Gouy-Chapman solution of the non-linear Poisson-Boltzmann equation. Thus the method is not restricted in accuracy and applicability as is the case for the linear Poisson-Boltzmann equation. The method can also provide solutions for cases where there are orders of magnitude changes in the ion densities; this has not been the case for previous studies where small perturbation analysis has been employed. The method developed here can readily be extended to two and three dimensions using time-splitting methods
A modified Poisson-Boltzmann surface excess calculation with a field dependent dielectric constant
International Nuclear Information System (INIS)
Gordillo, G.J.; Molina, F.V.; Posadas, D.
1990-01-01
The Unequal Radius Modified Gouy-Chapman (URMGC) was applied to mixtures of electrolytes. It was considered that the two anions, (1) and (2), have different radius, r 1 and r 2 , being r 2 smaller than r 1 . The dielectric constant was taken as a function of the electric field, using the theoretical Booth equation, or as a linear dependence varying between 6 and 78 when r 2 1 . The results show that the surface excess of anion 2 is much greater than the one predicted by Gouy-Chapman theory when the proportion of 2 increases in the mixture, while both the other anion and the cation show negative deviation. This effect is more evident in mixtures than in the case of single electrolytes, and has a maximum for a composition that depends on the chosen parameters for the model. (Author) [es
Molavi Tabrizi, Amirhossein; Goossens, Spencer; Mehdizadeh Rahimi, Ali; Cooper, Christopher D; Knepley, Matthew G; Bardhan, Jaydeep P
2017-06-13
We extend the linearized Poisson-Boltzmann (LPB) continuum electrostatic model for molecular solvation to address charge-hydration asymmetry. Our new solvation-layer interface condition (SLIC)/LPB corrects for first-shell response by perturbing the traditional continuum-theory interface conditions at the protein-solvent and the Stern-layer interfaces. We also present a GPU-accelerated treecode implementation capable of simulating large proteins, and our results demonstrate that the new model exhibits significant accuracy improvements over traditional LPB models, while reducing the number of fitting parameters from dozens (atomic radii) to just five parameters, which have physical meanings related to first-shell water behavior at an uncharged interface. In particular, atom radii in the SLIC model are not optimized but uniformly scaled from their Lennard-Jones radii. Compared to explicit-solvent free-energy calculations of individual atoms in small molecules, SLIC/LPB is significantly more accurate than standard parametrizations (RMS error 0.55 kcal/mol for SLIC, compared to RMS error of 3.05 kcal/mol for standard LPB). On parametrizing the electrostatic model with a simple nonpolar component for total molecular solvation free energies, our model predicts octanol/water transfer free energies with an RMS error 1.07 kcal/mol. A more detailed assessment illustrates that standard continuum electrostatic models reproduce total charging free energies via a compensation of significant errors in atomic self-energies; this finding offers a window into improving the accuracy of Generalized-Born theories and other coarse-grained models. Most remarkably, the SLIC model also reproduces positive charging free energies for atoms in hydrophobic groups, whereas standard PB models are unable to generate positive charging free energies regardless of the parametrized radii. The GPU-accelerated solver is freely available online, as is a MATLAB implementation.
Analytic solutions of hydrodynamics equations
International Nuclear Information System (INIS)
Coggeshall, S.V.
1991-01-01
Many similarity solutions have been found for the equations of one-dimensional (1-D) hydrodynamics. These special combinations of variables allow the partial differential equations to be reduced to ordinary differential equations, which must then be solved to determine the physical solutions. Usually, these reduced ordinary differential equations are solved numerically. In some cases it is possible to solve these reduced equations analytically to obtain explicit solutions. In this work a collection of analytic solutions of the 1-D hydrodynamics equations is presented. These can be used for a variety of purposes, including (i) numerical benchmark problems, (ii) as a basis for analytic models, and (iii) to provide insight into more complicated solutions
Weak solutions of magma equations
International Nuclear Information System (INIS)
Krishnan, E.V.
1999-01-01
Periodic solutions in terms of Jacobian cosine elliptic functions have been obtained for a set of values of two physical parameters for the magma equation which do not reduce to solitary-wave solutions. It was also obtained solitary-wave solutions for another set of these parameters as an infinite period limit of periodic solutions in terms of Weierstrass and Jacobian elliptic functions
Kumar, Bharat; Crittenden, Scott R
2013-11-01
We demonstrate the ability to measure Stern potential and Debye length in dilute ionic solution with atomic force microscopy. We develop an analytic expression for the second harmonic force component of the capacitive force in an ionic solution from the linearized Poisson-Boltzmann equation. This allows us to calibrate the AFM tip potential and, further, obtain the Stern potential of sample surfaces. In addition, the measured capacitive force is independent of van der Waals and double layer forces, thus providing a more accurate measure of Debye length.
Solution of the Baxter equation
International Nuclear Information System (INIS)
Janik, R.A.
1996-01-01
We present a method of construction of a family of solutions of the Baxter equation arising in the Generalized Leading Logarithmic Approximation (GLLA) of the QCD pomeron. The details are given for the exchange of N = 2 reggeons but everything can be generalized in a straightforward way to arbitrary N. A specific choice of solutions is shown to reproduce the correct energy levels for half integral conformal weights. It is shown that the Baxter's equation must be supplemented by an additional condition on the solution. (author)
Lou, Ping; Lee, Jin Yong
2009-04-14
For a simple modified Poisson-Boltzmann (SMPB) theory, taking into account the finite ionic size, we have derived the exact analytic expression for the contact values of the difference profile of the counterion and co-ion, as well as of the sum (density) and product profiles, near a charged planar electrode that is immersed in a binary symmetric electrolyte. In the zero ionic size or dilute limit, these contact values reduce to the contact values of the Poisson-Boltzmann (PB) theory. The analytic results of the SMPB theory, for the difference, sum, and product profiles were compared with the results of the Monte-Carlo (MC) simulations [ Bhuiyan, L. B.; Outhwaite, C. W.; Henderson, D. J. Electroanal. Chem. 2007, 607, 54 ; Bhuiyan, L. B.; Henderson, D. J. Chem. Phys. 2008, 128, 117101 ], as well as of the PB theory. In general, the analytic expression of the SMPB theory gives better agreement with the MC data than the PB theory does. For the difference profile, as the electrode charge increases, the result of the PB theory departs from the MC data, but the SMPB theory still reproduces the MC data quite well, which indicates the importance of including steric effects in modeling diffuse layer properties. As for the product profile, (i) it drops to zero as the electrode charge approaches infinity; (ii) the speed of the drop increases with the ionic size, and these behaviors are in contrast with the predictions of the PB theory, where the product is identically 1.
The charge effect on the hindrance factors for diffusion and convection of a solute in pores: II
Energy Technology Data Exchange (ETDEWEB)
Akinaga, Takeshi; O-tani, Hideyuki; Sugihara-Seki, Masako, E-mail: r091077@kansai-u.ac.jp [Department of Pure and Applied Physics, Kansai University, Yamate-cho, Suita, Osaka 564-8680 (Japan)
2012-10-15
The diffusion and convection of a solute suspended in a fluid across porous membranes are known to be reduced compared to those in a bulk solution, owing to the fluid mechanical interaction between the solute and the pore wall as well as steric restriction. If the solute and the pore wall are electrically charged, the electrostatic interaction between them could affect the hindrance to diffusion and convection. In this study, the transport of charged spherical solutes through charged circular cylindrical pores filled with an electrolyte solution containing small ions was studied numerically by using a fluid mechanical and electrostatic model. Based on a mean field theory, the electrostatic interaction energy between the solute and the pore wall was estimated from the Poisson-Boltzmann equation, and the charge effect on the solute transport was examined for the solute and pore wall of like charge. The results were compared with those obtained from the linearized form of the Poisson-Boltzmann equation, i.e. the Debye-Hueckel equation. (paper)
Approximate solutions to Mathieu's equation
Wilkinson, Samuel A.; Vogt, Nicolas; Golubev, Dmitry S.; Cole, Jared H.
2018-06-01
Mathieu's equation has many applications throughout theoretical physics. It is especially important to the theory of Josephson junctions, where it is equivalent to Schrödinger's equation. Mathieu's equation can be easily solved numerically, however there exists no closed-form analytic solution. Here we collect various approximations which appear throughout the physics and mathematics literature and examine their accuracy and regimes of applicability. Particular attention is paid to quantities relevant to the physics of Josephson junctions, but the arguments and notation are kept general so as to be of use to the broader physics community.
Solutions of Einstein's field equations
Energy Technology Data Exchange (ETDEWEB)
Tomonaga, Y [Utsunomiya Univ. (Japan). Faculty of Education
1978-12-01
In this paper the author investigates the Einstein's field equations of the non-vacuum case and generalizes the solution of Robertson-Walker by the three dimensional Einstein spaces. In Section 2 the author shortly generalizes the dynamic space-time of G. Lemetre and A. Friedmann by a simple transformation.
Numerical solution of Boltzmann's equation
International Nuclear Information System (INIS)
Sod, G.A.
1976-04-01
The numerical solution of Boltzmann's equation is considered for a gas model consisting of rigid spheres by means of Hilbert's expansion. If only the first two terms of the expansion are retained, Boltzmann's equation reduces to the Boltzmann-Hilbert integral equation. Successive terms in the Hilbert expansion are obtained by solving the same integral equation with a different source term. The Boltzmann-Hilbert integral equation is solved by a new very fast numerical method. The success of the method rests upon the simultaneous use of four judiciously chosen expansions; Hilbert's expansion for the distribution function, another expansion of the distribution function in terms of Hermite polynomials, the expansion of the kernel in terms of the eigenvalues and eigenfunctions of the Hilbert operator, and an expansion involved in solving a system of linear equations through a singular value decomposition. The numerical method is applied to the study of the shock structure in one space dimension. Numerical results are presented for Mach numbers of 1.1 and 1.6. 94 refs, 7 tables, 1 fig
A differential equation for the Generalized Born radii.
Fogolari, Federico; Corazza, Alessandra; Esposito, Gennaro
2013-06-28
The Generalized Born (GB) model offers a convenient way of representing electrostatics in complex macromolecules like proteins or nucleic acids. The computation of atomic GB radii is currently performed by different non-local approaches involving volume or surface integrals. Here we obtain a non-linear second-order partial differential equation for the Generalized Born radius, which may be solved using local iterative algorithms. The equation is derived under the assumption that the usual GB approximation to the reaction field obeys Laplace's equation. The equation admits as particular solutions the correct GB radii for the sphere and the plane. The tests performed on a set of 55 different proteins show an overall agreement with other reference GB models and "perfect" Poisson-Boltzmann based values.
Linear superposition solutions to nonlinear wave equations
International Nuclear Information System (INIS)
Liu Yu
2012-01-01
The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic, triangle, and exponential functions, and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics. The linear superposition solutions to the generalized KdV equation K(2,2,1), the Oliver water wave equation, and the k(n, n) equation are given. The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed, and the reason why the solutions with the forms of hyperbolic, triangle, and exponential functions can form the linear superposition solutions is also discussed
Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation
Directory of Open Access Journals (Sweden)
Hamidreza Rezazadeh
2014-05-01
Full Text Available In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.. So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.
On polynomial solutions of the Heun equation
International Nuclear Information System (INIS)
Gurappa, N; Panigrahi, Prasanta K
2004-01-01
By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before identifying the polynomial solutions. The Heun equation extended by the addition of a term, -σ/x, is also amenable for polynomial solutions. (letter to the editor)
STEADY STATE AND PSEUDO-TRANSIENT ELECTRIC POTENTIAL USING THE POISSONBOLTZMANN EQUATION
Directory of Open Access Journals (Sweden)
L. C. dos Santos
2015-03-01
Full Text Available A method for analysis of the electric potential profile in saline solutions was developed for systems with one or two infinite flat plates. A modified Poisson-Boltzmann equation, taking into account nonelectrostatic interactions between ions and surfaces, was used. To solve the stated problem in the steady-state approach the finite-difference method was used. For the formulated pseudo-transient problem, we solved the set of ordinary differential equations generated from the algebraic equations of the stationary case. A case study was also carried out in relation to temperature, solution concentration, surface charge and salt-type. The results were validated by the stationary problem solution, which had also been used to verify the ionic specificity for different salts. The pseudo-transient approach allowed a better understanding of the dynamic behavior of the ion-concentration profile and other properties due to the surface charge variation.
New solutions of Heun's general equation
International Nuclear Information System (INIS)
Ishkhanyan, Artur; Suominen, Kalle-Antti
2003-01-01
We show that in four particular cases the derivative of the solution of Heun's general equation can be expressed in terms of a solution to another Heun's equation. Starting from this property, we use the Gauss hypergeometric functions to construct series solutions to Heun's equation for the mentioned cases. Each of the hypergeometric functions involved has correct singular behaviour at only one of the singular points of the equation; the sum, however, has correct behaviour. (letter to the editor)
Special solutions of neutral functional differential equations
Directory of Open Access Journals (Sweden)
Győri István
2001-01-01
Full Text Available For a system of nonlinear neutral functional differential equations we prove the existence of an -parameter family of "special solutions" which characterize the asymptotic behavior of all solutions at infinity. For retarded functional differential equations the special solutions used in this paper were introduced by Ryabov.
EXACT TRAVELLING WAVE SOLUTIONS TO BBM EQUATION
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
Abundant new travelling wave solutions to the BBM (Benjamin-Bona-Mahoni) equation are obtained by the generalized Jacobian elliptic function method. This method can be applied to other nonlinear evolution equations.
Rational Solutions and Lump Solutions of the Potential YTSF Equation
Sun, Hong-Qian; Chen, Ai-Hua
2017-07-01
By using of the bilinear form, rational solutions and lump solutions of the potential Yu-Toda-Sasa-Fukuyama (YTSF) equation are derived. Dynamics of the fundamental lump solution, n1-order lump solutions, and N-lump solutions are studied for some special cases. We also find some interaction behaviours of solitary waves and one lump of rational solutions.
New exact solutions of the Dirac equation
International Nuclear Information System (INIS)
Bagrov, V.G.; Gitman, D.M.; Zadorozhnyj, V.N.; Lavrov, P.M.; Shapovalov, V.N.
1980-01-01
Search for new exact solutions of the Dirac and Klein-Gordon equations are in progress. Considered are general properties of the Dirac equation solutions for an electron in a purely magnetic field, in combination with a longitudinal magnetic and transverse electric fields. New solutions for the equations of charge motion in an electromagnetic field of axial symmetry and in a nonstationary field of a special form have been found for potentials selected concretely
Geometrical and Graphical Solutions of Quadratic Equations.
Hornsby, E. John, Jr.
1990-01-01
Presented are several geometrical and graphical methods of solving quadratic equations. Discussed are Greek origins, Carlyle's method, von Staudt's method, fixed graph methods and imaginary solutions. (CW)
Solutions to Arithmetic Convolution Equations
Czech Academy of Sciences Publication Activity Database
Glöckner, H.; Lucht, L.G.; Porubský, Štefan
2007-01-01
Roč. 135, č. 6 (2007), s. 1619-1629 ISSN 0002-9939 R&D Projects: GA ČR GA201/04/0381 Institutional research plan: CEZ:AV0Z10300504 Keywords : arithmetic functions * Dirichlet convolution * polynomial equations * analytic equations * topological algebras * holomorphic functional calculus Subject RIV: BA - General Mathematics Impact factor: 0.520, year: 2007
The Dirac equation and its solutions
Energy Technology Data Exchange (ETDEWEB)
Bagrov, Vladislav G. [Tomsk State Univ., Tomsk (Russian Federation). Dept. of Quantum Field Theroy; Gitman, Dmitry [Sao Paulo Univ. (Brazil). Inst. de Fisica; P.N. Lebedev Physical Institute, Moscow (Russian Federation); Tomsk State Univ., Tomsk (Russian Federation). Faculty of Physics
2013-07-01
The Dirac equation is of fundamental importance for relativistic quantum mechanics and quantum electrodynamics. In relativistic quantum mechanics, the Dirac equation is referred to as one-particle wave equation of motion for electron in an external electromagnetic field. In quantum electrodynamics, exact solutions of this equation are needed to treat the interaction between the electron and the external field exactly. In particular, all propagators of a particle, i.e., the various Green's functions, are constructed in a certain way by using exact solutions of the Dirac equation.
The Dirac equation and its solutions
Bagrov, Vladislav G
2014-01-01
Dirac equations are of fundamental importance for relativistic quantum mechanics and quantum electrodynamics. In relativistic quantum mechanics, the Dirac equation is referred to as one-particle wave equation of motion for electron in an external electromagnetic field. In quantum electrodynamics, exact solutions of this equation are needed to treat the interaction between the electron and the external field exactly.In particular, all propagators of a particle, i.e., the various Green's functions, are constructed in a certain way by using exact solutions of the Dirac equation.
The Dirac equation and its solutions
International Nuclear Information System (INIS)
Bagrov, Vladislav G.; Gitman, Dmitry; P.N. Lebedev Physical Institute, Moscow; Tomsk State Univ., Tomsk
2013-01-01
The Dirac equation is of fundamental importance for relativistic quantum mechanics and quantum electrodynamics. In relativistic quantum mechanics, the Dirac equation is referred to as one-particle wave equation of motion for electron in an external electromagnetic field. In quantum electrodynamics, exact solutions of this equation are needed to treat the interaction between the electron and the external field exactly. In particular, all propagators of a particle, i.e., the various Green's functions, are constructed in a certain way by using exact solutions of the Dirac equation.
Discrete Riccati equation solutions: Distributed algorithms
Directory of Open Access Journals (Sweden)
D. G. Lainiotis
1996-01-01
Full Text Available In this paper new distributed algorithms for the solution of the discrete Riccati equation are introduced. The algorithms are used to provide robust and computational efficient solutions to the discrete Riccati equation. The proposed distributed algorithms are theoretically interesting and computationally attractive.
New solutions of the confluent Heun equation
Directory of Open Access Journals (Sweden)
Harold Exton
1998-05-01
Full Text Available New compact triple series solutions of the confluent Heun equation (CHE are obtained by the appropriate applications of the Laplace transform and its inverse to a suitably constructed system of soluble differential equations. The computer-algebra package MAPLE V is used to tackle an auxiliary system of non-linear algebraic equations. This study is partly motivated by the relationship between the CHE and certain Schrödininger equations.
Solutions manual to accompany Ordinary differential equations
Greenberg, Michael D
2014-01-01
Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps
PARALLEL SOLUTION METHODS OF PARTIAL DIFFERENTIAL EQUATIONS
Directory of Open Access Journals (Sweden)
Korhan KARABULUT
1998-03-01
Full Text Available Partial differential equations arise in almost all fields of science and engineering. Computer time spent in solving partial differential equations is much more than that of in any other problem class. For this reason, partial differential equations are suitable to be solved on parallel computers that offer great computation power. In this study, parallel solution to partial differential equations with Jacobi, Gauss-Siedel, SOR (Succesive OverRelaxation and SSOR (Symmetric SOR algorithms is studied.
Exact Solutions for Two Equation Hierarchies
International Nuclear Information System (INIS)
Song-Lin, Zhao; Da-Jun, Zhang; Jie, Ji
2010-01-01
Bilinear forms and double-Wronskian solutions are given for two hierarchies, the (2+1)-dimensional breaking Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy and the negative order AKNS hierarchy. According to some choices of the coefficient matrix in the Wronskian condition equation set, we obtain some kinds of solutions for these two hierarchies, such as solitons, Jordan block solutions, rational solutions, complexitons and mixed solutions. (general)
Numerical Solution of Parabolic Equations
DEFF Research Database (Denmark)
Østerby, Ole
These lecture notes are designed for a one-semester course on finite-difference methods for parabolic equations. These equations which traditionally are used for describing diffusion and heat-conduction problems in Geology, Physics, and Chemistry have recently found applications in Finance Theory...... ? and how do boundary value approximations affect the overall order of the method. Knowledge of a reliable order and error estimate enables us to determine (near-)optimal step sizes to meet a prescribed error tolerance, and possibly to extrapolate to get (higher order and) better accuracy at a minimal...... expense. Problems in two space dimensions are effectively handled using the Alternating Direction Implicit (ADI) technique. We present a systematic way of incorporating inhomogeneous terms and derivative boundary conditions in ADI methods as well as mixed derivative terms....
Homoclinic solutions for Davey-Stewartson equation
International Nuclear Information System (INIS)
Huang Jian; Dai Zhengde
2008-01-01
In this paper, we firstly prove the existence of homoclinic solutions for Davey-Stewartson I equation (DSI) with the periodic boundary condition. Then we obtain a set of exact homoclinic solutions by the novel method-Hirota's method. Moreover, the structure of homoclinic solutions has been investigated. At the same time, we give some numerical simulations which validate these theoretical results
Numerical solutions of diffusive logistic equation
International Nuclear Information System (INIS)
Afrouzi, G.A.; Khademloo, S.
2007-01-01
In this paper we investigate numerically positive solutions of a superlinear Elliptic equation on bounded domains. The study of Diffusive logistic equation continues to be an active field of research. The subject has important applications to population migration as well as many other branches of science and engineering. In this paper the 'finite difference scheme' will be developed and compared for solving the one- and three-dimensional Diffusive logistic equation. The basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from many authors these years
Quantized Lax equations and their solutions
Jurco, B
1997-01-01
Integrable systems on quantum groups are investigated. The Heisenberg equations possessing the Lax form are solved in terms of the solution to the factorization problem on the corresponding quantum group.
New exact solutions for two nonlinear equations
International Nuclear Information System (INIS)
Wang Quandi; Tang Minying
2008-01-01
In this Letter, we investigate two nonlinear equations given by u t -u xxt +3u 2 u x =2u x u xx +uu xxx and u t -u xxt +4u 2 u x =3u x u xx +uu xxx . Through some special phase orbits we obtain four new exact solutions for each equation above. Some previous results are extended
Symmetries and Dirac equation solutions
International Nuclear Information System (INIS)
Souza, Marcio Lima de.
1991-06-01
The purpose of this thesis is the extension to be relativistic case of a method that has proved useful for the solution of various potential problems in non relativistic situation. This method, the method of dynamical symmetries, is based on the Baker-Campbell-Hausdorf formulae and developed first for the particular example of the relativistic Coulomb problem. Here we generalize the method for a Hamiltonian that can be written as a linear combination of generators of the SO(2,1) group. As illustrative examples, we solve the problem of a charged particle in a constant magnetic field and the exponential magnetic field. (author). 21 refs
Real solutions to equations from geometry
Sottile, Frank
2011-01-01
Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a difficult problem with many applications outside of mathematics. While it is hopeless to expect much in general, we know a surprising amount about these questions for systems which possess additional structure often coming from geometry. This book focuses on equations from toric varieties and Grassmannians. Not only is much known about these, but such equations are common in applications. There are three main themes: upper bounds on the number of real solutions, lower bounds on the number of real solutions, and geometric problems that can have all solutions be real. The book begins with an overview, giving background on real solutions to univariate polynomials and the geometry of sparse polynomial systems. The first half of the book concludes with fewnomial upper bounds and with lower bounds to sparse polynomial systems. The second half of the book begins by sampling some geometric problems for which all ...
A fractional Dirac equation and its solution
International Nuclear Information System (INIS)
Muslih, Sami I; Agrawal, Om P; Baleanu, Dumitru
2010-01-01
This paper presents a fractional Dirac equation and its solution. The fractional Dirac equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives. By applying the variational principle to a fractional action S, we obtain the fractional Euler-Lagrange equations of motion. We present a Lagrangian and a Hamiltonian for the fractional Dirac equation of order α. We also use a fractional Klein-Gordon equation to obtain the fractional Dirac equation which is the same as that obtained using the fractional variational principle. Eigensolutions of this equation are presented which follow the same approach as that for the solution of the standard Dirac equation. We also provide expressions for the path integral quantization for the fractional Dirac field which, in the limit α → 1, approaches to the path integral for the regular Dirac field. It is hoped that the fractional Dirac equation and the path integral quantization of the fractional field will allow further development of fractional relativistic quantum mechanics.
Complex solutions for generalised fitzhughnagumo equation
International Nuclear Information System (INIS)
Neirameh, A.
2014-01-01
During present investigation, a direct algebraic method on complex solutions of nonlinear partial differential equation is developed and tested in the case of generalized Burgers-Huxley equation. The proposed scheme can be used in a wide class of nonlinear reaction-diffusion equations. These calculations demonstrate that the accuracy of the direct algebraic solutions is quite high even in the case of a small number of grid points. This method is a very reliable, simple, small computation costs, flexible, and convenient alternative method. (author)
Spurious solutions in few-body equations
International Nuclear Information System (INIS)
Adhikari, S.K.; Gloeckle, W.
1979-01-01
After Faddeev and Yakubovskii showed how to write connected few-body equations which are free from discrete spurious solutions various authors have proposed different connected few-body scattering equations. Federbush first pointed out that Weinberg's formulation admits the existence of discrete spurious solutions. In this paper we investigate the possibility and consequence of the existence of spurious solutions in some of the few-body formulations. Contrary to a proof by Hahn, Kouri, and Levin and by Bencze and Tandy the channel coupling array scheme of Kouri, Levin, and Tobocman which is also the starting point of a formulation by Hahn is shown to admit spurious solutions. We can show that the set of six coupled four-body equations proposed independently by Mitra, Gillespie, Sugar, and Panchapakesan, by Rosenberg, by Alessandrini, and by Takahashi and Mishima and the seven coupled four-body equations proposed by Sloan and related by matrix multipliers to basic sets which correspond uniquely to the Schroedinger equation. These multipliers are likely to give spurious solutions to these equations. In all these cases spuriosities are shown to have no hazardous consequence if one is interested in studying the scattering problem
Polynomial solutions of nonlinear integral equations
International Nuclear Information System (INIS)
Dominici, Diego
2009-01-01
We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials
Polynomial solutions of nonlinear integral equations
Energy Technology Data Exchange (ETDEWEB)
Dominici, Diego [Department of Mathematics, State University of New York at New Paltz, 1 Hawk Dr. Suite 9, New Paltz, NY 12561-2443 (United States)], E-mail: dominicd@newpaltz.edu
2009-05-22
We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials.
Power Series Solution to the Pendulum Equation
Benacka, Jan
2009-01-01
This note gives a power series solution to the pendulum equation that enables to investigate the system in an analytical way only, i.e. to avoid numeric methods. A method of determining the number of the terms for getting a required relative error is presented that uses bigger and lesser geometric series. The solution is suitable for modelling the…
Extremal solutions of measure differential equations
Czech Academy of Sciences Publication Activity Database
Monteiro, Giselle Antunes; Slavík, A.
2016-01-01
Roč. 444, č. 1 (2016), s. 568-597 ISSN 0022-247X Institutional support: RVO:67985840 Keywords : measure differential equations * extremal solution * lower solution Subject RIV: BA - General Mathematics Impact factor: 1.064, year: 2016 http://www.sciencedirect.com/science/article/pii/S0022247X16302724
Positive integer solutions of certain diophantine equations
Indian Academy of Sciences (India)
BIJAN KUMAR PATEL
2018-03-19
Mar 19, 2018 ... integer solutions. They also found all the positive integer solutions of the given equations in terms of Fibonacci and Lucas numbers. Another interesting number sequence which is closely related to the sequence of. Fibonacci numbers is the sequence of balancing numbers. In 1999, Behera et al. [1] intro-.
Spurious Numerical Solutions Of Differential Equations
Lafon, A.; Yee, H. C.
1995-01-01
Paper presents detailed study of spurious steady-state numerical solutions of differential equations that contain nonlinear source terms. Main objectives of this study are (1) to investigate how well numerical steady-state solutions of model nonlinear reaction/convection boundary-value problem mimic true steady-state solutions and (2) to relate findings of this investigation to implications for interpretation of numerical results from computational-fluid-dynamics algorithms and computer codes used to simulate reacting flows.
Singularly perturbed Burger-Huxley equation: Analytical solution ...
African Journals Online (AJOL)
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solutions of singularly perturbed nonlinear differential equations. ... for solving generalized Burgers-Huxley equation but this equation is not singularly ...... Solitary waves solutions of the generalized Burger Huxley equations, Journal of.
Crossing-symmetric solutions to low equations
International Nuclear Information System (INIS)
McLeod, R.J.; Ernst, D.J.
1985-01-01
Crossing symmetric models of the pion-nucleon interaction in which crossing symmetry is kept to lowest order in msub(π)/msub(N) are investigated. Two iterative techniques are developed to solve the crossing-symmetric Low equation. The techniques are used to solve the original Chew-Low equations and their generalizations to include the coupling to the pion-production channels. Small changes are found in comparison with earlier results which used an iterative technique proposed by Chew and Low and which did not produce crossing-symmetric results. The iterative technique of Chew and Low is shown to fail because of its inability to produce zeroes in the amplitude at complex energies while physical solutions to the model require such zeroes. We also prove that, within the class of solutions such that phase shifts approach zero for infinite energy, the solution to the Low equation is unique. (orig.)
Generalized solutions of nonlinear partial differential equations
Rosinger, EE
1987-01-01
During the last few years, several fairly systematic nonlinear theories of generalized solutions of rather arbitrary nonlinear partial differential equations have emerged. The aim of this volume is to offer the reader a sufficiently detailed introduction to two of these recent nonlinear theories which have so far contributed most to the study of generalized solutions of nonlinear partial differential equations, bringing the reader to the level of ongoing research.The essence of the two nonlinear theories presented in this volume is the observation that much of the mathematics concernin
Lim, Jongil; Whitcomb, John; Boyd, James; Varghese, Julian
2007-01-01
A finite element implementation of the transient nonlinear Nernst-Planck-Poisson (NPP) and Nernst-Planck-Poisson-modified Stern (NPPMS) models is presented. The NPPMS model uses multipoint constraints to account for finite ion size, resulting in realistic ion concentrations even at high surface potential. The Poisson-Boltzmann equation is used to provide a limited check of the transient models for low surface potential and dilute bulk solutions. The effects of the surface potential and bulk molarity on the electric potential and ion concentrations as functions of space and time are studied. The ability of the models to predict realistic energy storage capacity is investigated. The predicted energy is much more sensitive to surface potential than to bulk solution molarity.
General solution of string inspired nonlinear equations
International Nuclear Information System (INIS)
Bandos, I.A.; Ivanov, E.; Kapustnikov, A.A.; Ulanov, S.A.
1998-07-01
We present the general solution of the system of coupled nonlinear equations describing dynamics of D-dimensional bosonic string in the geometric (or embedding) approach. The solution is parametrized in terms of two sets of the left- and right-moving Lorentz harmonic variables providing a special coset space realization of the product of two (D-2) dimensional spheres S D-2 = SO(1,D-1)/SO(1,1)xSO(D-2) contained in K D-2 . (author)
Non-autonomous equations with unpredictable solutions
Akhmet, Marat; Fen, Mehmet Onur
2018-06-01
To make research of chaos more amenable to investigating differential and discrete equations, we introduce the concepts of an unpredictable function and sequence. The topology of uniform convergence on compact sets is applied to define unpredictable functions [1,2]. The unpredictable sequence is defined as a specific unpredictable function on the set of integers. The definitions are convenient to be verified as solutions of differential and discrete equations. The topology is metrizable and easy for applications with integral operators. To demonstrate the effectiveness of the approach, the existence and uniqueness of the unpredictable solution for a delay differential equation are proved as well as for quasilinear discrete systems. As a corollary of the theorem, a similar assertion for a quasilinear ordinary differential equation is formulated. The results are demonstrated numerically, and an application to Hopfield neural networks is provided. In particular, Poincaré chaos near periodic orbits is observed. The completed research contributes to the theory of chaos as well as to the theory of differential and discrete equations, considering unpredictable solutions.
Iterative solutions of finite difference diffusion equations
International Nuclear Information System (INIS)
Menon, S.V.G.; Khandekar, D.C.; Trasi, M.S.
1981-01-01
The heterogeneous arrangement of materials and the three-dimensional character of the reactor physics problems encountered in the design and operation of nuclear reactors makes it necessary to use numerical methods for solution of the neutron diffusion equations which are based on the linear Boltzmann equation. The commonly used numerical method for this purpose is the finite difference method. It converts the diffusion equations to a system of algebraic equations. In practice, the size of this resulting algebraic system is so large that the iterative methods have to be used. Most frequently used iterative methods are discussed. They include : (1) basic iterative methods for one-group problems, (2) iterative methods for eigenvalue problems, and (3) iterative methods which use variable acceleration parameters. Application of Chebyshev theorem to iterative methods is discussed. The extension of the above iterative methods to multigroup neutron diffusion equations is also considered. These methods are applicable to elliptic boundary value problems in reactor design studies in particular, and to elliptic partial differential equations in general. Solution of sample problems is included to illustrate their applications. The subject matter is presented in as simple a manner as possible. However, a working knowledge of matrix theory is presupposed. (M.G.B.)
Soliton solutions for a quasilinear Schrodinger equation
Directory of Open Access Journals (Sweden)
Duchao Liu
2013-12-01
Full Text Available In this article, critical point theory is used to show the existence of nontrivial weak solutions to the quasilinear Schrodinger equation $$ -\\Delta_p u-\\frac{p}{2^{p-1}}u\\Delta_p(u^2=f(x,u $$ in a bounded smooth domain $\\Omega\\subset\\mathbb{R}^{N}$ with Dirichlet boundary conditions.
Discrete fractional solutions of a Legendre equation
Yılmazer, Resat
2018-01-01
One of the most popular research interests of science and engineering is the fractional calculus theory in recent times. Discrete fractional calculus has also an important position in fractional calculus. In this work, we acquire new discrete fractional solutions of the homogeneous and non homogeneous Legendre differential equation by using discrete fractional nabla operator.
Approximate radiative solutions of the Einstein equations
International Nuclear Information System (INIS)
Kuusk, P.; Unt, V.
1976-01-01
In this paper the external field of a bounded source emitting gravitational radiation is considered. A successive approximation method is used to integrate the Einstein equations in Bondi's coordinates (Bondi et al, Proc. R. Soc.; A269:21 (1962)). A method of separation of angular variables is worked out and the approximate Einstein equations are reduced to key equations. The losses of mass, momentum, and angular momentum due to gravitational multipole radiation are found. It is demonstrated that in the case of proper treatment a real mass occurs instead of a mass aspect in a solution of the Einstein equations. In an appendix Bondi's new function is given in terms of sources. (author)
Polygons of differential equations for finding exact solutions
International Nuclear Information System (INIS)
Kudryashov, Nikolai A.; Demina, Maria V.
2007-01-01
A method for finding exact solutions of nonlinear differential equations is presented. Our method is based on the application of polygons corresponding to nonlinear differential equations. It allows one to express exact solutions of the equation studied through solutions of another equation using properties of the basic equation itself. The ideas of power geometry are used and developed. Our approach has a pictorial interpretation, which is illustrative and effective. The method can be also applied for finding transformations between solutions of differential equations. To demonstrate the method application exact solutions of several equations are found. These equations are: the Korteveg-de Vries-Burgers equation, the generalized Kuramoto-Sivashinsky equation, the fourth-order nonlinear evolution equation, the fifth-order Korteveg-de Vries equation, the fifth-order modified Korteveg-de Vries equation and the sixth-order nonlinear evolution equation describing turbulent processes. Some new exact solutions of nonlinear evolution equations are given
Analysis of numerical solutions for Bateman equations
International Nuclear Information System (INIS)
Loch, Guilherme G.; Bevilacqua, Joyce S.
2013-01-01
The implementation of stable and efficient numerical methods for solving problems involving nuclear transmutation and radioactive decay chains is the main scope of this work. The physical processes associated with irradiations of samples in particle accelerators, or the burning spent nuclear fuel in reactors, or simply the natural decay chains, can be represented by a set of first order ordinary differential equations with constant coefficients, for instance, the decay radioactive constants of each nuclide in the chain. Bateman proposed an analytical solution for a particular case of a linear chain with n nuclides decaying in series and with different decay constants. For more complex and realistic applications, the construction of analytical solutions is not viable and the introduction of numerical techniques is imperative. However, depending on the magnitudes of the decay radioactive constants, the matrix of coefficients could be almost singular, generating unstable and non convergent numerical solutions. In this work, different numerical strategies for solving systems of differential equations were implemented, the Runge-Kutta 4-4, Adams Predictor-Corrector (PC2) and the Rosenbrock algorithm, this last one more specific for stiff equations. Consistency, convergence and stability of the numerical solutions are studied and the performance of the methods is analyzed for the case of the natural decay chain of Uranium-235 comparing numerical with analytical solutions. (author)
Doubly periodic solutions of the modified Kawahara equation
International Nuclear Information System (INIS)
Zhang Dan
2005-01-01
Some doubly periodic (Jacobi elliptic function) solutions of the modified Kawahara equation are presented in closed form. Our approach is to introduce a new auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly periodic solutions of the modified Kawahara equation. When the module m → 1, these solutions degenerate to the exact solitary wave solutions of the equation. Then we reveal the relation of some exact solutions for the modified Kawahara equation obtained by other authors
Differential-algebraic solutions of the heat equation
Buchstaber, Victor M.; Netay, Elena Yu.
2014-01-01
In this work we introduce the notion of differential-algebraic ansatz for the heat equation and explicitly construct heat equation and Burgers equation solutions given a solution of a homogeneous non-linear ordinary differential equation of a special form. The ansatz for such solutions is called the $n$-ansatz, where $n+1$ is the order of the differential equation.
Quasi-exact solutions of nonlinear differential equations
Kudryashov, Nikolay A.; Kochanov, Mark B.
2014-01-01
The concept of quasi-exact solutions of nonlinear differential equations is introduced. Quasi-exact solution expands the idea of exact solution for additional values of parameters of differential equation. These solutions are approximate solutions of nonlinear differential equations but they are close to exact solutions. Quasi-exact solutions of the the Kuramoto--Sivashinsky, the Korteweg--de Vries--Burgers and the Kawahara equations are founded.
Iterative solution of the semiconductor device equations
Energy Technology Data Exchange (ETDEWEB)
Bova, S.W.; Carey, G.F. [Univ. of Texas, Austin, TX (United States)
1996-12-31
Most semiconductor device models can be described by a nonlinear Poisson equation for the electrostatic potential coupled to a system of convection-reaction-diffusion equations for the transport of charge and energy. These equations are typically solved in a decoupled fashion and e.g. Newton`s method is used to obtain the resulting sequences of linear systems. The Poisson problem leads to a symmetric, positive definite system which we solve iteratively using conjugate gradient. The transport equations lead to nonsymmetric, indefinite systems, thereby complicating the selection of an appropriate iterative method. Moreover, their solutions exhibit steep layers and are subject to numerical oscillations and instabilities if standard Galerkin-type discretization strategies are used. In the present study, we use an upwind finite element technique for the transport equations. We also evaluate the performance of different iterative methods for the transport equations and investigate various preconditioners for a few generalized gradient methods. Numerical examples are given for a representative two-dimensional depletion MOSFET.
Numerical solutions of the Vlasov equation
International Nuclear Information System (INIS)
Satofuka, Nobuyuki; Morinishi, Koji; Nishida, Hidetoshi
1985-01-01
A numerical procedure is derived for the solutions of the one- and two-dimensional Vlasov-Poisson system equations. This numerical procedure consists of the phase space discretization and the integration of the resulting set of ordinary differential equations. In the phase space discretization, derivatives with respect to the phase space variable are approximated by a weighted sum of the values of the distribution function at properly chosen neighboring points. Then, the resulting set of ordinary differential equations is solved by using an appropriate time integration scheme. The results for linear Landau damping, nonlinear Landau damping and counter-streaming plasmas are investigated and compared with those of the splitting scheme. The proposed method is found to be very accurate and efficient. (author)
Exact solutions to operator differential equations
International Nuclear Information System (INIS)
Bender, C.M.
1992-01-01
In this talk we consider the Heisenberg equations of motion q = -i(q, H), p = -i(p, H), for the quantum-mechanical Hamiltonian H(p, q) having one degree of freedom. It is a commonly held belief that such operator differential equations are intractable. However, a technique is presented here that allows one to obtain exact, closed-form solutions for huge classes of Hamiltonians. This technique, which is a generalization of the classical action-angle variable methods, allows us to solve, albeit formally and implicitly, the operator differential equations of two anharmonic oscillators whose Hamiltonians are H = p 2 /2 + q 4 /4 and H = p 4 /4 + q 4 /4
On the solution of fractional evolution equations
International Nuclear Information System (INIS)
Kilbas, Anatoly A; Pierantozzi, Teresa; Trujillo, Juan J; Vazquez, Luis
2004-01-01
This paper is devoted to the solution of the bi-fractional differential equation ( C D α t u)(t, x) = λ( L D β x u)(t, x) (t>0, -∞ 0 and λ ≠ 0, with the initial conditions lim x→±∞ u(t,x) = 0 u(0+,x)=g(x). Here ( C D α t u)(t, x) is the partial derivative coinciding with the Caputo fractional derivative for 0 L D β x u)(t, x)) is the Liouville partial fractional derivative ( L D β t u)(t, x)) of order β > 0. The Laplace and Fourier transforms are applied to solve the above problem in closed form. The fundamental solution of these problems is established and its moments are calculated. The special case α = 1/2 and β = 1 is presented, and its application is given to obtain the Dirac-type decomposition for the ordinary diffusion equation
On new solutions of fuzzy differential equations
International Nuclear Information System (INIS)
Chalco-Cano, Y.; Roman-Flores, H.
2008-01-01
We study fuzzy differential equations (FDE) using the concept of generalized H-differentiability. This concept is based in the enlargement of the class of differentiable fuzzy mappings and, for this, we consider the lateral Hukuhara derivatives. We will see that both derivatives are different and they lead us to different solutions from a FDE. Also, some illustrative examples are given and some comparisons with other methods for solving FDE are made
Spurious Solutions Of Nonlinear Differential Equations
Yee, H. C.; Sweby, P. K.; Griffiths, D. F.
1992-01-01
Report utilizes nonlinear-dynamics approach to investigate possible sources of errors and slow convergence and non-convergence of steady-state numerical solutions when using time-dependent approach for problems containing nonlinear source terms. Emphasizes implications for development of algorithms in CFD and computational sciences in general. Main fundamental conclusion of study is that qualitative features of nonlinear differential equations cannot be adequately represented by finite-difference method and vice versa.
Method of lines solution of Richards` equation
Energy Technology Data Exchange (ETDEWEB)
Kelley, C.T.; Miller, C.T.; Tocci, M.D.
1996-12-31
We consider the method of lines solution of Richard`s equation, which models flow through porous media, as an example of a situation in which the method can give incorrect results because of premature termination of the nonlinear corrector iteration. This premature termination arises when the solution has a sharp moving front and the Jacobian is ill-conditioned. While this problem can be solved by tightening the tolerances provided to the ODE or DAE solver used for the temporal integration, it is more efficient to modify the termination criteria of the nonlinear solver and/or recompute the Jacobian more frequently. In this paper we continue previous work on this topic by analyzing the modifications in more detail and giving a strategy on how the modifications can be turned on and off in response to changes in the character of the solution.
New types of exact solutions for a breaking soliton equation
International Nuclear Information System (INIS)
Mei Jianqin; Zhang Hongqing
2004-01-01
In this paper based on a system of Riccati equations, we present a newly generally projective Riccati equation expansion method and its algorithm, which can be used to construct more new exact solutions of nonlinear differential equations in mathematical physics. A typical breaking soliton equation is chosen to illustrate our algorithm such that more families of new exact solutions are obtained, which contain soliton-like solutions and periodic solutions. This algorithm can also be applied to other nonlinear differential equations
Iterative solution of the Helmholtz equation
Energy Technology Data Exchange (ETDEWEB)
Larsson, E.; Otto, K. [Uppsala Univ. (Sweden)
1996-12-31
We have shown that the numerical solution of the two-dimensional Helmholtz equation can be obtained in a very efficient way by using a preconditioned iterative method. We discretize the equation with second-order accurate finite difference operators and take special care to obtain non-reflecting boundary conditions. We solve the large, sparse system of equations that arises with the preconditioned restarted GMRES iteration. The preconditioner is of {open_quotes}fast Poisson type{close_quotes}, and is derived as a direct solver for a modified PDE problem.The arithmetic complexity for the preconditioner is O(n log{sub 2} n), where n is the number of grid points. As a test problem we use the propagation of sound waves in water in a duct with curved bottom. Numerical experiments show that the preconditioned iterative method is very efficient for this type of problem. The convergence rate does not decrease dramatically when the frequency increases. Compared to banded Gaussian elimination, which is a standard solution method for this type of problems, the iterative method shows significant gain in both storage requirement and arithmetic complexity. Furthermore, the relative gain increases when the frequency increases.
Analytic Solutions and Resonant Solutions of Hyperbolic Partial Differential Equations
Wagenmaker, Timothy Roger
This dissertation contains two main subject areas. The first deals with solutions to the wave equation Du/Dt + a Du/Dx = 0, where D/Dt and D/Dx represent partial derivatives and a(t,x) is real valued. The question I studied, which arises in control theory, is whether solutions which are real analytic with respect to the time variable are dense in the space of all solutions. If a is real analytic in t and x, the Cauchy-Kovalevsky Theorem implies that the solutions real analytic in t and x are dense, since it suffices to approximate the initial data by polynomials. The same positive result is valid when a is continuously differentiable and independent of t. This is proved by regularization in time. The hypothesis that a is independent of t cannot be replaced by the weaker assumption that a is real analytic in t, even when it is infinitely smooth. I construct a(t,x) for which the solutions which are analytic in time are automatically periodic in time. In particular these solutions are not dense in the space of all solutions. The second area concerns the resonant interaction of oscillatory waves propagating in a compressible inviscid fluid. An asymptotic description given by Andrew Majda, Rodolfo Rosales, and Maria Schonbek (MRS) involves the genuinely nonlinear quasilinear hyperbolic system Du/Dt + D(uu/2)/Dt + v = 0, Dv/Dt - D(vv/2)/Dt - u = 0. They performed many numerical simulations which indicated that small amplitude solutions of this system tend to evade shock formation, and conjectured that "smooth initial data with a sufficiently small amplitude never develop shocks throughout a long time interval of integration.". I proved that for smooth periodic U(x), V(x) and initial data u(0,x) = epsilonU(x), v(0,x) = epsilonV(x), the solution is smooth for time at least constant times | ln epsilon| /epsilon. This is longer than the lifetime order 1/ epsilon of the solution to the decoupled Burgers equations. The decoupled equation describes nonresonant interaction of
Fundamental Solutions to Some Pell Equations
Directory of Open Access Journals (Sweden)
Merve Güney
2013-01-01
Full Text Available Let a,b and n are positive integers. In this paper, we find continued fraction expansion of ;#8730;d when d=a^2 b^2+2b, a^2 b^2+b,a^2±2,a^2±a. We will use continued fraction expansion of ;#8730;d in order to get the fundamental solutions to the equations x²-dy²=±1 when d=a^2 b^2+2b, a^2 b^2+b,a^2±2,a^2±a
Analytic Solutions of Special Functional Equations
Directory of Open Access Journals (Sweden)
Octav Olteanu
2013-07-01
Full Text Available We recall some of our earlier results on the construction of a mapping defined implicitly, without using the implicit function theorem. All these considerations work in the real case, for functions and operators. Then we consider the complex case, proving the analyticity of the function defined implicitly, under certain hypothesis. Some consequences are given. An approximating formula for the analytic form of the solution is also given. Finally, one illustrates the preceding results by an application to a concrete functional and operatorial equation. Some related examples are given.
Efimova, Olga Yu.
2010-01-01
The modification of simplest equation method to look for exact solutions of nonlinear partial differential equations is presented. Using this method we obtain exact solutions of generalized Korteweg-de Vries equation with cubic source and exact solutions of third-order Kudryashov-Sinelshchikov equation describing nonlinear waves in liquids with gas bubbles.
Numerical solution of the radionuclide transport equation
International Nuclear Information System (INIS)
Hadermann, J.; Roesel, F.
1983-11-01
A numerical solution of the one-dimensional geospheric radionuclide chain transport equation based on the pseudospectral method is developed. The advantages of this approach are flexibility in incorporating space and time dependent migration parameters, arbitrary boundary conditions and solute rock interactions as well as efficiency and reliability. As an application the authors investigate the impact of non-linear sorption isotherms on migration in crystalline rock. It is shown that non-linear sorption, in the present case a Freundlich isotherm, may reduce concentration at the geosphere outlet by orders of magnitude provided the migration time is comparable or larger than the half-life of the nuclide in question. The importance of fixing dispersivity within the continuum approach is stressed. (Auth.)
Minimal solution for inconsistent singular fuzzy matrix equations
Directory of Open Access Journals (Sweden)
M. Nikuie
2013-10-01
Full Text Available The fuzzy matrix equations $Ailde{X}=ilde{Y}$ is called a singular fuzzy matrix equations while the coefficients matrix of its equivalent crisp matrix equations be a singular matrix. The singular fuzzy matrix equations are divided into two parts: consistent singular matrix equations and inconsistent fuzzy matrix equations. In this paper, the inconsistent singular fuzzy matrix equations is studied and the effect of generalized inverses in finding minimal solution of an inconsistent singular fuzzy matrix equations are investigated.
Singularly perturbed Burger-Huxley equation: Analytical solution ...
African Journals Online (AJOL)
user
numbers, Navier-Stokes flows with large Reynolds numbers, chemical reactor ... It is to observe the layer behavior of the solution for smaller values of ε leading to singular ...... Burger equation, momentum gas equation and heat equation.
Differential constraints and exact solutions of nonlinear diffusion equations
International Nuclear Information System (INIS)
Kaptsov, Oleg V; Verevkin, Igor V
2003-01-01
The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining equations used in the search for classical Lie symmetries
Solution of differential equations by application of transformation groups
Driskell, C. N., Jr.; Gallaher, L. J.; Martin, R. H., Jr.
1968-01-01
Report applies transformation groups to the solution of systems of ordinary differential equations and partial differential equations. Lies theorem finds an integrating factor for appropriate invariance group or groups can be found and can be extended to partial differential equations.
Analytic structure of solutions to multiconfiguration equations
Energy Technology Data Exchange (ETDEWEB)
Fournais, Soeren [Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 1530, DK-8000 Arhus C (Denmark); Hoffmann-Ostenhof, Maria [Fakultaet fuer Mathematik, Universitaet Wien, Nordbergstrasse 15, A-1090 Vienna (Austria); Hoffmann-Ostenhof, Thomas [Institut fuer Theoretische Chemie, Waehringerstrasse 17, Universitaet Wien, A-1090 Vienna (Austria); Soerensen, Thomas Oestergaard [Department of Mathematics, Imperial College London, Huxley Building, 180 Queen' s Gate, London SW7 2AZ (United Kingdom)], E-mail: fournais@imf.au.dk, E-mail: Maria.Hoffmann-Ostenhof@univie.ac.at, E-mail: thoffman@esi.ac.at, E-mail: t.sorensen@imperial.ac.uk
2009-08-07
We study the regularity at the positions of the (fixed) nuclei of solutions to (non-relativistic) multiconfiguration equations (including Hartree-Fock) of Coulomb systems. We prove the following: let {l_brace}{psi}{sub 1}, ..., {psi}{sub M}{r_brace} be any solution to the rank-M multiconfiguration equations for a molecule with L fixed nuclei at R{sub 1},...,R{sub L} element of R{sup 3}. Then, for any j in {l_brace}1, ..., M{r_brace}, k in {l_brace}1, ..., L{r_brace}, there exists a neighborhood U{sub j,k} subset or equal R{sup 3} of R{sub k}, and functions {psi}{sup (1)}{sub j,k}, {psi}{sup (2)}{sub j,k}, real analytic in U{sub j,k}, such that {phi}{sub j}(x)={phi}{sub j,k}{sup (1)}(x)+|x-R{sub k}|{phi}{sub j,k}{sup (2)}(x), x element of U{sub j,k}. A similar result holds for the corresponding electron density. The proof uses the Kustaanheimo-Stiefel transformation, as applied in [9] to the study of the eigenfunctions of the Schroedinger operator of atoms and molecules near two-particle coalescence points.
On the solution of fractional evolution equations
Energy Technology Data Exchange (ETDEWEB)
Kilbas, Anatoly A [Department of Mathematics and Mechanics, Belarusian State University, 220050 Minsk (Belarus); Pierantozzi, Teresa [Departamento de Matematica Aplicada, Facultad de Informatica, Universidad Complutense, E-28040 Madrid (Spain); Trujillo, Juan J [Departamento de Analisis Matematico, Universidad de la Laguna, 38271 La Laguna-Tenerife (Spain); Vazquez, Luis [Departamento de Matematica Aplicada, Facultad de Informatica, Universidad Complutense, E-28040 Madrid (Spain)
2004-03-05
This paper is devoted to the solution of the bi-fractional differential equation ({sup C}D{sup {alpha}}{sub t}u)(t, x) = {lambda}({sup L}D{sup {beta}}{sub x}u)(t, x) (t>0, -{infinity}
A class of exact solutions to the Einstein field equations
International Nuclear Information System (INIS)
Goyal, Nisha; Gupta, R K
2012-01-01
The Einstein-Rosen metric is considered and a class of new exact solutions of the field equations for stationary axisymmetric Einstein-Maxwell fields is obtained. The Lie classical approach is applied to obtain exact solutions. 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 Einstein-Maxwell equations. (paper)
ALMOST PERIODIC SOLUTIONS TO SOME NONLINEAR DELAY DIFFERENTIAL EQUATION
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
The existence of an almost periodic solutions to a nonlinear delay diffierential equation is considered in this paper. A set of sufficient conditions for the existence and uniqueness of almost periodic solutions to some delay diffierential equations is obtained.
Exact solutions to the Lienard equation and its applications
International Nuclear Information System (INIS)
Feng Zhaosheng
2004-01-01
In this paper, a kind of explicit exact solutions to the Lienard equation is obtained, and the applications of the result in seeking traveling solitary wave solution of the nonlinear Schroedinger equation are presented
Pure soliton solutions of some nonlinear partial differential equations
International Nuclear Information System (INIS)
Fuchssteiner, B.
1977-01-01
A general approach is given to obtain the system of ordinary differential equations which determines the pure soliton solutions for the class of generalized Korteweg-de Vries equations. This approach also leads to a system of ordinary differential equations for the pure soliton solutions of the sine-Gordon equation. (orig.) [de
New exact travelling wave solutions for the Ostrovsky equation
International Nuclear Information System (INIS)
Kangalgil, Figen; Ayaz, Fatma
2008-01-01
In this Letter, auxiliary equation method is proposed for constructing more general exact solutions of nonlinear partial differential equation with the aid of symbolic computation. In order to illustrate the validity and the advantages of the method we choose the Ostrovsky equation. As a result, many new and more general exact solutions have been obtained for the equation
Explicit solutions of two nonlinear dispersive equations with variable coefficients
International Nuclear Information System (INIS)
Lai Shaoyong; Lv Xiumei; Wu Yonghong
2008-01-01
A mathematical technique based on an auxiliary equation and the symbolic computation system Matlab is developed to construct the exact solutions for a generalized Camassa-Holm equation and a nonlinear dispersive equation with variable coefficients. It is shown that the variable coefficients of the derivative terms in the equations cause the qualitative change in the physical structures of the solutions
Exact traveling wave solutions of the Boussinesq equation
International Nuclear Information System (INIS)
Ding Shuangshuang; Zhao Xiqiang
2006-01-01
The repeated homogeneous balance method is used to construct exact traveling wave solutions of the Boussinesq equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions of the Boussinesq equation are successfully obtained
Exact, multiple soliton solutions of the double sine Gordon equation
International Nuclear Information System (INIS)
Burt, P.B.
1978-01-01
Exact, particular solutions of the double sine Gordon equation in n dimensional space are constructed. Under certain restrictions these solutions are N solitons, where N <= 2q - 1 and q is the dimensionality of space-time. The method of solution, known as the base equation technique, relates solutions of nonlinear partial differential equations to solutions of linear partial differential equations. This method is reviewed and its applicability to the double sine Gordon equation shown explicitly. The N soliton solutions have the remarkable property that they collapse to a single soliton when the wave vectors are parallel. (author)
Effect of ionic strength on the kinetics of ionic and micellar reactions in aqueous solution
International Nuclear Information System (INIS)
Dung, M.H.; Kozak, J.J.
1982-01-01
The effect of electrostatic forces on the rate of reaction between ions in aqueous solutions of intermediate ionic strength is studied in this paper. We consider the kinetics of reactions involving simple ionic species (1--1 and 2--2 electrolyte systems) as well as kinetic processes mediated by the presence of micellar ions (or other charged organizates). In the regime of ionic strength considered, dielectric saturation of the solvent in the vicinity of the reacting ions must be taken into account and this is done by introducing several models to describe the recovery of the solvent from saturation to its continuum dielectric behavior. To explore the effects of ion size, charge number, and ionic strength on the overall rate constant for the process considered, we couple the traditional theory of ionic reactions in aqueous solution with calculations of the electrostatic potential obtained via solution of the nonlinear Poisson--Boltzmann equation. The great flexibility of the nonlinear Poisson--Boltzmann theory allows us to explore quantitatively the influence of each of these effects, and our simulations show that the short-range properties of the electrostatic potential affect primarily kinetically controlled processes (to varying degrees, depending on the ionic system considered) whereas the down-range properties of the potential play a (somewhat) greater role in influencing diffusion-controlled processes. A detailed examination is made of ionic strength effects over a broad range of ionic concentrations. In the regime of low ionic strength, the limiting slope and intercept of the curve describing the dependence of log k/sub D/ on I/sup 1/2//(1+I/sup 1/2/) may differ considerably from the usual Debye--Hueckel limiting relations, depending on the particular model chosen to describe local saturation effects
Special function solutions of the free particle Dirac equation
International Nuclear Information System (INIS)
Strange, P
2012-01-01
The Dirac equation is one of the fundamental equations in physics. Here we present and discuss two novel solutions of the free particle Dirac equation. These solutions have an exact analytical form in terms of Airy or Mathieu functions and exhibit unexpected properties including an enhanced Doppler effect, accelerating wavefronts and solutions with a degree of localization. (paper)
Exact solution for the generalized Telegraph Fisher's equation
International Nuclear Information System (INIS)
Abdusalam, H.A.; Fahmy, E.S.
2009-01-01
In this paper, we applied the factorization scheme for the generalized Telegraph Fisher's equation and an exact particular solution has been found. The exact particular solution for the generalized Fisher's equation was obtained as a particular case of the generalized Telegraph Fisher's equation and the two-parameter solution can be obtained when n=2.
Solutions for a non-Markovian diffusion equation
International Nuclear Information System (INIS)
Lenzi, E.K.; Evangelista, L.R.; Lenzi, M.K.; Ribeiro, H.V.; Oliveira, E.C. de
2010-01-01
Solutions for a non-Markovian diffusion equation are investigated. For this equation, we consider a spatial and time dependent diffusion coefficient and the presence of an absorbent term. The solutions exhibit an anomalous behavior which may be related to the solutions of fractional diffusion equations and anomalous diffusion.
Exact solution of the neutron transport equation in spherical geometry
Energy Technology Data Exchange (ETDEWEB)
Anli, Fikret; Akkurt, Abdullah; Yildirim, Hueseyin; Ates, Kemal [Kahramanmaras Suetcue Imam Univ. (Turkey). Faculty of Sciences and Letters
2017-03-15
Solution of the neutron transport equation in one dimensional slab geometry construct a basis for the solution of neutron transport equation in a curvilinear geometry. Therefore, in this work, we attempt to derive an exact analytical benchmark solution for both neutron transport equations in slab and spherical medium by using P{sub N} approximation which is widely used in neutron transport theory.
Solution of diffusion equation in deformable spheroids
Energy Technology Data Exchange (ETDEWEB)
Ayyoubzadeh, Seyed Mohsen [Department of Energy Engineering, Sharif University of Technology, Tehran (Iran, Islamic Republic of); Safari, Mohammad Javad, E-mail: iFluka@gmail.com [Department of Energy Engineering, Sharif University of Technology, Tehran (Iran, Islamic Republic of); Vosoughi, Naser [Department of Energy Engineering, Sharif University of Technology, Tehran (Iran, Islamic Republic of)
2011-05-15
Research highlights: > Developing an explicit solution for the diffusion equation in spheroidal geometry. > Proving an orthogonality relation for spheroidal eigenfunctions. > Developing a relation for the extrapolation distance in spheroidal geometry. > Considering the sphere and slab as limiting cases for a spheroid. > Cross-validation of the analytical solution with Monte Carlo simulations. - Abstract: The time-dependent diffusion of neutrons in a spheroid as a function of the focal distance has been studied. The solution is based on an orthogonal basis and an extrapolation distanced related boundary condition for the spheroidal geometry. It has been shown that spheres and disks are two limiting cases for the spheroids, for which there is a smooth transition for the systems properties between these two limits. Furthermore, it is demonstrated that a slight deformation from a sphere does not affect the fundamental mode properties, to the first order. The calculations for both multiplying and non-multiplying media have been undertaken, showing good agreement with direct Monte Carlo simulations.
Exact solutions to sine-Gordon-type equations
International Nuclear Information System (INIS)
Liu Shikuo; Fu Zuntao; Liu Shida
2006-01-01
In this Letter, sine-Gordon-type equations, including single sine-Gordon equation, double sine-Gordon equation and triple sine-Gordon equation, are systematically solved by Jacobi elliptic function expansion method. It is shown that different transformations for these three sine-Gordon-type equations play different roles in obtaining exact solutions, some transformations may not work for a specific sine-Gordon equation, while work for other sine-Gordon equations
New exact solutions of the mBBM equation
International Nuclear Information System (INIS)
Zhang Zhe; Li Desheng
2013-01-01
The enhanced modified simple equation method presented in this article is applied to construct the exact solutions of modified Benjamin-Bona-Mahoney equation. Some new exact solutions are derived by using this method. When some parameters are taken as special values, the solitary wave solutions can be got from the exact solutions. It is shown that the method introduced in this paper has general significance in searching for exact solutions to the nonlinear evolution equations. (authors)
New analytic solutions of stochastic coupled KdV equations
International Nuclear Information System (INIS)
Dai Chaoqing; Chen Junlang
2009-01-01
In this paper, firstly, we use the exp-function method to seek new exact solutions of the Riccati equation. Then, with the help of Hermit transformation, we employ the Riccati equation and its new exact solutions to find new analytic solutions of the stochastic coupled KdV equation in the white noise environment. As some special examples, some analytic solutions can degenerate into these solutions reported in open literatures.
A new auxiliary equation and exact travelling wave solutions of nonlinear equations
International Nuclear Information System (INIS)
Sirendaoreji
2006-01-01
A new auxiliary ordinary differential equation and its solutions are used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the auxiliary equation which has more new exact solutions. More new exact travelling wave solutions are obtained for the quadratic nonlinear Klein-Gordon equation, the combined KdV and mKdV equation, the sine-Gordon equation and the Whitham-Broer-Kaup equations
Exact solutions for the cubic-quintic nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Zhu Jiamin; Ma Zhengyi
2007-01-01
In this paper, the cubic-quintic nonlinear Schroedinger equation is solved through the extended elliptic sub-equation method. As a consequence, many types of exact travelling wave solutions are obtained which including bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions
Iterative solution of a nonlinear operator equation
International Nuclear Information System (INIS)
Chidume, C.E.
1988-01-01
Suppose X=L p , p ≥ 2, and K is a non-empty closed convex subset of X. Suppose T:K → X is a monotonic Lipschitzian mapping with Lipschitz constant L ≥ 1 such that, for x in K and fixed f in X, the equation x+Tx=f has a solution in K. Define the sequence (x n ) ∞ n=0 by x 0 is an element of K, x n+1 =x n +λr n , for n ≥ 1, where λ=((p-1)L 2 ) -1 and r n =f-x n -Tx n . Then, (x n ) ∞ n=0 converges strongly to a solution of x+Tx=f in K. Convergence is at least as fast as a geometric progression with ratio (1-λ) 1/2 . A related result deals with convergence of the sequence (x n ) ∞ n=0 when T is monotone and locally Lipschitzian. (author). 19 refs
Analytical Solution of Pantograph Equation with Incommensurate Delay
Patade, Jayvant; Bhalekar, Sachin
2017-08-01
Pantograph equation is a delay differential equation (DDE) arising in electrodynamics. This paper studies the pantograph equation with two delays. The existence, uniqueness, stability and convergence results for DDEs are presented. The series solution of the proposed equation is obtained by using Daftardar-Gejji and Jafari method and given in terms of a special function. This new special function has several properties and relations with other functions. Further, we generalize the proposed equation to fractional-order case and obtain its solution.
Exact solutions for modified Korteweg-de Vries equation
International Nuclear Information System (INIS)
Sarma, Jnanjyoti
2009-01-01
Using the simple wave or traveling wave solution technique, many different types of solutions are derived for modified Korteweg-de Vries (KdV) equation. The solutions are obtained from the set of nonlinear algebraic equations, which can be derived from the modified Korteweg-de Vries (KdV) equation by using the hyperbolic transformation method. The method can be applicable for similar nonlinear wave equations.
Decay Mode Solutions for Kadomtsev-Petviashvili Equation
International Nuclear Information System (INIS)
Fan Guohao; Deng Shufang; Zhang Meng
2012-01-01
The decay mode solutions for the Kadomtsev-Petviashvili (KP) equation are derived by Hirota method (direct method). The decay mode solution is a new set of analytical solutions with Airy function. (general)
Stability and instability of stationary solutions for sublinear parabolic equations
Kajikiya, Ryuji
2018-01-01
In the present paper, we study the initial boundary value problem of the sublinear parabolic equation. We prove the existence of solutions and investigate the stability and instability of stationary solutions. We show that a unique positive and a unique negative stationary solutions are exponentially stable and give the exact exponent. We prove that small stationary solutions are unstable. For one space dimensional autonomous equations, we elucidate the structure of stationary solutions and study the stability of all stationary solutions.
Semianalytic Solution of Space-Time Fractional Diffusion Equation
Directory of Open Access Journals (Sweden)
A. Elsaid
2016-01-01
Full Text Available We study the space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. The series solution of this problem is obtained via the optimal homotopy analysis method (OHAM. Numerical simulations are presented to validate the method and to show the effect of changing the fractional derivative parameters on the solution behavior.
Exact Solutions to a Combined sinh-cosh-Gordon Equation
International Nuclear Information System (INIS)
Wei Long
2010-01-01
Based on a transformed Painleve property and the variable separated ODE method, a function transformation method is proposed to search for exact solutions of some partial differential equations (PDEs) with hyperbolic or exponential functions. This approach provides a more systematical and convenient handling of the solution process of this kind of nonlinear equations. Its key point is to eradicate the hyperbolic or exponential terms by a transformed Painleve property and reduce the given PDEs to a variable-coefficient ordinary differential equations, then we seek for solutions to the resulting equations by some methods. As an application, exact solutions for the combined sinh-cosh-Gordon equation are formally derived. (general)
Seed and soliton solutions for Adler's lattice equation
International Nuclear Information System (INIS)
Atkinson, James; Hietarinta, Jarmo; Nijhoff, Frank
2007-01-01
Adler's lattice equation has acquired the status of a master equation among 2D discrete integrable systems. In this paper we derive what we believe are the first explicit solutions of this equation. In particular it turns out to be necessary to establish a non-trivial seed solution from which soliton solutions can subsequently be constructed using the Baecklund transformation. As a corollary we find the corresponding solutions of the Krichever-Novikov equation which is obtained from Adler's equation in a continuum limit. (fast track communication)
New exact solutions of the Dirac equation. 8
International Nuclear Information System (INIS)
Bagrov, V.G.; Gitman, D.M.; Zadorozhnyj, V.N.; Sukhomlin, N.B.; Shapovalov, V.N.
1978-01-01
The paper continues the investigation into the exact solutions of the Dirac, Klein-Gordon, and Lorentz equations for a charge in an external electromagnetic field. The fields studied do not allow for separation of variables in the Dirac equation, but solutions to the Dirac equation are obtained
Nontrivial Periodic Solutions for Nonlinear Second-Order Difference Equations
Directory of Open Access Journals (Sweden)
Tieshan He
2011-01-01
Full Text Available This paper is concerned with the existence of nontrivial periodic solutions and positive periodic solutions to a nonlinear second-order difference equation. Under some conditions concerning the first positive eigenvalue of the linear equation corresponding to the nonlinear second-order equation, we establish the existence results by using the topological degree and fixed point index theories.
Intuitive Understanding of Solutions of Partially Differential Equations
Kobayashi, Y.
2008-01-01
This article uses diagrams that help the observer see how solutions of the wave equation and heat conduction equation are obtained. The analytical approach cannot necessarily show the mechanisms of the key to the solution without transforming the differential equation into a more convenient form by separation of variables. The visual clues based…
New exact solutions of the Dirac equation. 11
International Nuclear Information System (INIS)
Bagrov, V.G.; Noskov, M.D.
1984-01-01
Investigations into determining new exact solutions of relativistic wave equations started in another paper were continued. Exact solutions of the Dirac, Klein-Gordon equations and classical relativistic equations of motion in four new types of external electromagnetic fields were found
The fundamental solutions for fractional evolution equations of parabolic type
Directory of Open Access Journals (Sweden)
Mahmoud M. El-Borai
2004-01-01
Full Text Available The fundamental solutions for linear fractional evolution equations are obtained. The coefficients of these equations are a family of linear closed operators in the Banach space. Also, the continuous dependence of solutions on the initial conditions is studied. A mixed problem of general parabolic partial differential equations with fractional order is given as an application.
Analytical Solutions of the KDV-KZK Equation
Gan, W. S.
The KdV-KZK equation for fluids developed by me was presented at the ICSV 11 in St. Petersburg in July 2004. In this paper, I made an attempt on the analytical solutions of this equation using the perturbation method. Some physical interpretation of the solutions is given. A brief introduction to KdV-KZK equation for solids is given
Properties of meromorphic solutions to certain differential-difference equations
Directory of Open Access Journals (Sweden)
Xiaoguang Qi
2013-06-01
Full Text Available We consider the properties of meromorphic solutions to certain type of non-linear difference equations. Also we show the existence of meromorphic solutions with finite order for differential-difference equations related to the Fermat type functional equation. This article extends earlier results by Liu et al [12].
Growth of meromorphic solutions of delay differential equations
Halburd, Rod; Korhonen, Risto
2016-01-01
Necessary conditions are obtained for certain types of rational delay differential equations to admit a non-rational meromorphic solution of hyper-order less than one. The equations obtained include delay Painlev\\'e equations and equations solved by elliptic functions.
Exact solutions of generalized Zakharov and Ginzburg-Landau equations
International Nuclear Information System (INIS)
Zhang Jinliang; Wang Mingliang; Gao Kequan
2007-01-01
By using the homogeneous balance principle, the exact solutions of the generalized Zakharov equations and generalized Ginzburg-Landau equation are obtained with the aid of a set of subsidiary higher-order ordinary differential equations (sub-equations for short)
Analytic method for solitary solutions of some partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Ugurlu, Yavuz [Firat University, Department of Mathematics, 23119 Elazig (Turkey); Kaya, Dogan [Firat University, Department of Mathematics, 23119 Elazig (Turkey)], E-mail: dkaya@firat.edu.tr
2007-10-22
In this Letter by considering an improved tanh function method, we found some exact solutions of the clannish random walker's parabolic equation, the modified Korteweg-de Vries (KdV) equation, and the Sharma-Tasso-Olver (STO) equation with its fission and fusion, the Jaulent-Miodek equation.
Analytic method for solitary solutions of some partial differential equations
International Nuclear Information System (INIS)
Ugurlu, Yavuz; Kaya, Dogan
2007-01-01
In this Letter by considering an improved tanh function method, we found some exact solutions of the clannish random walker's parabolic equation, the modified Korteweg-de Vries (KdV) equation, and the Sharma-Tasso-Olver (STO) equation with its fission and fusion, the Jaulent-Miodek equation
International Nuclear Information System (INIS)
Tian Lixin; Yin Jiuli
2004-01-01
In this paper, we introduce the fully nonlinear generalized Camassa-Holm equation C(m,n,p) and by using four direct ansatzs, we obtain abundant solutions: compactons (solutions with the absence of infinite wings), solitary patterns solutions having infinite slopes or cups, solitary waves and singular periodic wave solutions and obtain kink compacton solutions and nonsymmetry compacton solutions. We also study other forms of fully nonlinear generalized Camassa-Holm equation, and their compacton solutions are governed by linear equations
Perturbation Solutions of the Quintic Duffing Equation with Strong Nonlinearities
Directory of Open Access Journals (Sweden)
Mehmet Pakdemirli
Full Text Available The quintic Duffing equation with strong nonlinearities is considered. Perturbation solutions are constructed using two different techniques: The classical multiple scales method (MS and the newly developed multiple scales Lindstedt Poincare method (MSLP. The validity criteria for admissible solutions are derived. Both approximate solutions are contrasted with the numerical solutions. It is found that MSLP provides compatible solution with the numerical solution for strong nonlinearities whereas MS solution fail to produce physically acceptable solution for large perturbation parameters.
Exact solutions of some nonlinear partial differential equations using ...
Indian Academy of Sciences (India)
The functional variable method is a powerful solution method for obtaining exact solutions of some nonlinear partial differential equations. In this paper, the functional variable method is used to establish exact solutions of the generalized forms of Klein–Gordon equation, the (2 + 1)-dimensional Camassa–Holm ...
Exact analytical solutions for nonlinear reaction-diffusion equations
International Nuclear Information System (INIS)
Liu Chunping
2003-01-01
By using a direct method via the computer algebraic system of Mathematica, some exact analytical solutions to a class of nonlinear reaction-diffusion equations are presented in closed form. Subsequently, the hyperbolic function solutions and the triangular function solutions of the coupled nonlinear reaction-diffusion equations are obtained in a unified way
Solutions of the finite type of Sine-Gordon equation
International Nuclear Information System (INIS)
Zhao Guosong
1998-01-01
We use the technique of differential geometry to prove that the solutions of finite type of the sine-Gordon equation φ xx - φ yy = sin φ cosφ can be obtained from a system of ordinary differential equations
Differential and difference equations a comparison of methods of solution
Maximon, Leonard C
2016-01-01
This book, intended for researchers and graduate students in physics, applied mathematics and engineering, presents a detailed comparison of the important methods of solution for linear differential and difference equations - variation of constants, reduction of order, Laplace transforms and generating functions - bringing out the similarities as well as the significant differences in the respective analyses. Equations of arbitrary order are studied, followed by a detailed analysis for equations of first and second order. Equations with polynomial coefficients are considered and explicit solutions for equations with linear coefficients are given, showing significant differences in the functional form of solutions of differential equations from those of difference equations. An alternative method of solution involving transformation of both the dependent and independent variables is given for both differential and difference equations. A comprehensive, detailed treatment of Green’s functions and the associat...
Bounded solutions for fuzzy differential and integral equations
Energy Technology Data Exchange (ETDEWEB)
Nieto, Juan J. [Departamento de Analisis Matematico Facultad de Matematicas Universidad de Santiago de Compostela, 15782 (Spain)] e-mail: amnieto@usc.es; Rodriguez-Lopez, Rosana [Departamento de Analisis Matematico Facultad de Matematicas Universidad de Santiago de Compostela, 15782 (Spain)] e-mail: amrosana@usc.es
2006-03-01
We find sufficient conditions for the boundness of every solution of first-order fuzzy differential equations as well as certain fuzzy integral equations. Our results are based on several theorems concerning crisp differential and integral inequalities.
Adjoint P1 equations solution for neutron slowing down
International Nuclear Information System (INIS)
Cardoso, Carlos Eduardo Santos; Martinez, Aquilino Senra; Silva, Fernando Carvalho da
2002-01-01
In some applications of perturbation theory, it is necessary know the adjoint neutron flux, which is obtained by the solution of adjoint neutron diffusion equation. However, the multigroup constants used for this are weighted in only the direct neutron flux, from the solution of direct P1 equations. In this work, the adjoint P1 equations are derived by the neutron transport equation, the reversion operators rules and analogies between direct and adjoint parameters. The direct and adjoint neutron fluxes resulting from the solution of P 1 equations were used to three different weighting processes, to obtain the macrogroup macroscopic cross sections. It was found out noticeable differences among them. (author)
Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics
Directory of Open Access Journals (Sweden)
Khaled A. Gepreel
2013-01-01
Full Text Available We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.
Exact solution of some linear matrix equations using algebraic methods
Djaferis, T. E.; Mitter, S. K.
1977-01-01
A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.
Exact solutions to some modified sine-Gordon equations
International Nuclear Information System (INIS)
Saermark, K.
1983-01-01
Exact, translational solutions to a number of modified sine-Gordon equations are presented. In deriving the equations and the solutions use is made of results from the theory of ordinary differential equations without moving critical points as given by Ince. It is found that kink-like solutions exist also in cases where the coefficients of the trigonometric terms are space- and time-dependent. (Auth.)
Numerical Asymptotic Solutions Of Differential Equations
Thurston, Gaylen A.
1992-01-01
Numerical algorithms derived and compared with classical analytical methods. In method, expansions replaced with integrals evaluated numerically. Resulting numerical solutions retain linear independence, main advantage of asymptotic solutions.
Exact Solutions to Nonlinear Schroedinger Equation and Higher-Order Nonlinear Schroedinger Equation
International Nuclear Information System (INIS)
Ren Ji; Ruan Hangyu
2008-01-01
We study solutions of the nonlinear Schroedinger equation (NLSE) and higher-order nonlinear Schroedinger equation (HONLSE) with variable coefficients. By considering all the higher-order effect of HONLSE as a new dependent variable, the NLSE and HONLSE can be changed into one equation. Using the generalized Lie group reduction method (GLGRM), the abundant solutions of NLSE and HONLSE are obtained
Symmetry and exact solutions of nonlinear spinor equations
International Nuclear Information System (INIS)
Fushchich, W.I.; Zhdanov, R.Z.
1989-01-01
This review is devoted to the application of algebraic-theoretical methods to the problem of constructing exact solutions of the many-dimensional nonlinear systems of partial differential equations for spinor, vector and scalar fields widely used in quantum field theory. Large classes of nonlinear spinor equations invariant under the Poincare group P(1, 3), Weyl group (i.e. Poincare group supplemented by a group of scale transformations), and the conformal group C(1, 3) are described. Ansaetze invariant under the Poincare and the Weyl groups are constructed. Using these we reduce the Poincare-invariant nonlinear Dirac equations to systems of ordinary differential equations and construct large families of exact solutions of the nonlinear Dirac-Heisenberg equation depending on arbitrary parameters and functions. In a similar way we have obtained new families of exact solutions of the nonlinear Maxwell-Dirac and Klein-Gordon-Dirac equations. The obtained solutions can be used for quantization of nonlinear equations. (orig.)
Null solution of the Yang-Mills equations
International Nuclear Information System (INIS)
Tafel, J.
1986-05-01
We investigate the correspondence between null solutions of the Yang-Mills equations and shearfree geodesic null congruences. We give an example of a non-Abelian null solution with twisting rays. (orig.)
Exact solutions for some discrete models of the Boltzmann equation
International Nuclear Information System (INIS)
Cabannes, H.; Hong Tiem, D.
1987-01-01
For the simplest of the discrete models of the Boltzmann equation: the Broadwell model, exact solutions have been obtained by Cornille in the form of bisolitons. In the present Note, we build exact solutions for more complex models [fr
Jacobian elliptic function expansion solutions of nonlinear stochastic equations
International Nuclear Information System (INIS)
Wei Caimin; Xia Zunquan; Tian Naishuo
2005-01-01
Jacobian elliptic function expansion method is extended and applied to construct the exact solutions of the nonlinear Wick-type stochastic partial differential equations (SPDEs) and some new exact solutions are obtained via this method and Hermite transformation
Entire solutions of Fermat type q-difference differential equations
Directory of Open Access Journals (Sweden)
Kai Liu
2013-02-01
Full Text Available In this article, we describe the finite-order transcendental entire solutions of Fermat type $q$-difference and $q$-difference differential equations. In addition, we investigate the similarities and other properties among those solutions.
A new solution of Einstein's vacuum field equations
Indian Academy of Sciences (India)
The motivation for the new solution ensues ... terms of singularity, does not seem to work universally as there also exist other solutions of eq. ..... the field equations and not necessarily a contribution to the energy–stress tensor, rather just.
Localized solutions of non-linear Klein--Gordon equations
International Nuclear Information System (INIS)
Werle, J.
1977-05-01
Nondissipative, stationary solutions for a class of nonlinear Klein-Gordon equations for a scalar field were found explicitly. Since the field is different from zero only inside a sphere of definite radius, the solutions are called quantum droplets
Classes of general axisymmetric solutions of Einstein-Maxwell equations
International Nuclear Information System (INIS)
Krori, K.D.; Choudhury, T.
1981-01-01
An exact solution of the Einstein equations for a stationary axially symmetric distribution of mass composed of all types of multipoles is obtained. Following Ernst (1968), from this vacuum solution the corresponding solution of the coupled Einstein-Maxwell equations is derived. A solution of Einstein-Maxwell fields for a static axially symmetric system composed of all types of multipoles is also obtained. (author)
Kinetic equation solution by inverse kinetic method
International Nuclear Information System (INIS)
Salas, G.
1983-01-01
We propose a computer program (CAMU) which permits to solve the inverse kinetic equation. The CAMU code is written in HPL language for a HP 982 A microcomputer with a peripheral interface HP 9876 A ''thermal graphic printer''. The CAMU code solves the inverse kinetic equation by taking as data entry the output of the ionization chambers and integrating the equation with the help of the Simpson method. With this program we calculate the evolution of the reactivity in time for a given disturbance
The Investigation of Solutions to the Coupled Schrödinger-Boussinesq Equations
Directory of Open Access Journals (Sweden)
Xin Huang
2013-01-01
equations. The hyperbolic function solutions, trigonometric function solutions, and rational function solutions to the equations are obtained. The decaying properties of several solutions are analyzed.
New exact wave solutions for Hirota equation
Indian Academy of Sciences (India)
2Department of Engineering Sciences, Faculty of Technology and Engineering,. University ... of nonlinear partial differential equations (NPDEs) in mathematical physics. Keywords. ... This method has been successfully applied to obtain exact.
Unsteady Stokes equations: Some complete general solutions
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
homogeneous unsteady Stokes equations are examined. A necessary and sufficient condition for a divergence-free vector to represent the velocity field of a possible unsteady Stokes flow in the absence of body forces is derived. Keywords. Complete ...
Solution of the porous media equation by Adomian's decomposition method
International Nuclear Information System (INIS)
Pamuk, Serdal
2005-01-01
The particular exact solutions of the porous media equation that usually occurs in nonlinear problems of heat and mass transfer, and in biological systems are obtained using Adomian's decomposition method. Also, numerical comparison of particular solutions in the decomposition method indicate that there is a very good agreement between the numerical solutions and particular exact solutions in terms of efficiency and accuracy
Determination of multiple solutions of load flow equations
Indian Academy of Sciences (India)
This paper is concerned with the problem of finding all the real solutions (all components of the solution vector must be real values) of load flow equations. Solutions in which some of the components are complex values are of no interest as they have no physical significance as a load flow solution. This problem issignificant ...
The existence of solutions of q-difference-differential equations.
Wang, Xin-Li; Wang, Hua; Xu, Hong-Yan
2016-01-01
By using the Nevanlinna theory of value distribution, we investigate the existence of solutions of some types of non-linear q-difference differential equations. In particular, we generalize the Rellich-Wittich-type theorem and Malmquist-type theorem about differential equations to the case of q-difference differential equations (system).
Global existence of small solutions to semilinear Schroedinger equations
International Nuclear Information System (INIS)
Chihara, Hiroyuki
1996-01-01
We present global existence theorem for semilinear Schrodinger equations. In general, Schrodinger-type equations do not admit the classical energy estimates. To avoid this difficulty, we use S. Doi's method for linear Schrodinger-type equations. Combining his method and L p -L q estimates, we prove the global existence of solutions with small initial data
Exact solutions to two higher order nonlinear Schroedinger equations
International Nuclear Information System (INIS)
Xu Liping; Zhang Jinliang
2007-01-01
Using the homogeneous balance principle and F-expansion method, the exact solutions to two higher order nonlinear Schroedinger equations which describe the propagation of femtosecond pulses in nonlinear fibres are obtained with the aid of a set of subsidiary higher order ordinary differential equations (sub-equations for short)
Weierstrass Elliptic Function Solutions to Nonlinear Evolution Equations
International Nuclear Information System (INIS)
Yu Jianping; Sun Yongli
2008-01-01
This paper is based on the relations between projection Riccati equations and Weierstrass elliptic equation, combined with the Groebner bases in the symbolic computation. Then the novel method for constructing the Weierstrass elliptic solutions to the nonlinear evolution equations is given by using the above relations
Graphical Solution of the Monic Quadratic Equation with Complex Coefficients
Laine, A. D.
2015-01-01
There are many geometrical approaches to the solution of the quadratic equation with real coefficients. In this article it is shown that the monic quadratic equation with complex coefficients can also be solved graphically, by the intersection of two hyperbolas; one hyperbola being derived from the real part of the quadratic equation and one from…
Soliton solutions of some nonlinear evolution equations with time ...
Indian Academy of Sciences (India)
Abstract. In this paper, we obtain exact soliton solutions of the modified KdV equation, inho- mogeneous nonlinear Schrödinger equation and G(m, n) equation with variable coefficients using solitary wave ansatz. The constraint conditions among the time-dependent coefficients turn out as necessary conditions for the ...
Analytic solution of integral equations for molecular fluids
International Nuclear Information System (INIS)
Cummings, P.T.
1984-01-01
We review some recent progress in the analytic solution of integral equations for molecular fluids. The site-site Ornstein-Zernike (SSOZ) equation with approximate closures appropriate to homonuclear diatomic fluids both with and without attractive dispersion-like interactions has recently been solved in closed form analytically. In this paper, the close relationship between the SSOZ equation for homonuclear dumbells and the usual Ornstein-Zernike (OZ) equation for atomic fluids is carefully elucidated. This relationship is a key motivation for the analytic solutions of the SSOZ equation that have been obtained to date. (author)
Self-similar solutions of the modified nonlinear schrodinger equation
International Nuclear Information System (INIS)
Kitaev, A.V.
1986-01-01
This paper considers a 2 x 2 matrix linear ordinary differential equation with large parameter t and irregular singular point of fourth order at infinity. The leading order of the monodromy data of this equation is calculated in terms of its coefficients. Isomonodromic deformations of the equation are self-similar solutions of the modified nonlinear Schrodinger equation, and therefore inversion of the expressions obtained for the monodromy data gives the leading term in the time-asymptotic behavior of the self-similar solution. The application of these results to the type IV Painleve equation is considered in detail
Islam, Md Hamidul; Khan, Kamruzzaman; Akbar, M Ali; Salam, Md Abdus
2014-01-01
Mathematical modeling of many physical systems leads to nonlinear evolution equations because most physical systems are inherently nonlinear in nature. The investigation of traveling wave solutions of nonlinear partial differential equations (NPDEs) plays a significant role in the study of nonlinear physical phenomena. In this article, we construct the traveling wave solutions of modified KDV-ZK equation and viscous Burgers equation by using an enhanced (G '/G) -expansion method. A number of traveling wave solutions in terms of unknown parameters are obtained. Derived traveling wave solutions exhibit solitary waves when special values are given to its unknown parameters. 35C07; 35C08; 35P99.
New solutions of Heun's general equation
Energy Technology Data Exchange (ETDEWEB)
Ishkhanyan, Artur [Engineering Center of Armenian National Academy of Sciences, Ashtarak (Armenia); Suominen, Kalle-Antti [Helsinki Institute of Physics, PL 64, Helsinki (Finland)
2003-02-07
We show that in four particular cases the derivative of the solution of Heun's general equation can be expressed in terms of a solution to another Heun's equation. Starting from this property, we use the Gauss hypergeometric functions to construct series solutions to Heun's equation for the mentioned cases. Each of the hypergeometric functions involved has correct singular behaviour at only one of the singular points of the equation; the sum, however, has correct behaviour. (letter to the editor)
On the solution of the Liouville equation over a rectangle
Directory of Open Access Journals (Sweden)
A. M. Arthurs
1996-01-01
Full Text Available Methods for integral equations are used to derive pointwise bounds for the solution of a boundary value problem for the nonlinear Liouville partial differential equation over a rectangle. Several test calculations are performed and the resulting solutions are more accurate than those obtained previously by other methods.
Exact Solution of a Generalized Nonlinear Schrodinger Equation Dimer
DEFF Research Database (Denmark)
Christiansen, Peter Leth; Maniadis, P.; Tsironis, G.P.
1998-01-01
We present exact solutions for a nonlinear dimer system defined throught a discrete nonlinear Schrodinger equation that contains also an integrable Ablowitz-Ladik term. The solutions are obtained throught a transformation that maps the dimer into a double Sine-Gordon like ordinary nonlinear...... differential equation....
New solitary wave solutions to the modified Kawahara equation
International Nuclear Information System (INIS)
Wazwaz, Abdul-Majid
2007-01-01
In this work we use the sine-cosine method, the tanh method, the extended tanh method, and ansatze of hyperbolic functions for analytic treatment for the modified Kawahara equation. New solitons solutions and periodic solutions are formally derived. The change of the parameters, that will drastically change the characteristics of the equation, is examined. The employed approaches are reliable and manageable
Exact Solutions of the Harry-Dym Equation
International Nuclear Information System (INIS)
Mokhtari, Reza
2011-01-01
The aim of this paper is to generate exact travelling wave solutions of the Harry-Dym equation through the methods of Adomian decomposition, He's variational iteration, direct integration, and power series. We show that the two later methods are more successful than the two former to obtain more solutions of the equation. (general)
Exact soliton-like solutions of perturbed phi4-equation
International Nuclear Information System (INIS)
Gonzalez, J.A.
1986-05-01
Exact soliton-like solutions of damped, driven phi 4 -equation are found. The exact expressions for the velocities of solitons are given. It is non-perturbatively proved that the perturbed phi 4 -equation has stable kink-like solutions of a new type. (author)
Approximate variational solutions of the Grad-Shafranov equation
International Nuclear Information System (INIS)
Ludwig, G.O.
2001-01-01
Approximate solutions of the Grad-Schlueter-Shafranov equation based on variational methods are developed. The power series solutions of the Euler-Lagrange equations for equilibrium are compared with direct variational results for a low aspect ratio tokamak equilibrium. (author)
Remarks about singular solutions to the Dirac equation
International Nuclear Information System (INIS)
Uhlir, M.
1975-01-01
In the paper singular solutions of the Dirac equation are investigated. They are derived in the Lorentz-covariant way of functions proportional to static multipole fields of scalar and (or) electromagnetic fields and of regular solutions of the Dirac equations. The regularization procedure excluding divergences of total energy, momentum and angular momentum of the spinor field considered is proposed
On genus-two solutions for the ILW equation
Tutiya, Y.
2018-02-01
The existence of theta function solutions of genus two for the intermediate long-wave equation is established. A numerical example is also presented. The method basically goes along with Krichever's construction of theta function solutions for soliton equations, such as the Kronecker product equation. This idea leads us to a question whether a Riemann surface exists which allows a peculiar abelian integral of the third kind. The answer is affirmative at least for genus-two curves.
Similarity Solutions for Multiterm Time-Fractional Diffusion Equation
Elsaid, A.; Abdel Latif, M. S.; Maneea, M.
2016-01-01
Similarity method is employed to solve multiterm time-fractional diffusion equation. The orders of the fractional derivatives belong to the interval (0,1] and are defined in the Caputo sense. We illustrate how the problem is reduced from a multiterm two-variable fractional partial differential equation to a multiterm ordinary fractional differential equation. Power series solution is obtained for the resulting ordinary problem and the convergence of the series solution is discussed. Based on ...
Rational approximations to solutions of linear differential equations.
Chudnovsky, D V; Chudnovsky, G V
1983-08-01
Rational approximations of Padé and Padé type to solutions of differential equations are considered. One of the main results is a theorem stating that a simultaneous approximation to arbitrary solutions of linear differential equations over C(x) cannot be "better" than trivial ones implied by the Dirichlet box principle. This constitutes, in particular, the solution in the linear case of Kolchin's problem that the "Roth's theorem" holds for arbitrary solutions of algebraic differential equations. Complete effective proofs for several valuations are presented based on the Wronskian methods and graded subrings of Picard-Vessiot extensions.
Lipschitz Regularity of Solutions for Mixed Integro-Differential Equations
Barles, Guy; Chasseigne, Emmanuel; Ciomaga, Adina; Imbert, Cyril
2011-01-01
We establish new Hoelder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii-Lions's method. We thus extend the Hoelder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the loc...
New Exact Solutions for New Model Nonlinear Partial Differential Equation
Maher, A.; El-Hawary, H. M.; Al-Amry, M. S.
2013-01-01
In this paper we propose a new form of Padé-II equation, namely, a combined Padé-II and modified Padé-II equation. The mapping method is a promising method to solve nonlinear evaluation equations. Therefore, we apply it, to solve the combined Padé-II and modified Padé-II equation. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions, and elliptic functions.
Variational solution of the loop equation in QCD
International Nuclear Information System (INIS)
Agishtein, M.E.; Migdal, A.A.
1988-01-01
A new technique for the large N loop equation of QCD is worked out. The Wilson loop W(C) is approximated by a Gaussian functional. The parameters are fitted to the loop equation, after which the equation is statisfied up to 0.2%. The resulting Wilson loop corresponds to linearly rising Regge trajectories. The problem of tachyon is still present, but it could be cured by iteration of the loop equation starting from this variational solution. (orig.)
Variational solution of the loop equation in QCD
International Nuclear Information System (INIS)
Agishtein, M.E.; Migdal, A.A.
1988-01-01
A new technique for the large N loop equation of QCD is worked out. The Wilson loop W(C) is approximated by a Gaussian functional. The parameters are fitted to the loop equation, after which the equation is satisfied up to 0.2%. The resulting Wilson loop corresponds to linearly rising Regge trajectories. The problem of tachyon is still present, but it could be cured by iteration of the loop equation starting from this variational solution
International Nuclear Information System (INIS)
Chen Yong; Wang Qi; Li Biao
2005-01-01
Based on a new general ansatz and a general subepuation, a new general algebraic method named elliptic equation rational expansion method is devised for constructing multiple travelling wave solutions in terms of rational special function for nonlinear evolution equations (NEEs). We apply the proposed method to solve Whitham-Broer-Kaup equation and explicitly construct a series of exact solutions which include rational form solitary wave solution, rational form triangular periodic wave solutions and rational wave solutions as special cases. In addition, the links among our proposed method with the method by Fan [Chaos, Solitons and Fractals 2004;20:609], are also clarified generally
Numerical integration of asymptotic solutions of ordinary differential equations
Thurston, Gaylen A.
1989-01-01
Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.
Solution of Fuzzy Differential Equations Using Fuzzy Sumudu Transforms
Directory of Open Access Journals (Sweden)
Raheleh Jafari
2018-01-01
Full Text Available The uncertain nonlinear systems can be modeled with fuzzy differential equations (FDEs and the solutions of these equations are applied to analyze many engineering problems. However, it is very difficult to obtain solutions of FDEs. In this paper, the solutions of FDEs are approximated by utilizing the fuzzy Sumudu transform (FST method. Significant theorems are suggested in order to explain the properties of FST. The proposed method is validated with three real examples.
Analytical solution of population balance equation involving ...
Indian Academy of Sciences (India)
This paper presents an effective analytical simulation to solve population balance equation (PBE), involving particulate aggregation and breakage, by making use ... The domain part of the email address of all email addresses used by the office of Indian Academy of Sciences, including those of the staff, the journals, various ...
Probabilistic solution of the Dirac equation
International Nuclear Information System (INIS)
Blanchard, P.; Combe, P.
1985-01-01
Various probabilistic representations of the 2, 3 and 4 dimensional Dirac equation are given in terms of expectation with respect to stochastic jump processes and are used to derive the nonrelativistic limit even in the presence of an external electromagnetic field. (orig.)
Mallet, D. G.; McCue, S. W.
2009-01-01
The solution of linear ordinary differential equations (ODEs) is commonly taught in first-year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognizing what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to…
Tisdell, C. C.
2017-01-01
Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem…
Calculation of similarity solutions of partial differential equations
International Nuclear Information System (INIS)
Dresner, L.
1980-08-01
When a partial differential equation in two independent variables is invariant to a group G of stretching transformations, it has similarity solutions that can be found by solving an ordinary differential equation. Under broad conditions, this ordinary differential equation is also invariant to another stretching group G', related to G. The invariance of the ordinary differential equation to G' can be used to simplify its solution, particularly if it is of second order. Then a method of Lie's can be used to reduce it to a first-order equation, the study of which is greatly facilitated by analysis of its direction field. The method developed here is applied to three examples: Blasius's equation for boundary layer flow over a flat plate and two nonlinear diffusion equations, cc/sub t/ = c/sub zz/ and c/sub t/ = (cc/sub z/)/sub z/
Solutions of hyperbolic equations with the CIP-BS method
International Nuclear Information System (INIS)
Utsumi, Takayuki; Koga, James; Yamagiwa, Mitsuru; Yabe, Takashi; Aoki, Takayuki
2004-01-01
In this paper, we show that a new numerical method, the Constrained Interpolation Profile - Basis Set (CIP-BS) method, can solve general hyperbolic equations efficiently. This method uses a simple polynomial basis set that is easily extendable to any desired higher-order accuracy. The interpolating profile is chosen so that the subgrid scale solution approaches the local real solution owing to the constraints from the spatial derivatives of the master equations. Then, introducing scalar products, the linear and nonlinear partial differential equations are uniquely reduced to the ordinary differential equations for values and spatial derivatives at the grid points. The method gives stable, less diffusive, and accurate results. It is successfully applied to the continuity equation, the Burgers equation, the Korteweg-de Vries equation, and one-dimensional shock tube problems. (author)
Soliton-like solutions to the ordinary Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Zamboni-Rached, Michel [Universidade Estadual de Campinas (DMO/FEEC/UNICAMP), Campinas, SP (Brazil). Fac. de Engenharia Eletrica e de Computacao. Dept. de Microondas e Optica; Recami, Erasmo, E-mail: recami@mi.infn.i [Universita Statale di Bergamo, Bergamo (Italy). Facolta di Ingegneria
2011-07-01
In recent times it has been paid attention to the fact that (linear) wave equations admit of soliton-like solutions, known as Localized Waves or Non-diffracting Waves, which propagate without distortion in one direction. Such Localized Solutions (existing also for K-G or Dirac equations) are a priori suitable, more than Gaussian's, for describing elementary particle motion. In this paper we show that, mutatis mutandis, Localized Solutions exist even for the ordinary Schroedinger equation within standard Quantum Mechanics; and we obtain both approximate and exact solutions, also setting forth for them particular examples. In the ideal case such solutions bear infinite energy, as well as plane or spherical waves: we show therefore how to obtain nite-energy solutions. At last, we briefly consider solutions for a particle moving in the presence of a potential. (author)
Soliton-like solutions to the ordinary Schroedinger equation
International Nuclear Information System (INIS)
Zamboni-Rached, Michel; Recami, Erasmo
2011-01-01
In recent times it has been paid attention to the fact that (linear) wave equations admit of soliton-like solutions, known as Localized Waves or Non-diffracting Waves, which propagate without distortion in one direction. Such Localized Solutions (existing also for K-G or Dirac equations) are a priori suitable, more than Gaussian's, for describing elementary particle motion. In this paper we show that, mutatis mutandis, Localized Solutions exist even for the ordinary Schroedinger equation within standard Quantum Mechanics; and we obtain both approximate and exact solutions, also setting forth for them particular examples. In the ideal case such solutions bear infinite energy, as well as plane or spherical waves: we show therefore how to obtain nite-energy solutions. At last, we briefly consider solutions for a particle moving in the presence of a potential. (author)
Bifurcations of traveling wave solutions for an integrable equation
International Nuclear Information System (INIS)
Li Jibin; Qiao Zhijun
2010-01-01
This paper deals with the following equation m t =(1/2)(1/m k ) xxx -(1/2)(1/m k ) x , which is proposed by Z. J. Qiao [J. Math. Phys. 48, 082701 (2007)] and Qiao and Liu [Chaos, Solitons Fractals 41, 587 (2009)]. By adopting the phase analysis method of planar dynamical systems and the theory of the singular traveling wave systems to the traveling wave solutions of the equation, it is shown that for different k, the equation may have infinitely many solitary wave solutions, periodic wave solutions, kink/antikink wave solutions, cusped solitary wave solutions, and breaking loop solutions. We discuss in a detail the cases of k=-2,-(1/2),(1/2),2, and parametric representations of all possible bounded traveling wave solutions are given in the different (c,g)-parameter regions.
Explicit solutions of the Camassa-Holm equation
International Nuclear Information System (INIS)
Parkes, E.J.; Vakhnenko, V.O.
2005-01-01
Explicit travelling-wave solutions of the Camassa-Holm equation are sought. The solutions are characterized by two parameters. For propagation in the positive x-direction, both periodic and solitary smooth-hump, peakon, cuspon and inverted-cuspon waves are found. For propagation in the negative x-direction, there are solutions which are just the mirror image in the x-axis of the aforementioned solutions. Some composite wave solutions of the Degasperis-Procesi equation are given in an appendix
Unsteady analytical solutions to the Poisson–Nernst–Planck equations
International Nuclear Information System (INIS)
Schönke, Johannes
2012-01-01
It is shown that the Poisson–Nernst–Planck equations for a single ion species can be formulated as one equation in terms of the electric field. This previously not analyzed equation shows similarities to the vector Burgers equation and is identical with it in the one dimensional case. Several unsteady exact solutions for one and multidimensional cases are presented. Besides new mathematical insights which these first known unsteady solutions give, they can serve as test cases in computer simulations to analyze numerical algorithms and to verify code. (paper)
On solutions of variable-order fractional differential equations
Directory of Open Access Journals (Sweden)
Ali Akgül
2017-01-01
solutions to fractional differential equations are compelling to get in real applications, due to the nonlocality and complexity of the fractional differential operators, especially for variable-order fractional differential equations. Therefore, it is significant to enhanced numerical methods for fractional differential equations. In this work, we consider variable-order fractional differential equations by reproducing kernel method. There has been much attention in the use of reproducing kernels for the solutions to many problems in the recent years. We give two examples to demonstrate how efficiently our theory can be implemented in practice.
Positive Solutions for Coupled Nonlinear Fractional Differential Equations
Directory of Open Access Journals (Sweden)
Wenning Liu
2014-01-01
Full Text Available We consider the existence of positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary values. Assume the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two cones K1, K2 and computing the fixed point index in product cone K1×K2, we obtain that the system has a pair of positive solutions. It is remarkable that it is established on the Cartesian product of two cones, in which the feature of two equations can be opposite.
Closed form solutions of two time fractional nonlinear wave equations
Directory of Open Access Journals (Sweden)
M. Ali Akbar
2018-06-01
Full Text Available In this article, we investigate the exact traveling wave solutions of two nonlinear time fractional wave equations. The fractional derivatives are described in the sense of conformable fractional derivatives. In addition, the traveling wave solutions are accomplished in the form of hyperbolic, trigonometric, and rational functions involving free parameters. To investigate such types of solutions, we implement the new generalized (G′/G-expansion method. The extracted solutions are reliable, useful and suitable to comprehend the optimal control problems, chaotic vibrations, global and local bifurcations and resonances, furthermore, fission and fusion phenomena occur in solitons, the relativistic energy-momentum relation, scalar electrodynamics, quantum relativistic one-particle theory, electromagnetic interactions etc. The results reveal that the method is very fruitful and convenient for exploring nonlinear differential equations of fractional order treated in theoretical physics. Keywords: Traveling wave solution, Soliton, Generalized (G′/G-expansion method, Time fractional Duffing equation, Time fractional Riccati equation
Approximate solution fuzzy pantograph equation by using homotopy perturbation method
Jameel, A. F.; Saaban, A.; Ahadkulov, H.; Alipiah, F. M.
2017-09-01
In this paper, Homotopy Perturbation Method (HPM) is modified and formulated to find the approximate solution for its employment to solve (FDDEs) involving a fuzzy pantograph equation. The solution that can be obtained by using HPM is in the form of infinite series that converge to the actual solution of the FDDE and this is one of the benefits of this method In addition, it can be used for solving high order fuzzy delay differential equations directly without reduction to a first order system. Moreover, the accuracy of HPM can be detected without needing the exact solution. The HPM is studied for fuzzy initial value problems involving pantograph equation. Using the properties of fuzzy set theory, we reformulate the standard approximate method of HPM and obtain the approximate solutions. The effectiveness of the proposed method is demonstrated for third order fuzzy pantograph equation.
Topological soliton solutions for some nonlinear evolution equations
Directory of Open Access Journals (Sweden)
Ahmet Bekir
2014-03-01
Full Text Available In this paper, the topological soliton solutions of nonlinear evolution equations are obtained by the solitary wave ansatz method. Under some parameter conditions, exact solitary wave solutions are obtained. Note that it is always useful and desirable to construct exact solutions especially soliton-type (dark, bright, kink, anti-kink, etc. envelope for the understanding of most nonlinear physical phenomena.
Rotationally symmetric numerical solutions to the sine-Gordon equation
DEFF Research Database (Denmark)
Olsen, O. H.; Samuelsen, Mogens Rugholm
1981-01-01
We examine numerically the properties of solutions to the spherically symmetric sine-Gordon equation given an initial profile which coincides with the one-dimensional breather solution and refer to such solutions as ring waves. Expanding ring waves either exhibit a return effect or expand towards...
Existence of extremal periodic solutions for quasilinear parabolic equations
Directory of Open Access Journals (Sweden)
Siegfried Carl
1997-01-01
bounded domain under periodic Dirichlet boundary conditions. Our main goal is to prove the existence of extremal solutions among all solutions lying in a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof of our result are recently obtained abstract results on nonlinear evolution equations, comparison and truncation techniques and suitably constructed special testfunction.
Static Solutions of Einstein's Equations with Cylindrical Symmetry
Trendafilova, C. S.; Fulling, S. A.
2011-01-01
In analogy with the standard derivation of the Schwarzschild solution, we find all static, cylindrically symmetric solutions of the Einstein field equations for vacuum. These include not only the well-known cone solution, which is locally flat, but others in which the metric coefficients are powers of the radial coordinate and the spacetime is…
Abundant Interaction Solutions of Sine-Gordon Equation
Directory of Open Access Journals (Sweden)
DaZhao Lü
2012-01-01
Full Text Available With the help of computer symbolic computation software (e.g., Maple, abundant interaction solutions of sine-Gordon equation are obtained by means of a constructed Wronskian form expansion method. The method is based upon the forms and structures of Wronskian solutions of sine-Gordon equation, and the functions used in the Wronskian determinants do not satisfy linear partial differential equations. Such interaction solutions are difficultly obtained via other methods. And the method can be automatically carried out in computer.
An asymptotic formula for Weyl solutions of the dirac equations
International Nuclear Information System (INIS)
Misyura, T.V.
1995-01-01
In the spectral analysis of differential operators and its applications an important role is played by the investigation of the behavior of the Weyl solutions of the corresponding equations when the spectral parameter tends to infinity. Elsewhere an exact asymptotic formula for the Weyl solutions of a large class of Sturm-Liouville equations has been obtained. A decisve role in the proof of this formula has been the semiboundedness property of the corresponding Sturm-Liouville operators. In this paper an analogous formula is obtained for the Weyl solutions of the Dirac equations
On the Solution of the Rational Matrix Equation
Directory of Open Access Journals (Sweden)
Faßbender Heike
2007-01-01
Full Text Available We study numerical methods for finding the maximal symmetric positive definite solution of the nonlinear matrix equation , where is symmetric positive definite and is nonsingular. Such equations arise for instance in the analysis of stationary Gaussian reciprocal processes over a finite interval. Its unique largest positive definite solution coincides with the unique positive definite solution of a related discrete-time algebraic Riccati equation (DARE. We discuss how to use the butterfly algorithm to solve the DARE. This approach is compared to several fixed-point and doubling-type iterative methods suggested in the literature.
Conformally flat spaces and solutions to Yang-Mills equations
International Nuclear Information System (INIS)
Chaohao, G.
1980-01-01
Using the conformal invariance of Yang-Mills equations in four-dimensional manifolds, it is proved that in a simply connected space of negative constant curvature Yang-Mills equations admit solutions with any real number as their Pontryagin number. It is also shown that the space S 3 x S 1 which is the regular counterpart of the meron solution is one example of a class of solutions to Yang-Mills equations on compact manifolds that are neither self-dual nor anti-self-dual
Periodic solutions of Volterra integral equations
Directory of Open Access Journals (Sweden)
M. N. Islam
1988-01-01
Full Text Available Consider the system of equationsx(t=f(t+∫−∞tk(t,sx(sds, (1andx(t=f(t+∫−∞tk(t,sg(s,x(sds. (2Existence of continuous periodic solutions of (1 is shown using the resolvent function of the kernel k. Some important properties of the resolvent function including its uniqueness are obtained in the process. In obtaining periodic solutions of (1 it is necessary that the resolvent of k is integrable in some sense. For a scalar convolution kernel k some explicit conditions are derived to determine whether or not the resolvent of k is integrable. Finally, the existence and uniqueness of continuous periodic solutions of (1 and (2 are btained using the contraction mapping principle as the basic tool.
The recursive solution of the Schroedinger equation
International Nuclear Information System (INIS)
Haydock, R.
The transformation of an arbitrary quantum model and its subsequent analysis is proposed. The chain expresses mathematically the physical concept of local environment. The recursive transformation yields analytic chains for some systems, but it is also convenient and efficient for constructing numerical chain models enabling the solution of problems which are too big for numerical matrix methods. The chain model sugests new approach to quantum mechanical models. Because of the simple solution of chain models, the qualitative behaviour of different physical properties can be determined. Unlike many methods for solving quantum models, one has rigorous results about the convergence of approximation. Because they are defined recursively, the approsimations are suited to computation. (Ha)
Spurious solutions in few-body equations. II. Numerical investigations
International Nuclear Information System (INIS)
Adhikari, S.K.
1979-01-01
A recent analytic study of spurious solutions in few-body equations by Adhikari and Gloeckle is here complemented by numerical investigations. As proposed by Adhikari and Gloeckle we study numerically the spurious solutions in the three-body Weinberg type equations and draw some general conclusions about the existence of spurious solutions in three-body equations with the Weinberg kernel and in other few-body formulations. In particular we conclude that for most of the potentials we encounter in problems of nuclear physics the three-body Weinberg type equation will not have a spurious solution which may interfere with the bound state or scattering calculation. Hence, if proven convenient, the three-body Weinberg type equation can be used in practical calculations. The same conclusion is true for the three-body channel coupling array scheme of Kouri, Levin, and Tobocman. In the case of the set of six coupled four-body equations proposed by Rosenberg et al. and the set of the Bencze-Redish-Sloan equations a careful study of the possible spurious solutions is needed before using these equations in practical calculations
Numerical solution of distributed order fractional differential equations
Katsikadelis, John T.
2014-02-01
In this paper a method for the numerical solution of distributed order FDEs (fractional differential equations) of a general form is presented. The method applies to both linear and nonlinear equations. The Caputo type fractional derivative is employed. The distributed order FDE is approximated with a multi-term FDE, which is then solved by adjusting appropriately the numerical method developed for multi-term FDEs by Katsikadelis. Several example equations are solved and the response of mechanical systems described by such equations is studied. The convergence and the accuracy of the method for linear and nonlinear equations are demonstrated through well corroborated numerical results.
Analytical exact solution of the non-linear Schroedinger equation
International Nuclear Information System (INIS)
Martins, Alisson Xavier; Rocha Filho, Tarcisio Marciano da
2011-01-01
Full text: In this work we present how to classify and obtain analytical solutions of the Schroedinger equation with a generic non-linearity in 1+1 dimensions. Our approach is based on the determination of Lie symmetry transformation mapping solutions into solutions, and non-classical symmetry transformations, mapping a given solution into itself. From these symmetries it is then possible to reduce the equation to a system of ordinary differential equations which can then be solved using standard methods. The generic non-linearity is handled by considering it as an additional unknown in the determining equations for the symmetry transformations. This results in an over-determined system of non-linear partial differential equations. Its solution can then be determined in some cases by reducing it to the so called involutive (triangular) form, and then solved. This reduction is very tedious and can only performed using a computer algebra system. Once the determining system is solved, we obtain the explicit form for the non-linearity admitting a Lie or non-classical symmetry. The analytical solutions are then derived by solving the reduced ordinary differential equations. The non-linear determining system for the non-classical symmetry transformations and Lie symmetry generators are obtaining using the computer algebra package SADE (symmetry analysis of differential equations), developed at our group. (author)
On the solutions of the second heavenly and Pavlov equations
Manakov, S. V.; Santini, P. M.
2009-10-01
We have recently solved the inverse scattering problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields, such as the heavenly equation of Plebanski, the dispersionless Kadomtsev-Petviashvili (dKP) equation and the two-dimensional dispersionless Toda (2ddT) equation, as well as with the commutation of one-dimensional vector fields, such as the Pavlov equation. We also showed that the associated Riemann-Hilbert inverse problems are powerful tools to establish if the solutions of the Cauchy problem break at finite time, to construct their long-time behaviour and characterize classes of implicit solutions. In this paper, using the above theory, we concentrate on the heavenly and Pavlov equations, (i) establishing that their localized solutions evolve without breaking, unlike the cases of dKP and 2ddT; (ii) constructing the long-time behaviour of the solutions of their Cauchy problems; (iii) characterizing a distinguished class of implicit solutions of the heavenly equation.
On the solutions of the second heavenly and Pavlov equations
International Nuclear Information System (INIS)
Manakov, S V; Santini, P M
2009-01-01
We have recently solved the inverse scattering problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields, such as the heavenly equation of Plebanski, the dispersionless Kadomtsev-Petviashvili (dKP) equation and the two-dimensional dispersionless Toda (2ddT) equation, as well as with the commutation of one-dimensional vector fields, such as the Pavlov equation. We also showed that the associated Riemann-Hilbert inverse problems are powerful tools to establish if the solutions of the Cauchy problem break at finite time, to construct their long-time behaviour and characterize classes of implicit solutions. In this paper, using the above theory, we concentrate on the heavenly and Pavlov equations, (i) establishing that their localized solutions evolve without breaking, unlike the cases of dKP and 2ddT; (ii) constructing the long-time behaviour of the solutions of their Cauchy problems; (iii) characterizing a distinguished class of implicit solutions of the heavenly equation.
Travelling wave solutions to the Kuramoto-Sivashinsky equation
International Nuclear Information System (INIS)
Nickel, J.
2007-01-01
Combining the approaches given by Baldwin [Baldwin D et al. Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs. J Symbol Comput 2004;37:669-705], Peng [Peng YZ. A polynomial expansion method and new general solitary wave solutions to KS equation. Comm Theor Phys 2003;39:641-2] and by Schuermann [Schuermann HW, Serov VS. Weierstrass' solutions to certain nonlinear wave and evolution equations. Proc progress electromagnetics research symposium, 28-31 March 2004, Pisa. p. 651-4; Schuermann HW. Traveling-wave solutions to the cubic-quintic nonlinear Schroedinger equation. Phys Rev E 1996;54:4312-20] leads to a method for finding exact travelling wave solutions of nonlinear wave and evolution equations (NLWEE). The first idea is to generalize ansaetze given by Baldwin and Peng to find elliptic solutions of NLWEEs. Secondly, conditions used by Schuermann to find physical (real and bounded) solutions and to discriminate between periodic and solitary wave solutions are used. The method is shown in detail by evaluating new solutions of the Kuramoto-Sivashinsky equation
Analytical solution of population balance equation involving ...
Indian Academy of Sciences (India)
This paper presents an effective analytical simulation to solve population .... considering spatial dependence and growth, based on the so-called LPA formulation as .... But the particle size distribution is defined so that n(v,t) dx is the number of ..... that was made beforehand in the construction of the analytical solutions ...
Symbolic computation of exact solutions for a nonlinear evolution equation
International Nuclear Information System (INIS)
Liu Yinping; Li Zhibin; Wang Kuncheng
2007-01-01
In this paper, by means of the Jacobi elliptic function method, exact double periodic wave solutions and solitary wave solutions of a nonlinear evolution equation are presented. It can be shown that not only the obtained solitary wave solutions have the property of loop-shaped, cusp-shaped and hump-shaped for different values of parameters, but also different types of double periodic wave solutions are possible, namely periodic loop-shaped wave solutions, periodic hump-shaped wave solutions or periodic cusp-shaped wave solutions. Furthermore, periodic loop-shaped wave solutions will be degenerated to loop-shaped solitary wave solutions for the same values of parameters. So do cusp-shaped solutions and hump-shaped solutions. All these solutions are new and first reported here
Lump solutions to the Kadomtsev–Petviashvili equation
Energy Technology Data Exchange (ETDEWEB)
Ma, Wen-Xiu, E-mail: mawx@cas.usf.edu
2015-09-25
Through symbolic computation with Maple, a class of lump solutions, rationally localized in all directions in the space, to the (2 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation is presented, making use of its Hirota bilinear form. The resulting lump solutions contain six free parameters, two of which are due to the translation invariance of the KP equation and the other four of which satisfy a non-zero determinant condition guaranteeing analyticity and rational localization of the solutions. Three contour plots with different determinant values are sequentially made to show that the corresponding lump solution tends to zero when the determinant approaches zero. Two particular lump solutions with specific values of the involved parameters are plotted, as illustrative examples. - Highlights: • Positive quadratic function solutions. • Solitons rationally-localized in all directions in the space. • Solving systems of nonlinear algebraic equations by symbolic computation with Maple.
Explicit solution for a wave equation with nonlocal condition
Bazhlekova, Emilia; Dimovski, Ivan
2012-11-01
An initial-boundary value problem with a nonlocal boundary condition for one-dimensional wave equation is studied. Applying spectral projections, we find a series solution of the problem. The character of the solution found shows that the oscillation amplitude of the system described by this equation increases with time at any fixed x in absence of external forces. To find a representation of the solution more convenient for numerical calculation we develop a two-dimensional operational calculus for the problem. The solution is expressed as a sum of non-classical convolution products of particular solutions and the arbitrary initial functions. This result is an extension of the classical Duhamel principle for the space variable. The representation is used successfully for numerical computation and visualization of the solution. Numerical results obtained for specific test problems with known exact solutions indicate that the present technique provides accurate numerical solutions.
Soliton solutions for ABS lattice equations: I. Cauchy matrix approach
Nijhoff, Frank; Atkinson, James; Hietarinta, Jarmo
2009-10-01
In recent years there have been new insights into the integrability of quadrilateral lattice equations, i.e. partial difference equations which are the natural discrete analogues of integrable partial differential equations in 1+1 dimensions. In the scalar (i.e. single-field) case, there now exist classification results by Adler, Bobenko and Suris (ABS) leading to some new examples in addition to the lattice equations 'of KdV type' that were known since the late 1970s and early 1980s. In this paper, we review the construction of soliton solutions for the KdV-type lattice equations and use those results to construct N-soliton solutions for all lattice equations in the ABS list except for the elliptic case of Q4, which is left to a separate treatment.
Approximated solutions to the Schroedinger equation
International Nuclear Information System (INIS)
Rico, J.F.; Fernandez-Alonso, J.I.
1977-01-01
The authors are currently working on a couple of the well-known deficiencies of the variation method and present here some of the results that have been obtained so far. The variation method does not give information a priori on the trial functions best suited for a particular problem nor does it give information a posteriori on the degree of precision attained. In order to clarify the origin of both difficulties, a geometric interpretation of the variation method is presented. This geometric interpretation is the starting point for the exact formal solution to the fundamental state and for the step-by-step approximations to the exact solution which are also given. Some comments on these results are included. (Auth.)
Thin-Layer Solutions of the Helmholtz and Related Equations
Ockendon, J. R.
2012-01-01
This paper concerns a certain class of two-dimensional solutions to four generic partial differential equations-the Helmholtz, modified Helmholtz, and convection-diffusion equations, and the heat conduction equation in the frequency domain-and the connections between these equations for this particular class of solutions.S pecifically, we consider thin-layer solutions, valid in narrow regions across which there is rapid variation, in the singularly perturbed limit as the coefficient of the Laplacian tends to zero.F or the wellstudied Helmholtz equation, this is the high-frequency limit and the solutions in question underpin the conventional ray theory/WKB approach in that they provide descriptions valid in some of the regions where these classical techniques fail.E xamples are caustics, shadow boundaries, whispering gallery, and creeping waves and focusing and bouncing ball modes.It transpires that virtually all such thin-layer models reduce to a class of generalized parabolic wave equations, of which the heat conduction equation is a special case. Moreover, in most situations, we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the Helmholtz equation.W e also show how reasonably well-understood thin-layer phenomena associated with any one of the four generic equations may translate into less well-known effects associated with the others.In addition, our considerations also shed some light on the relationship between the methods of matched asymptotic, WKB, and multiple-scales expansions. © 2012 Society for Industrial and Applied Mathematics.
Existence criterion of spurious solutions of Faddeev equations
International Nuclear Information System (INIS)
Pupyshev, V.V.
1995-01-01
The Faddeev differential equations for a system of three different particles interacting via central two-body potentials are investigated within the hyperharmonics approach. A simple method for classification and construction of these solutions is proposed. 25 refs
Lectures on the practical solution of differential equations
International Nuclear Information System (INIS)
Dresner, L.
1979-11-01
This report comprises lectures on the practical solution of ordinary and partial differential equations given in the In-Hours Continuing Education Program for Scientific and Technical Personnel at Oak Ridge National Laboratory
Convex solutions of systems arising from Monge-Ampere equations
Directory of Open Access Journals (Sweden)
Haiyan Wang
2009-10-01
Full Text Available We establish two criteria for the existence of convex solutions to a boundary value problem for weakly coupled systems arising from the Monge-Ampère equations. We shall use fixed point theorems in a cone.
Existence of Solutions of Nonlinear Integrodifferential Equations of ...
Indian Academy of Sciences (India)
Abstract. In this paper we prove the existence of mild and strong solutions of a nonlinear integrodifferential equation of Sobolev type with nonlocal condition. The results are obtained by using semigroup theory and the Schauder fixed point theorem.
ALMOST AUTOMORPHIC MILD SOLUTIONS TO SOME FRACTIONAL DELAY DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
In this paper,a new and general existence and uniqueness theorem of almost automorphic mild solutions is obtained for some fractional delay differential equations,using sectorial operators and the Banach contraction principle.
Normal solutions of the Boltzmann equation with small Knudsen number
International Nuclear Information System (INIS)
Ding, E.J.; Huang, Z.Q.
1986-01-01
A singular perturbation method is used to find the normal solutions of the Boltzmann equation with small Knudsen number. It is proved that the secular terms may be removed by improving the Hilbert expansion and the Enskog expansion
New exact travelling wave solutions of bidirectional wave equations
Indian Academy of Sciences (India)
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea. ∗ ... exact travelling wave solutions of system (1) using the modified tanh–coth function method ... The ordinary differential equation is then integrated.
Subroutine for series solutions of linear differential equations
International Nuclear Information System (INIS)
Tasso, H.; Steuerwald, J.
1976-02-01
A subroutine for Taylor series solutions of systems of ordinary linear differential equations is descriebed. It uses the old idea of Lie series but allows simple implementation and is time-saving for symbolic manipulations. (orig.) [de
Exact solutions for a system of nonlinear plasma fluid equations
International Nuclear Information System (INIS)
Prahovic, M.G.; Hazeltine, R.D.; Morrison, P.J.
1991-04-01
A method is presented for constructing exact solutions to a system of nonlinear plasma fluid equations that combines the physics of reduced magnetohydrodynamics and the electrostatic drift-wave description of the Charney-Hasegawa-Mima equation. The system has nonlinearities that take the form of Poisson brackets involving the fluid field variables. The method relies on modifying a class of simple equilibrium solutions, but no approximations are made. A distinguishing feature is that the original nonlinear problem is reduced to the solution of two linear partial differential equations, one fourth-order and the other first-order. The first-order equation has Hamiltonian characteristics and is easily integrated, supplying information about the general structure of solutions. 6 refs
Solitary wave solutions to nonlinear evolution equations in ...
Indian Academy of Sciences (India)
1Computer Engineering Technique Department, Al-Rafidain University College, Baghdad, ... applied to extract solutions are tan–cot method and functional variable approaches. ... Consider the nonlinear partial differential equation in the form.
Symmetry of solutions of differential equations Mythily Ramaswamy ...
Indian Academy of Sciences (India)
2007-11-02
.. • crystals, plants, flowers, insects....... • yet, there are symmetry break ups ! • When is a profile symmetric ? • If a physical phenomenon is modelled by a differential equation, when is the solution symmetric? • Can we ...
EXISTENCE OF POSITIVE SOLUTIONS FOR AN ELASTIC CURVED BEAM EQUATION
Directory of Open Access Journals (Sweden)
Béla Kovacs
2017-06-01
Full Text Available This paper investigates the existence of positive solutions for a sixth-order differential equations. By using the Leggett-Williams fixed point theorem we give some new existence results.
Massively Parallel Algorithms for Solution of Schrodinger Equation
Fijany, Amir; Barhen, Jacob; Toomerian, Nikzad
1994-01-01
In this paper massively parallel algorithms for solution of Schrodinger equation are developed. Our results clearly indicate that the Crank-Nicolson method, in addition to its excellent numerical properties, is also highly suitable for massively parallel computation.
Solution to random differential equations with boundary conditions
Directory of Open Access Journals (Sweden)
Fairouz Tchier
2017-04-01
Full Text Available We study a family of random differential equations with boundary conditions. Using a random fixed point theorem, we prove an existence theorem that yields a unique random solution.
Exponential decay for solutions to semilinear damped wave equation
Gerbi, Sté phane; Said-Houari, Belkacem
2011-01-01
This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Intro- ducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data
Solution of heat equation with variable coefficient using derive
CSIR Research Space (South Africa)
Lebelo, RS
2008-09-01
Full Text Available In this paper, the method of approximating solutions of partial differential equations with variable coefficients is studied. This is done by considering heat flow through a one-dimensional model with variable cross-sections. Two cases...
Global solution branches for a nonlocal Allen-Cahn equation
Kuto, Kousuke; Mori, Tatsuki; Tsujikawa, Tohru; Yotsutani, Shoji
2018-05-01
We consider the Neumann problem of a 1D stationary Allen-Cahn equation with nonlocal term. Our previous paper [4] obtained a local branch of asymmetric solutions which bifurcates from a point on the branch of odd-symmetric solutions. This paper derives the global behavior of the branch of asymmetric solutions, and moreover, determines the set of all solutions to the nonlocal Allen-Cahn equation. Our proof is based on a level set analysis for an integral map associated with the nonlocal term.
Numerical solution of the kinetic equation in reactor shielding
International Nuclear Information System (INIS)
Germogenova, T.A.
1975-01-01
A review is made of methods of solving marginal problems of multi-group systems of equations of neutron and γ radiation transfer. The first stage of the solution - the quantification of the basic task, is determined by the qualitative behaviour of the solution - is the nature of its performance and asymptotics. In the second stage - solution of the approximating system, various modifications of the iterative method are as a rule used. A description is given of the features of the major Soviet complexes of programmes (ROZ and RADUGA) for the solution of multi-group systems of transfer equations and some methodological research findings are presented. (author)
Real-time optical laboratory solution of parabolic differential equations
Casasent, David; Jackson, James
1988-01-01
An optical laboratory matrix-vector processor is used to solve parabolic differential equations (the transient diffusion equation with two space variables and time) by an explicit algorithm. This includes optical matrix-vector nonbase-2 encoded laboratory data, the combination of nonbase-2 and frequency-multiplexed data on such processors, a high-accuracy optical laboratory solution of a partial differential equation, new data partitioning techniques, and a discussion of a multiprocessor optical matrix-vector architecture.
Solutions of the spin coefficient equations with nongeodesic eigenrays
International Nuclear Information System (INIS)
Kota, J.; Lukacs, B.; Perjes, Z.
1982-01-01
Among the many significant results obtained by spin coefficient techniques in general relativity, the exact integrals of gravitational equations have enjoyed particular attention. These integration procedures were first carried out with respect to a congruence of null geodesic curves. The authors show that spin coefficient equations can sometimes be exactly solved when the selected null congruence is nongeodesic. They derive metrics with this property and, among them, a new solution of the coupled Einstein-Maxwell equations. (Auth.)
Iterative solution of linear equations in ODE codes. [Krylov subspaces
Energy Technology Data Exchange (ETDEWEB)
Gear, C. W.; Saad, Y.
1981-01-01
Each integration step of a stiff equation involves the solution of a nonlinear equation, usually by a quasi-Newton method that leads to a set of linear problems. Iterative methods for these linear equations are studied. Of particular interest are methods that do not require an explicit Jacobian, but can work directly with differences of function values using J congruent to f(x + delta) - f(x). Some numerical experiments using a modification of LSODE are reported. 1 figure, 2 tables.
Exact solutions of a nonpolynomially nonlinear Schrodinger equation
International Nuclear Information System (INIS)
Parwani, R.; Tan, H.S.
2007-01-01
A nonlinear generalisation of Schrodinger's equation had previously been obtained using information-theoretic arguments. The nonlinearities in that equation were of a nonpolynomial form, equivalent to the occurrence of higher-derivative nonlinear terms at all orders. Here we construct some exact solutions to that equation in 1+1 dimensions. On the half-line, the solutions resemble (exponentially damped) Bloch waves even though no external periodic potential is included. The solutions are nonperturbative as they do not reduce to solutions of the linear theory in the limit that the nonlinearity parameter vanishes. An intriguing feature of the solutions is their infinite degeneracy: for a given energy, there exists a very large arbitrariness in the normalisable wavefunctions. We also consider solutions to a q-deformed version of the nonlinear equation and discuss a natural discretisation implied by the nonpolynomiality. Finally, we contrast the properties of our solutions with other solutions of nonlinear Schrodinger equations in the literature and suggest some possible applications of our results in the domains of low-energy and high-energy physics
Probabilistic Representations of Solutions to the Heat Equation
Indian Academy of Sciences (India)
In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition , is given by the convolution of with the heat kernel (Gaussian density). Our results also extend the ...
Modulation equations for spatially periodic systems: derivation and solutions
Schielen, R.; Doelman, A.
1996-01-01
We study a class of partial dierential equations in one spatial dimension, which can be seen as model equations for the analysis of pattern formation in physical systems dened on unbounded, weakly oscillating domains. We perform a linear and weakly nonlinear stability analysis for solutions that
Galois action on solutions of a differential equation
Hendriks, Peter A.; Put, Marius van der
Consider a second-order differential equation of the form y '' + ay' + by = O with a,b is an element of Q(x). Kovacic's algorithm tries to compute a solution of the associated Riccati equation that is algebraic and of minimal degree over (Q) over bar(x). The coefficients of the monic irreducible
Existence of solutions of abstract fractional impulsive semilinear evolution equations
Directory of Open Access Journals (Sweden)
K. Balachandran
2010-01-01
Full Text Available In this paper we prove the existence of solutions of fractional impulsive semilinear evolution equations in Banach spaces. A nonlocal Cauchy problem is discussed for the evolution equations. The results are obtained using fractional calculus and fixed point theorems. An example is provided to illustrate the theory.
The Numerical Solution of an Abelian Ordinary Differential Equation ...
African Journals Online (AJOL)
In this paper we present a relatively new technique call theNew Hybrid of Adomian decomposition method (ADM) for solution of an Abelian Differential equation. The numerical results of the equation have been obtained in terms of convergent series with easily computable component. These methods are applied to solve ...
On monotonic solutions of an integral equation of Abel type
International Nuclear Information System (INIS)
Darwish, Mohamed Abdalla
2007-08-01
We present an existence theorem of monotonic solutions for a quadratic integral equation of Abel type in C[0, 1]. The famous Chandrasekhar's integral equation is considered as a special case. The concept of measure of noncompactness and a fi xed point theorem due to Darbo are the main tools in carrying out our proof. (author)
Global weak solution for a equations in plasma
International Nuclear Information System (INIS)
Guo Boling; Huang Daiwen
2010-01-01
The existence of global weak solutions of the initial boundary value problem to a simplified equations, derived from the equations of two fluid system in plasma, is proven by using energy method and some embedding theorems in Sobolev-Orlicz space.
Multiple solutions to some singular nonlinear Schrodinger equations
Directory of Open Access Journals (Sweden)
Monica Lazzo
2001-01-01
Full Text Available We consider the equation $$ - h^2 Delta u + V_varepsilon(x u = |u|^{p-2} u $$ which arises in the study of standing waves of a nonlinear Schrodinger equation. We allow the potential $V_varepsilon$ to be unbounded below and prove existence and multiplicity results for positive solutions.
Asymptotic behaviour of solutions of a degenerate quasilinear hyperbolic equation
International Nuclear Information System (INIS)
Pereira, D.C.
1988-10-01
The decay as t->∞ of the solutions of equation u''(t)|A 1/2 u(t)| 2 Au(t)+Au'(t)=0 where A is a self-adjoint operator in a Hilbert space H with norm |.| is studied. A decay of algebraic rate for the energy associated to the studied equation is obtained. (author) [pt
General solution of Bateman equations for nuclear transmutations
International Nuclear Information System (INIS)
Cetnar, Jerzy
2006-01-01
The paper concerns the linear chain method of solving Bateman equations for nuclear transmutation in derivation of the general solution for linear chain with repeated transitions and thus elimination of existing numerical problems. In addition, applications of derived equations for transmutation trajectory analysis method is presented
Solution of partial differential equations by agent-based simulation
International Nuclear Information System (INIS)
Szilagyi, Miklos N
2014-01-01
The purpose of this short note is to demonstrate that partial differential equations can be quickly solved by agent-based simulation with high accuracy. There is no need for the solution of large systems of algebraic equations. This method is especially useful for quick determination of potential distributions and demonstration purposes in teaching electromagnetism. (letters and comments)
Dressing symmetry of the uniformization solution of Liouville equation
International Nuclear Information System (INIS)
Shen Jianmin.
1994-10-01
In this paper, the relations between monodromy group and dressing group for Liouville equation in uniformization theorem are discussed. The representation of monodromy transformation, acting on the chiral components of the solution of Liouville equation, is obtained. The non-trivial exchange algebra for monodromy transformation is calculated. (author). 14 refs
Entire solutions of nonlinear differential-difference equations.
Li, Cuiping; Lü, Feng; Xu, Junfeng
2016-01-01
In this paper, we describe the properties of entire solutions of a nonlinear differential-difference equation and a Fermat type equation, and improve several previous theorems greatly. In addition, we also deduce a uniqueness result for an entire function f(z) that shares a set with its shift [Formula: see text], which is a generalization of a result of Liu.
Analysis of the Numerical Solution of the Shallow Water Equations
National Research Council Canada - National Science Library
Hamrick, Thomas
1997-01-01
.... The two schemes are finite difference method (FDM) and the finite element method (FEM). After presenting the shallow water equations in several formulations, some examples will be presented. The use of the Fourier transform to find the solution of a semidiscrete analog of the shallow water equations is also demonstrated.
Trigonometric solutions of triangle equations. Simple Lie superalgebras
International Nuclear Information System (INIS)
Bazhanov, V.V.; Shadrikov, A.G.
1988-01-01
Trigonometric solutions of the graded triangle equation are constructed for the fundamental representations of all simple (nonexceptional) Lie superalgebras with nondegenerate metric. In Sec. 1, we introduce the concept of Z 2 graded spaces and give the basic definitions. In Sec. 2, we determine fundamental representations of the Lie superalgebras sl(mn) and osp(2rs) and give explicit realizations of the Coxeter automorphisms. In secs. 3 and 4, we give the trigonometric solutions of the graded triangle equation (quantum R matrices)
Towards the general solution of the Yang-Mills equations
International Nuclear Information System (INIS)
Helfer, A.D.
1985-01-01
The author presents a new non-perturbative technique for finding arbitrary self-dual solutions to the Yang-Mills equations, and of describing massless fields minimally coupled to them. The approach uses techniques of complex analysis in several variables, and is complementary to Ward's: it is expected that a combination of the two techniques will yield general, non-self-dual solutions to the Yang-Mills equations. This has been verified to first order in perturbation theory
Existence of solutions to quasilinear Schrodinger equations with indefinite potential
Directory of Open Access Journals (Sweden)
Zupei Shen
2015-04-01
Full Text Available In this article, we study the existence and multiplicity of solutions of the quasilinear Schrodinger equation $$ -u''+V(xu-(|u| ^2''u=f(u $$ on $\\mathbb{R}$, where the potential $V$ allows sign changing and the nonlinearity satisfies conditions weaker than the classical Ambrosetti-Rabinowitz condition. By a local linking theorem and the fountain theorem, we obtain the existence and multiplicity of solutions for the equation.
On the solution of the nonlinear Schrodinger equation
International Nuclear Information System (INIS)
Zayed, E.M.E.; Zedan, Hassan A.
2003-01-01
In this paper we study the nonlinear Schrodinger equation with respect to the unknown function S(x,t). New dimensional reduction and exact solution for a nonlinear Schrodinger equation are presented and a complete group classification is given with respect to the function S(x,t). Moreover, specializing the potential function S(x,t), new classes of invariant solution and group classification are obtained in the cases of physical interest
Stable solutions of nonlocal electron heat transport equations
International Nuclear Information System (INIS)
Prasad, M.K.; Kershaw, D.S.
1991-01-01
Electron heat transport equations with a nonlocal heat flux are in general ill-posed and intrinsically unstable, as proved by the present authors [Phys. Fluids B 1, 2430 (1989)]. A straightforward numerical solution of these equations will therefore lead to absurd results. It is shown here that by imposing a minimal set of constraints on the problem it is possible to arrive at a globally stable, consistent, and energy conserving numerical solution
New Poisson–Boltzmann type equations: one-dimensional solutions
International Nuclear Information System (INIS)
Lee, Chiun-Chang; Lee, Hijin; Hyon, YunKyong; Lin, Tai-Chia; Liu, Chun
2011-01-01
The Poisson–Boltzmann (PB) equation is conventionally used to model the equilibrium of bulk ionic species in different media and solvents. In this paper we study a new Poisson–Boltzmann type (PB n ) equation with a small dielectric parameter ε 2 and non-local nonlinearity which takes into consideration the preservation of the total amount of each individual ion. This equation can be derived from the original Poisson–Nernst–Planck system. Under Robin-type boundary conditions with various coefficient scales, we demonstrate the asymptotic behaviours of one-dimensional solutions of PB n equations as the parameter ε approaches zero. In particular, we show that in case of electroneutrality, i.e. α = β, solutions of 1D PB n equations have a similar asymptotic behaviour as those of 1D PB equations. However, as α ≠ β (non-electroneutrality), solutions of 1D PB n equations may have blow-up behaviour which cannot be found in 1D PB equations. Such a difference between 1D PB and PB n equations can also be verified by numerical simulations
Particlelike solutions of the Einstein-Dirac equations
Finster, Felix; Smoller, Joel; Yau, Shing-Tung
1999-05-01
The coupled Einstein-Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state are derived. Using numerical methods, we construct an infinite number of solitonlike solutions of these equations. The stability of the solutions is analyzed. For weak coupling (i.e., small rest mass of the fermions), all the solutions are linearly stable (with respect to spherically symmetric perturbations), whereas for stronger coupling, both stable and unstable solutions exist. For the physical interpretation, we discuss how the energy of the fermions and the (ADM) mass behave as functions of the rest mass of the fermions. Although gravitation is not renormalizable, our solutions of the Einstein-Dirac equations are regular and well behaved even for strong coupling.
Exact solution of nonsteady thermal boundary layer equation
International Nuclear Information System (INIS)
Dorfman, A.S.
1995-01-01
There are only a few exact solutions of the thermal boundary layer equation. Most of them are derived for a specific surface temperature distribution. The first exact solution of the steady-state boundary layer equation was given for a plate with constant surface temperature and free-stream velocity. The same problem for a plate with polynomial surface temperature distribution was solved by Chapmen and Rubesin. Levy gave the exact solution for the case of a power law distribution of both surface temperature and free-stream velocity. The exact solution of the steady-state boundary layer equation for an arbitrary surface temperature and a power law free-stream velocity distribution was given by the author in two forms: of series and of the integral with an influence function of unheated zone. A similar solution of the nonsteady thermal boundary layer equation for an arbitrary surface temperature and a power law free-stream velocity distribution is presented here. In this case, the coefficients of series depend on time, and in the limit t → ∞ they become the constant coefficients of a similar solution published before. This solution, unlike the one presented here, does not satisfy the initial conditions at t = 0, and, hence, can be used only in time after the beginning of the process. The solution in the form of a series becomes a closed-form exact solution for polynomial surface temperature and a power law free-stream velocity distribution. 7 refs., 2 figs
Numerical solution of ordinary differential equations
Fox, L
1987-01-01
Nearly 20 years ago we produced a treatise (of about the same length as this book) entitled Computing methods for scientists and engineers. It was stated that most computation is performed by workers whose mathematical training stopped somewhere short of the 'professional' level, and that some books are therefore needed which use quite simple mathematics but which nevertheless communicate the essence of the 'numerical sense' which is exhibited by the real computing experts and which is surely needed, at least to some extent, by all who use modern computers and modern numerical software. In that book we treated, at no great length, a variety of computational problems in which the material on ordinary differential equations occupied about 50 pages. At that time it was quite common to find books on numerical analysis, with a little on each topic ofthat field, whereas today we are more likely to see similarly-sized books on each major topic: for example on numerical linear algebra, numerical approximation, numeri...
Solution of K-V envelope equations
International Nuclear Information System (INIS)
Anderson, O.A.
1995-04-01
The envelope equations for a KV beam with space charge have been analyzed systematically by an e expansion followed by integrations. The focusing profile as a function of axial length is assumed to be symmetric but otherwise arbitrary. Given the bean current, emittance, and peak focusing field, we find the envelopes a(s) and b(s) and obtain , a max , σ, and σ 0 . Explicit results are presented for various truncations of the expansion. The zeroth order results correspond to those from the well-known smooth approximation; the same convenient format is retained for the higher order cases. The first order results, involving single correction terms, give 3--10 times better accuracy and are good to ∼1% at σ 0 = 70 degree. Third order gives a factor of 10--30 improvement over the smooth approximation and derived quantities accurate to ∼1% at σ 0 = 112 degree. The first order expressions are convenient design tools. They lend themselves to variable energy problems and have been applied to the design, construction, and testing of ESQ accelerators at LBL
Solution of Deformed Einstein Equations and Quantum Black Holes
International Nuclear Information System (INIS)
Dil, Emre; Kolay, Erdinç
2016-01-01
Recently, one- and two-parameter deformed Einstein equations have been studied for extremal quantum black holes which have been proposed to obey deformed statistics by Strominger. In this study, we give a deeper insight into the deformed Einstein equations and consider the solutions of these equations for the extremal quantum black holes. We then represent the implications of the solutions, such that the deformation parameters lead the charged black holes to have a smaller mass than the usual Reissner-Nordström black holes. This reduction in mass of a usual black hole can be considered as a transition from classical to quantum black hole regime.
New solutions of the generalized ellipsoidal wave equation
Directory of Open Access Journals (Sweden)
Harold Exton
1999-10-01
Full Text Available Certain aspects and a contribution to the theory of new forms of solutions of an algebraic form of the generalized ellipsoidal wave equation are deduced by considering the Laplace transform of a soluble system of linear differential equations. An ensuing system of non-linear algebraic equations is shown to be consistent and is numerically implemented by means of the computer algebra package MAPLE V. The main results are presented as series of hypergeometric type of there and four variables which readily lend themselves to numerical handling although this does not indicate all of the detailedanalytic properties of the solutions under consideration.
Similarity Solutions for Multiterm Time-Fractional Diffusion Equation
Directory of Open Access Journals (Sweden)
A. Elsaid
2016-01-01
Full Text Available Similarity method is employed to solve multiterm time-fractional diffusion equation. The orders of the fractional derivatives belong to the interval (0,1] and are defined in the Caputo sense. We illustrate how the problem is reduced from a multiterm two-variable fractional partial differential equation to a multiterm ordinary fractional differential equation. Power series solution is obtained for the resulting ordinary problem and the convergence of the series solution is discussed. Based on the obtained results, we propose a definition for a multiterm error function with generalized coefficients.
Automatic computation and solution of generalized harmonic balance equations
Peyton Jones, J. C.; Yaser, K. S. A.; Stevenson, J.
2018-02-01
Generalized methods are presented for generating and solving the harmonic balance equations for a broad class of nonlinear differential or difference equations and for a general set of harmonics chosen by the user. In particular, a new algorithm for automatically generating the Jacobian of the balance equations enables efficient solution of these equations using continuation methods. Efficient numeric validation techniques are also presented, and the combined algorithm is applied to the analysis of dc, fundamental, second and third harmonic response of a nonlinear automotive damper.
Solution of the Burgers Equation in the Time Domain
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M. Bednařík
2002-01-01
Full Text Available This paper deals with a theoretical description of the propagation of a finite amplitude acoustic waves. The theory based on the homogeneous Burgers equation of the second order of accuracy is presented here. This equation takes into account both nonlinear effects and dissipation. The method for solving this equation, using the well-known Cole-Hopf transformation, is presented. Two methods for numerical solution of these equations in the time domain are presented. The first is based on the simple Simpson method, which is suitable for smaller Goldberg numbers. The second uses the more advanced saddle point method, and is appropriate for large Goldberg numbers.
FORSIM-6, Automatic Solution of Coupled Differential Equation System
International Nuclear Information System (INIS)
Carver, M.B.; Stewart, D.G.; Blair, J.M.; Selander, W.N.
1983-01-01
1 - Description of problem or function: The FORSIM program is a versatile package which automates the solution of coupled differential equation systems. The independent variables are time, and up to three space coordinates, and the equations may be any mixture of partial and/or ordinary differential equations. The philosophy of the program is to provide a tool which will solve a system of differential equations for a user who has basic but unspecialized knowledge of numerical analysis and FORTRAN. The equations to be solved, together with the initial conditions and any special instructions, may be specified by the user in a single FORTRAN subroutine, although he may write a number of routines if this is more suitable. These are then loaded with the control routines, which perform the solution and any requested input and output. 2 - Method of solution: Partial differential equations are automatically converted into sets of coupled ordinary differential equations by variable order discretization in the spatial dimensions. These and other ordinary differential equations are integrated continuously in time using efficient variable order, variable step, error-controlled algorithms
New correct solutions of the Dirac equation. 5
International Nuclear Information System (INIS)
Bagrov, V.G.; Byzov, N.N.; Gitman, D.M.; Klimenko, Yu.I.; Meshkov, A.G.; Shapovalov, V.N.; Shakhmatov, V.M.
1975-01-01
Some exact solutions for the Dirac equation, Klein-Gordon equation and classical relativistic equations of motion of an electron in external electromagnetic fields of a special type are considered. When fields E vector and H vector are related by the expression H vector=[n vector E vector]+n vector H 3 , where n vector is a constant unit vector, it turns out that among fields permitting the separation of variables in the Klein-Gordon equation more than half satisfy this relationship. For such fields the solution of the Dirac equation may be simplified considerably. Four specific kinds of fields are examined. The character of electron motion in such fields is peculiar but in the mathematical aspect, part of the problem is reduced to those considered previously
Unbounded solutions of quasi-linear difference equations
Czech Academy of Sciences Publication Activity Database
Cecchi, M.; Došlá, Zuzana; Marini, M.
2003-01-01
Roč. 45, 10-11 (2003), s. 1113-1123 ISSN 0898-1221 Institutional research plan: CEZ:AV0Z1019905 Keywords : nonlinear difference equation * possitive increasing solution * strongly increasing solution Subject RIV: BA - General Mathematics Impact factor: 0.498, year: 2003
Analytical solutions of advection-dispersion equation for varying ...
African Journals Online (AJOL)
Analytical solutions are obtained for a one-dimensional advection–dispersion equation with variable coefficients in a longitudinal domain. Two cases are considered. In the first one the solute dispersion is time dependent along a uniform flow in a semi-infinite domain while in the second case the dispersion and the velocity ...
Steady state solution of the Poisson-Nernst-Planck equations
International Nuclear Information System (INIS)
Golovnev, A.; Trimper, S.
2010-01-01
The exact steady state solution of the Poisson-Nernst-Planck equations (PNP) is given in terms of Jacobi elliptic functions. A more tractable approximate solution is derived which can be used to compare the results with experimental observations in binary electrolytes. The breakdown of the PNP for high concentration and high applied voltage is discussed.
On the Schauder estimates of solutions to parabolic equations
International Nuclear Information System (INIS)
Han Qing
1998-01-01
This paper gives a priori estimates on asymptotic polynomials of solutions to parabolic differential equations at any points. This leads to a pointwise version of Schauder estimates. The result improves the classical Schauder estimates in a way that the estimates of solutions and their derivatives at one point depend on the coefficient and nonhomogeneous terms at this particular point
Perturbation method for periodic solutions of nonlinear jerk equations
International Nuclear Information System (INIS)
Hu, H.
2008-01-01
A Lindstedt-Poincare type perturbation method with bookkeeping parameters is presented for determining accurate analytical approximate periodic solutions of some third-order (jerk) differential equations with cubic nonlinearities. In the process of the solution, higher-order approximate angular frequencies are obtained by Newton's method. A typical example is given to illustrate the effectiveness and simplicity of the proposed method
Crossing symmetric solution of the Chew-Low equation
International Nuclear Information System (INIS)
McLeod, R.J.; Ernst, D.J.
1982-01-01
An N/D dispersion theory is developed which solves crossing symmetric Low equations. The method is used to generate crossing symmetric solutions to the Chew-Low model. We show why the technique originally proposed by Chew and Low was incapable of producing solutions. (orig.)
Invariant Solutions for a Class of Perturbed Nonlinear Wave Equations
Directory of Open Access Journals (Sweden)
Waheed A. Ahmed
2017-11-01
Full Text Available Approximate symmetries of a class of perturbed nonlinear wave equations are computed using two newly-developed methods. Invariant solutions associated with the approximate symmetries are constructed for both methods. Symmetries and solutions are compared through discussing the advantages and disadvantages of each method.
Four-state solution of the Yang-Baxter equation
International Nuclear Information System (INIS)
Kashaev, R.M.; Mangazeev, V.V.
1990-01-01
A new four-state solution of the Yang-Baxter equation is constructed with the help of the lowest dimensional cyclic L-operator related to a 3-state R-matrix. Some special choice of parameters which this solution depends on, leads to the exactly solvable spin model on the chain with Hermitian Hamiltonian. 8 refs
Exponential estimates for solutions of half-linear differential equations
Czech Academy of Sciences Publication Activity Database
Řehák, Pavel
2015-01-01
Roč. 147, č. 1 (2015), s. 158-171 ISSN 0236-5294 Institutional support: RVO:67985840 Keywords : half-linear differential equation * decreasing solution * increasing solution * asymptotic behavior Subject RIV: BA - General Mathematics Impact factor: 0.469, year: 2015 http://link.springer.com/article/10.1007%2Fs10474-015-0522-9
Exponential decay for solutions to semilinear damped wave equation
Gerbi, Stéphane
2011-10-01
This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Intro- ducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in [4].
A Gradient Based Iterative Solutions for Sylvester Tensor Equations
Directory of Open Access Journals (Sweden)
Zhen Chen
2013-01-01
proposed by Ding and Chen, 2005, and by using tensor arithmetic concepts, an iterative algorithm and its modification are established to solve the Sylvester tensor equation. Convergence analysis indicates that the iterative solutions always converge to the exact solution for arbitrary initial value. Finally, some examples are provided to show that the proposed algorithms are effective.
On periodic solutions to second-order Duffing type equations
Czech Academy of Sciences Publication Activity Database
Lomtatidze, Alexander; Šremr, Jiří
2018-01-01
Roč. 40, April (2018), s. 215-242 ISSN 1468-1218 Institutional support: RVO:67985840 Keywords : periodic solution * Duffing type equation * positive solution Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 1.659, year: 2016 http://www. science direct.com/ science /article/pii/S1468121817301335?via%3Dihub
Travelling wave solutions to the perturbed Π4 equation
International Nuclear Information System (INIS)
Geicke, J.
1985-01-01
Exact travelling wave solutions to the Π 4 equation, perturbed by a dissipative force and a constant external field η, are presented. For |η| 3 -λ 2 and λ 2 -λ 1 where λ 1 2 3 are the real roots of λ 3 -λ+η=O. The class with |v/ 3 -λ 1 . The stability of the solutions is discussed. (author) [pt
Exact and numerical solutions of generalized Drinfeld-Sokolov equations
Energy Technology Data Exchange (ETDEWEB)
Ugurlu, Yavuz [Firat University, Department of Mathematics, 23119 Elazig (Turkey); Kaya, Dogan [Firat University, Department of Mathematics, 23119 Elazig (Turkey)], E-mail: dkaya36@yahoo.com
2008-04-14
In this Letter, we consider a system of generalized Drinfeld-Sokolov (gDS) equations which models one-dimensional nonlinear wave processes in two-component media. We find some exact solutions of gDS by using tanh function method and we also obtain a numerical solution by using the Adomian's Decomposition Method (ADM)
Exact and numerical solutions of generalized Drinfeld-Sokolov equations
International Nuclear Information System (INIS)
Ugurlu, Yavuz; Kaya, Dogan
2008-01-01
In this Letter, we consider a system of generalized Drinfeld-Sokolov (gDS) equations which models one-dimensional nonlinear wave processes in two-component media. We find some exact solutions of gDS by using tanh function method and we also obtain a numerical solution by using the Adomian's Decomposition Method (ADM)
Exact solutions of nonlinear differential equations using continued fractions
International Nuclear Information System (INIS)
Ditto, W.L.; Pickett, T.J.
1990-01-01
The continued-fraction conversion method (J. Math. Phys. (N.Y.), 29, 1761 (1988)) is used to generate a homologous family of exact solutions to the Lane-Emden equation φ(r) '' + 2φ(r)'/r + αφ(r) p = 0, for p=5. An exact solution is also obtained for a generalization of the Lane-Emden equation of the form -φ '' (r) -2φ(r)'/r + αφ(r) 2p+1 + λφ(r) 4p+1 = 0 for arbitrary α, γ and p. A condition is established for the generation of exact solutions from the method
International Nuclear Information System (INIS)
Marczynski, Slawomir
2011-01-01
The integro-differential Berk-Breizman (BB) equation, describing the evolution of particle-driven wave mode is transformed into a simple delayed differential equation form ν∂a(τ)/∂τ=a(τ) -a 2 (τ- 1) a(τ- 2). This transformation is also applied to the two modes extension of the BB theory. The obtained solutions are presented together with the derived asymptotic analytical solutions and the numerical results.
International Nuclear Information System (INIS)
Chen Huaitang; Zhang Hongqing
2004-01-01
A generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Riccati equation which has more new solutions. More new multiple soliton solutions are obtained for the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation
A class of solutions for the strong gravity equations
International Nuclear Information System (INIS)
Salam, A.; Strathdee, J.
1976-12-01
We solve the Einstein equation for strong gravity in the limit that weak gravity is neglected. The class of solutions we find reduces to the Schwarzschild solution (with the weak gravity Newtonian constant replaced by a strong coupling parameter) in the limit M 2 →0 where M is the mass of the strong gravity spin-2 meson. These solutions may be of relevance for the problem of defining temperature in hadronic physics
International Nuclear Information System (INIS)
Zhang Huiqun
2009-01-01
By using a new coupled Riccati equations, a direct algebraic method, which was applied to obtain exact travelling wave solutions of some complex nonlinear equations, is improved. And the exact travelling wave solutions of the complex KdV equation, Boussinesq equation and Klein-Gordon equation are investigated using the improved method. The method presented in this paper can also be applied to construct exact travelling wave solutions for other nonlinear complex equations.
Time-periodic solutions of the Benjamin-Ono equation
Energy Technology Data Exchange (ETDEWEB)
Ambrose , D.M.; Wilkening, Jon
2008-04-01
We present a spectrally accurate numerical method for finding non-trivial time-periodic solutions of non-linear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which, in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase and the real part of one of the Fourier modes at t = 0. We use our method to study global paths of non-trivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached (or until the solution blows up). By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODE's governing the evolution of solitons using the ansatz suggested by the numerical simulations.
Time-periodic solutions of the Benjamin-Ono equation
International Nuclear Information System (INIS)
Ambrose, D.M.; Wilkening, Jon
2008-01-01
We present a spectrally accurate numerical method for finding non-trivial time-periodic solutions of non-linear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which, in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase and the real part of one of the Fourier modes at t = 0. We use our method to study global paths of non-trivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached (or until the solution blows up). By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODE's governing the evolution of solitons using the ansatz suggested by the numerical simulations
Poisson-Fermi Formulation of Nonlocal Electrostatics in Electrolyte Solutions
Directory of Open Access Journals (Sweden)
Liu Jinn-Liang
2017-10-01
Full Text Available We present a nonlocal electrostatic formulation of nonuniform ions and water molecules with interstitial voids that uses a Fermi-like distribution to account for steric and correlation efects in electrolyte solutions. The formulation is based on the volume exclusion of hard spheres leading to a steric potential and Maxwell’s displacement field with Yukawa-type interactions resulting in a nonlocal electric potential. The classical Poisson-Boltzmann model fails to describe steric and correlation effects important in a variety of chemical and biological systems, especially in high field or large concentration conditions found in and near binding sites, ion channels, and electrodes. Steric effects and correlations are apparent when we compare nonlocal Poisson-Fermi results to Poisson-Boltzmann calculations in electric double layer and to experimental measurements on the selectivity of potassium channels for K+ over Na+.
Shallow water equations: viscous solutions and inviscid limit
Chen, Gui-Qiang; Perepelitsa, Mikhail
2012-12-01
We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C 2 test-functions, are confined in a compact set in H -1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.
Efficient traveltime solutions of the acoustic TI eikonal equation
Waheed, Umair bin
2015-02-01
Numerical solutions of the eikonal (Hamilton-Jacobi) equation for transversely isotropic (TI) media are essential for imaging and traveltime tomography applications. Such solutions, however, suffer from the inherent higher-order nonlinearity of the TI eikonal equation, which requires solving a quartic polynomial for every grid point. Analytical solutions of the quartic polynomial yield numerically unstable formulations. Thus, it requires a numerical root finding algorithm, adding significantly to the computational load. Using perturbation theory we approximate, in a first order discretized form, the TI eikonal equation with a series of simpler equations for the coefficients of a polynomial expansion of the eikonal solution, in terms of the anellipticity anisotropy parameter. Such perturbation, applied to the discretized form of the eikonal equation, does not impose any restrictions on the complexity of the perturbed parameter field. Therefore, it provides accurate traveltime solutions even for models with complex distribution of velocity and anisotropic anellipticity parameter, such as that for the complicated Marmousi model. The formulation allows for large cost reduction compared to using the direct TI eikonal solver. Furthermore, comparative tests with previously developed approximations illustrate remarkable gain in accuracy in the proposed algorithm, without any addition to the computational cost.
International Nuclear Information System (INIS)
Zhang Liang; Zhang Lifeng; Li Chongyin
2008-01-01
By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions
Approximation of entropy solutions to degenerate nonlinear parabolic equations
Abreu, Eduardo; Colombeau, Mathilde; Panov, Evgeny Yu
2017-12-01
We approximate the unique entropy solutions to general multidimensional degenerate parabolic equations with BV continuous flux and continuous nondecreasing diffusion function (including scalar conservation laws with BV continuous flux) in the periodic case. The approximation procedure reduces, by means of specific formulas, a system of PDEs to a family of systems of the same number of ODEs in the Banach space L^∞, whose solutions constitute a weak asymptotic solution of the original system of PDEs. We establish well posedness, monotonicity and L^1-stability. We prove that the sequence of approximate solutions is strongly L^1-precompact and that it converges to an entropy solution of the original equation in the sense of Carrillo. This result contributes to justify the use of this original method for the Cauchy problem to standard multidimensional systems of fluid dynamics for which a uniqueness result is lacking.
The numerical solution of linear multi-term fractional differential equations: systems of equations
Edwards, John T.; Ford, Neville J.; Simpson, A. Charles
2002-11-01
In this paper, we show how the numerical approximation of the solution of a linear multi-term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity. We begin by showing how our method applies to a simple class of problems and we give a convergence result. We solve the Bagley Torvik equation as an example. We show how the method can be applied to a general linear multi-term equation and give two further examples.
Exact multi-line soliton solutions of noncommutative KP equation
International Nuclear Information System (INIS)
Wang, Ning; Wadati, Miki
2003-01-01
A method of solving noncommutative linear algebraic equations plays a key role in the extension of the ∂-bar -dressing on the noncommutative space-time manifold. In this paper, a solution-generating method of noncommutative linear algebraic equations is proposed. By use of the proposed method, a class of multi-line soliton solutions of noncommutative KP (ncKP) equation is constructed explicitly. The method is expected to be of use for constructions of noncommutative soliton equations. The significance of the noncommutativity of coordinates is investigated. It is found that the noncommutativity of the space-time coordinate has a role to split the spatial waveform of the classical multi-line solitons and reform it to a new configuration. (author)
Asymptotic Behavior of Solutions of Delayed Difference Equations
Directory of Open Access Journals (Sweden)
J. Diblík
2011-01-01
Full Text Available This contribution is devoted to the investigation of the asymptotic behavior of delayed difference equations with an integer delay. We prove that under appropriate conditions there exists at least one solution with its graph staying in a prescribed domain. This is achieved by the application of a more general theorem which deals with systems of first-order difference equations. In the proof of this theorem we show that a good way is to connect two techniques—the so-called retract-type technique and Liapunov-type approach. In the end, we study a special class of delayed discrete equations and we show that there exists a positive and vanishing solution of such equations.
Solution methods for large systems of linear equations in BACCHUS
International Nuclear Information System (INIS)
Homann, C.; Dorr, B.
1993-05-01
The computer programme BACCHUS is used to describe steady state and transient thermal-hydraulic behaviour of a coolant in a fuel element with intact geometry in a fast breeder reactor. In such computer programmes generally large systems of linear equations with sparse matrices of coefficients, resulting from discretization of coolant conservation equations, must be solved thousands of times giving rise to large demands of main storage and CPU time. Direct and iterative solution methods of the systems of linear equations, available in BACCHUS, are described, giving theoretical details and experience with their use in the programme. Besides use of a method of lines, a Runge-Kutta-method, for solution of the partial differential equation is outlined. (orig.) [de
Small data global solutions for the Camassa–Choi equations
Harrop-Griffiths, Benjamin; Marzuola, Jeremy L.
2018-05-01
We consider solutions to the Cauchy problem for an internal-wave model derived by Camassa–Choi (1996 J. Fluid Mech. 313 83–103). This model is a natural generalization of the Benjamin–Ono and intermediate long wave equations for weak transverse effects as in the case of the Kadomtsev–Petviashvili equations for the Korteweg-de Vries equation. For that reason they are often referred to as the KP-ILW or the KP–Benjamin–Ono equations regarding finite or infinite depth respectively. We prove the existence and long-time dynamics of global solutions from small, smooth, spatially localized initial data on . The techniques applied here involve testing by wave packet techniques developed by Ifrim and Tataru in (2015 Nonlinearity 28 2661–75 2016 Bull. Soc. Math. France 144 369–94).
Approximate Solution of LR Fuzzy Sylvester Matrix Equations
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Xiaobin Guo
2013-01-01
Full Text Available The fuzzy Sylvester matrix equation AX~+X~B=C~ in which A,B are m×m and n×n crisp matrices, respectively, and C~ is an m×n LR fuzzy numbers matrix is investigated. Based on the Kronecker product of matrices, we convert the fuzzy Sylvester matrix equation into an LR fuzzy linear system. Then we extend the fuzzy linear system into two systems of linear equations according to the arithmetic operations of LR fuzzy numbers. The fuzzy approximate solution of the original fuzzy matrix equation is obtained by solving the crisp linear systems. The existence condition of the LR fuzzy solution is also discussed. Some examples are given to illustrate the proposed method.
Soliton solutions of coupled nonlinear Klein-Gordon equations
International Nuclear Information System (INIS)
Alagesan, T.; Chung, Y.; Nakkeeran, K.
2004-01-01
The coupled nonlinear Klein-Gordon equations are analyzed for their integrability properties in a systematic manner through Painleve test. From the Painleve test, by truncating the Laurent series at the constant level term, the Hirota bilinear form is identified, from which one-soliton solutions are derived. Then, the results are generalized to the two, three and N-coupled Klein-Gordon equations
Almost Periodic Solutions for Impulsive Fractional Stochastic Evolution Equations
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Toufik Guendouzi
2014-08-01
Full Text Available In this paper, we consider the existence of square-mean piecewise almost periodic solutions for impulsive fractional stochastic evolution equations involving Caputo fractional derivative. The main results are obtained by means of the theory of operators semi-group, fractional calculus, fixed point technique and stochastic analysis theory and methods adopted directly from deterministic fractional equations. Some known results are improved and generalized.
Solution of the Bagley Torvik equation by fractional DTM
Arora, Geeta; Pratiksha
2017-07-01
In this paper, fractional differential transform method(DTM) is implemented on the Bagley Torvik equation. This equation models the viscoelastic behavior of geological strata, metals, glasses etc. It explains the motion of a rigid plate immersed in a Newtonian fluid. DTM is a simple, reliable and efficient method that gives a series solution. Caputo fractional derivative is considered throughout this work. Two examples are given to demonstrate the validity and applicability of the method and comparison is made with the existing results.
Analytical Solution of General Bagley-Torvik Equation
William Labecca; Osvaldo Guimarães; José Roberto C. Piqueira
2015-01-01
Bagley-Torvik equation appears in viscoelasticity problems where fractional derivatives seem to play an important role concerning empirical data. There are several works treating this equation by using numerical methods and analytic formulations. However, the analytical solutions presented in the literature consider particular cases of boundary and initial conditions, with inhomogeneous term often expressed in polynomial form. Here, by using Laplace transform methodology, the general inhomoge...
Analytical solution to the hybrid diffusion-transport equation
International Nuclear Information System (INIS)
Nanneh, M.M.; Williams, M.M.R.
1986-01-01
A special integral equation was derived in previous work using a hybrid diffusion-transport theory method for calculating the flux distribution in slab lattices. In this paper an analytical solution of this equation has been carried out on a finite reactor lattice. The analytical results of disadvantage factors are shown to be accurate in comparison with the numerical results and accurate transport theory calculations. (author)
Solution of the Schroedinger equation by a spectral method
International Nuclear Information System (INIS)
Feit, M.D.; Fleck, J.A. Jr.; Steiger, A.
1982-01-01
A new computational method for determining the eigenvalues and eigenfunctions of the Schroedinger equation is described. Conventional methods for solving this problem rely on diagonalization of a Hamiltonian matrix or iterative numerical solutions of a time independent wave equation. The new method, in contrast, is based on the spectral properties of solutions to the time-dependent Schroedinger equation. The method requires the computation of a correlation function from a numerical solution psi(r, t). Fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system. Analysis of the location of these peaks reveals the eigenvalues with high accuracy. Additional Fourier transforms of psi(r, t) with respect to time generate the eigenfunctions. The effectiveness of the method is demonstrated for a one-dimensional asymmetric double well potential and for the two-dimensional Henon--Heiles potential
Numerical solution of a reaction-diffusion equation
International Nuclear Information System (INIS)
Moyano, Edgardo A.; Scarpettini, Alberto F.
2000-01-01
The purpose of the present work to continue the observations and the numerical experiences on a reaction-diffusion model, that is a simplified form of the neutronic flux equation. The model is parabolic, nonlinear, with Dirichlet boundary conditions. The purpose is to approximate non trivial solutions, asymptotically stables for t → ∞, that is solutions that tend to the elliptic problem, in the Lyapunov sense. It belongs to the so-called reaction-diffusion equations of semi linear kind, that is, linear equations in the heat operator and they have a nonlinear reaction function, in this case f (u, a, b) = u (a - b u), being u concentration, a and b parameters. The study of the incidence of these parameters take an interest to the neutronic flux physics. So that we search non trivial, positive and bounded solutions. The used algorithm is based on the concept of monotone and ordered sequences, and on the existence theorem of Amann and Sattinger. (author)
Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations
International Nuclear Information System (INIS)
Indekeu, Joseph O; Smets, Ruben
2017-01-01
Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically. (paper)
Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations
Indekeu, Joseph O.; Smets, Ruben
2017-08-01
Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically.
Error estimates in projective solutions of the radon equation
International Nuclear Information System (INIS)
Lubuma, M.S.
1991-04-01
The model Radon equation is the integral equation of the second kind defined by the interior limits of the electrostatic double layer potential relative to a curve with one angular point and characterized by the non compactness of the operator with respect to the maximum norm. It is shown that the solution to this equation is decomposable into a regular part and a finite linear combination of intrinsic singular functions. The maximal regularity of the solution and explicit formulae for the coefficients of the singular functions are given. The regularity permits to specify how slow the convergence of the classical projection method is, while the above mentioned formulae lead to modified projection methods of the Dual Singular Function Method type, with better approximations for the solution and for the coefficients of singularities. (author). 23 refs
On a new solution of Chew-Low type equations
International Nuclear Information System (INIS)
Rerikh, K.V.
1985-01-01
The system is investigated of Chew-Low type equations, defined by the crossing symmetry matrix A(1, 1) for S-matrix elements as dfunctions of the uniformmizing variable w, in terms of which this system is a system of nonlinear difference equations. The quadratic Cremona transformation for unknown functions reducing the initial equations to a vey simple form is found. New particular solutions are obtained as functions of variable exp(cw). Existence of the new first integral γ(w) that is an even periodical function of w is estalished. The structure of the general solution depending on γ(w) and the relation of the found particular solutions with the first integral are discussed
Global Solutions to the Coupled Chemotaxis-Fluid Equations
Duan, Renjun
2010-08-10
In this paper, we are concerned with a model arising from biology, which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations through transport and external forcing. The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the Chemotaxis-Navier-Stokes system over three space dimensions, we obtain global existence and rates of convergence on classical solutions near constant states. When the fluid motion is described by the simpler Stokes equations, we prove global existence of weak solutions in two space dimensions for cell density with finite mass, first-order spatial moment and entropy provided that the external forcing is weak or the substrate concentration is small. © Taylor & Francis Group, LLC.
International Nuclear Information System (INIS)
Zhang Huiqun
2009-01-01
By using some exact solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact complex solutions for nonlinear partial differential equations. The method is implemented for the NLS equation, a new Hamiltonian amplitude equation, the coupled Schrodinger-KdV equations and the Hirota-Maccari equations. New exact complex solutions are obtained.
Solutions of Navier-Stokes Equation with Coriolis Force
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Sunggeun Lee
2017-01-01
Full Text Available We investigate the Navier-Stokes equation in the presence of Coriolis force in this article. First, the vortex equation with the Coriolis effect is discussed. It turns out that the vorticity can be generated due to a rotation coming from the Coriolis effect, Ω. In both steady state and two-dimensional flow, the vorticity vector ω gets shifted by the amount of -2Ω. Second, we consider the specific expression of the velocity vector of the Navier-Stokes equation in two dimensions. For the two-dimensional potential flow v→=∇→ϕ, the equation satisfied by ϕ is independent of Ω. The remaining Navier-Stokes equation reduces to the nonlinear partial differential equations with respect to the velocity and the corresponding exact solution is obtained. Finally, the steady convective diffusion equation is considered for the concentration c and can be solved with the help of Navier-Stokes equation for two-dimensional potential flow. The convective diffusion equation can be solved in three dimensions with a simple choice of c.
On Approximate Solutions of Functional Equations in Vector Lattices
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Bogdan Batko
2014-01-01
Full Text Available We provide a method of approximation of approximate solutions of functional equations in the class of functions acting into a Riesz space (algebra. The main aim of the paper is to provide a general theorem that can act as a tool applicable to a possibly wide class of functional equations. The idea is based on the use of the Spectral Representation Theory for Riesz spaces. The main result will be applied to prove the stability of an alternative Cauchy functional equation F(x+y+F(x+F(y≠0⇒F(x+y=F(x+F(y in Riesz spaces, the Cauchy equation with squares F(x+y2=(F(x+F(y2 in f-algebras, and the quadratic functional equation F(x+y+F(x-y=2F(x+2F(y in Riesz spaces.
Comparative analysis of solution methods of the punctual kinetic equations
International Nuclear Information System (INIS)
Hernandez S, A.
2003-01-01
The following one written it presents a comparative analysis among different analytical solutions for the punctual kinetics equation, which present two variables of interest: a) the temporary behavior of the neutronic population, and b) The temporary behavior of the different groups of precursors of delayed neutrons. The first solution is based on a method that solves the transfer function of the differential equation for the neutronic population, in which intends to obtain the different poles that give the stability of this transfer function. In this section it is demonstrated that the temporary variation of the reactivity of the system can be managed as it is required, since the integration time for this method doesn't affect the result. However, the second solution is based on an iterative method like that of Runge-Kutta or the Euler method where the algorithm was only used to solve first order differential equations giving this way solution to each differential equation that conforms the equations of punctual kinetics. In this section it is demonstrated that only it can obtain a correct temporary behavior of the neutronic population when it is integrated on an interval of very short time, forcing to the temporary variation of the reactivity to change very quick way without one has some control about the time. In both methods the same change is used so much in the reactivity of the system like in the integration times, giving validity to the results graph the one the temporary behavior of the neutronic population vs. time. (Author)
Efficient Traveltime Solutions of the TI Acoustic Eikonal Equation
Waheed, Umair bin
2014-10-22
Numerical solutions of the eikonal (Hamilton-Jacobi) equation for transversely isotropic (TI) media are essential for integral imaging and traveltime tomography applications. Such solutions, however, suffer from the inherent higher-order nonlinearity of the TI eikonal equation, which requires solving a quartic polynomial at each computational step. Using perturbation theory, we approximate the first-order discretized form of the TI eikonal equation with a series of simpler equations for the coefficients of a polynomial expansion of the eikonal solution in terms of the anellipticity anisotropy parameter. Such perturbation, applied to the discretized form of the eikonal equation, does not impose any restrictions on the complexity of the perturbed parameter field. Therefore, it provides accurate traveltime solutions even for the anisotropic Marmousi model, with complex distribution of velocity and anellipticity anisotropy parameter. The formulation allows tremendous cost reduction compared to using the exact TI eikonal solver. Furthermore, comparative tests with previously developed approximations illustrate remarkable gain in accuracy of the proposed approximation, without any addition to the computational cost.
Efficient Traveltime Solutions of the TI Acoustic Eikonal Equation
Waheed, Umair bin; Alkhalifah, Tariq Ali
2014-01-01
Numerical solutions of the eikonal (Hamilton-Jacobi) equation for transversely isotropic (TI) media are essential for integral imaging and traveltime tomography applications. Such solutions, however, suffer from the inherent higher-order nonlinearity of the TI eikonal equation, which requires solving a quartic polynomial at each computational step. Using perturbation theory, we approximate the first-order discretized form of the TI eikonal equation with a series of simpler equations for the coefficients of a polynomial expansion of the eikonal solution in terms of the anellipticity anisotropy parameter. Such perturbation, applied to the discretized form of the eikonal equation, does not impose any restrictions on the complexity of the perturbed parameter field. Therefore, it provides accurate traveltime solutions even for the anisotropic Marmousi model, with complex distribution of velocity and anellipticity anisotropy parameter. The formulation allows tremendous cost reduction compared to using the exact TI eikonal solver. Furthermore, comparative tests with previously developed approximations illustrate remarkable gain in accuracy of the proposed approximation, without any addition to the computational cost.
Analytical solutions for systems of partial differential-algebraic equations.
Benhammouda, Brahim; Vazquez-Leal, Hector
2014-01-01
This work presents the application of the power series method (PSM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-one and index-three are solved to show that PSM can provide analytical solutions of PDAEs in convergent series form. What is more, we present the post-treatment of the power series solutions with the Laplace-Padé (LP) resummation method as a useful strategy to find exact solutions. The main advantage of the proposed methodology is that the procedure is based on a few straightforward steps and it does not generate secular terms or depends of a perturbation parameter.
Semi-analytic solution to planar Helmholtz equation
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Tukač M.
2013-06-01
Full Text Available Acoustic solution of interior domains is of great interest. Solving acoustic pressure fields faster with lower computational requirements is demanded. A novel solution technique based on the analytic solution to the Helmholtz equation in rectangular domain is presented. This semi-analytic solution is compared with the finite element method, which is taken as the reference. Results show that presented method is as precise as the finite element method. As the semi-analytic method doesn’t require spatial discretization, it can be used for small and very large acoustic problems with the same computational costs.
Einstein equation solutions with axial symmetry, conical and essential singularities
International Nuclear Information System (INIS)
Oliveira, S.R. de.
1986-01-01
New classes of exact solutions to the Einstein equations of a static axisymetric space-time associated with rings and disks are found. Also, the solutions associated to a axisymetric superposition of punctual bodies, bars, rings and disks are obtained. These solutions have a strut singularities to keep the bodies apart. When one of the bodies of the superposition is a ring, the ring interior is covered with a membrane that serve as a support for the strut that hold the other body. Furthermore, the curvature singularities for different solutions ae analised. (author) [pt
An Efficient Series Solution for Nonlinear Multiterm Fractional Differential Equations
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Moh’d Khier Al-Srihin
2017-01-01
Full Text Available In this paper, we introduce an efficient series solution for a class of nonlinear multiterm fractional differential equations of Caputo type. The approach is a generalization to our recent work for single fractional differential equations. We extend the idea of the Taylor series expansion method to multiterm fractional differential equations, where we overcome the difficulty of computing iterated fractional derivatives, which are difficult to be computed in general. The terms of the series are obtained sequentially using a closed formula, where only integer derivatives have to be computed. Several examples are presented to illustrate the efficiency of the new approach and comparison with the Adomian decomposition method is performed.
Solution of wave-like equation based on Haar wavelet
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Naresh Berwal
2012-11-01
Full Text Available Wavelet transform and wavelet analysis are powerful mathematical tools for many problems. Wavelet also can be applied in numerical analysis. In this paper, we apply Haar wavelet method to solve wave-like equation with initial and boundary conditions known. The fundamental idea of Haar wavelet method is to convert the differential equations into a group of algebraic equations, which involves a finite number or variables. The results and graph show that the proposed way is quite reasonable when compared to exact solution.
Family of electrovac colliding wave solutions of Einstein's equations
International Nuclear Information System (INIS)
Li, W.; Ernst, F.J.
1989-01-01
Beginning with any colliding wave solution of the vacuum Einstein equations, a corresponding electrified colliding wave solution can be generated through the use of a transformation due to Harrison [J. Math. Phys. 9, 1744 (1968)]. The method, long employed in the context of stationary axisymmetric fields, is equally applicable to colliding wave solutions. Here it is applied to a large family of vacuum metrics derived by applying a generalized Ehlers transformation to solutions published recently by Ernst, Garcia, and Hauser (EGH) [J. Math. Phys. 28, 2155, 2951 (1987); 29, 681 (1988)]. Those EGH solutions were themselves a generalization of solutions first derived by Ferrari, Ibanez, and Bruni [Phys. Rev. D 36, 1053 (1987)]. Among the electrovac solutions that are obtained is a charged version of the Nutku--Halil [Phys. Rev. Lett. 39, 1379 (1977)] metric that possesses an arbitrary complex charge parameter
Iterative Solutions of Nonlinear Integral Equations of Hammerstein Type
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Abebe R. Tufa
2015-11-01
Full Text Available Let H be a real Hilbert space. Let F,K : H → H be Lipschitz monotone mappings with Lipschtiz constants L1and L2, respectively. Suppose that the Hammerstein type equation u + KFu = 0 has a solution in H. It is our purpose in this paper to construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized Hammerstein type equation. The results obtained in this paper improve and extend known results in the literature.
Analytical Solution of General Bagley-Torvik Equation
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William Labecca
2015-01-01
Full Text Available Bagley-Torvik equation appears in viscoelasticity problems where fractional derivatives seem to play an important role concerning empirical data. There are several works treating this equation by using numerical methods and analytic formulations. However, the analytical solutions presented in the literature consider particular cases of boundary and initial conditions, with inhomogeneous term often expressed in polynomial form. Here, by using Laplace transform methodology, the general inhomogeneous case is solved without restrictions in boundary and initial conditions. The generalized Mittag-Leffler functions with three parameters are used and the solutions presented are expressed in terms of Wiman’s functions and their derivatives.
Covariant solutions of the Bethe-Salpeter equation
International Nuclear Information System (INIS)
Williams, A.G.; Kusaka, K.; Simpson, K.M.
1997-01-01
There is a need for covariant solutions of bound state equations in order to construct realistic QCD based models of mesons and baryons. Furthermore, we ideally need to know the structure of these bound states in all kinematical regimes, which makes a direct solution in Minkowski space (without any 3-dimensional reductions) desirable. The Bethe-Salpeter equation (BSE) for bound states in scalar theories is reformulated and solved for arbitrary scattering kernels in terms of a generalized spectral representation directly in Minkowski space. This differs from the conventional Euclidean approach, where the BSE can only be solved in ladder approximation after a Wick rotation. (author)
A New Solution for Einstein Field Equation in General Relativity
Mousavi, Sadegh
2006-05-01
There are different solutions for Einstein field equation in general relativity that they have been proposed by different people the most important solutions are Schwarzchild, Reissner Nordstrom, Kerr and Kerr Newmam. However, each one of these solutions limited to special case. I've found a new solution for Einstein field equation which is more complete than all previous ones and this solution contains the previous solutions as its special forms. In this talk I will present my new metric for Einstein field equation and the Christofel symbols and Richi and Rieman tensor components for the new metric that I have calculated them by GR TENSOR software. As a result I will determine the actual movement of black holes which is different From Kerr black hole's movement. Finally this new solution predicts, existence of a new and constant field in the nature (that nobody can found it up to now), so in this talk I will introduce this new field and even I will calculate the amount of this field. SADEGH MOUSAVI, Amirkabir University of Technology.
Perturbational blowup solutions to the compressible Euler equations with damping.
Cheung, Ka Luen
2016-01-01
The N-dimensional isentropic compressible Euler system with a damping term is one of the most fundamental equations in fluid dynamics. Since it does not have a general solution in a closed form for arbitrary well-posed initial value problems. Constructing exact solutions to the system is a useful way to obtain important information on the properties of its solutions. In this article, we construct two families of exact solutions for the one-dimensional isentropic compressible Euler equations with damping by the perturbational method. The two families of exact solutions found include the cases [Formula: see text] and [Formula: see text], where [Formula: see text] is the adiabatic constant. With analysis of the key ordinary differential equation, we show that the classes of solutions include both blowup type and global existence type when the parameters are suitably chosen. Moreover, in the blowup cases, we show that the singularities are of essential type in the sense that they cannot be smoothed by redefining values at the odd points. The two families of exact solutions obtained in this paper can be useful to study of related numerical methods and algorithms such as the finite difference method, the finite element method and the finite volume method that are applied by scientists to simulate the fluids for applications.
Closed form solutions of two time fractional nonlinear wave equations
Akbar, M. Ali; Ali, Norhashidah Hj. Mohd.; Roy, Ripan
2018-06-01
In this article, we investigate the exact traveling wave solutions of two nonlinear time fractional wave equations. The fractional derivatives are described in the sense of conformable fractional derivatives. In addition, the traveling wave solutions are accomplished in the form of hyperbolic, trigonometric, and rational functions involving free parameters. To investigate such types of solutions, we implement the new generalized (G‧ / G) -expansion method. The extracted solutions are reliable, useful and suitable to comprehend the optimal control problems, chaotic vibrations, global and local bifurcations and resonances, furthermore, fission and fusion phenomena occur in solitons, the relativistic energy-momentum relation, scalar electrodynamics, quantum relativistic one-particle theory, electromagnetic interactions etc. The results reveal that the method is very fruitful and convenient for exploring nonlinear differential equations of fractional order treated in theoretical physics.
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Oyelami, Benjamin Oyediran
2013-09-01
Full Text Available In this paper, criteria for the existence of weak solutions and uniformly weak bounded solution of impulsive heat equation containing maximum temperature are investigated and results obtained. An example is given for heat flow system with impulsive temperature using maximum temperature simulator and criteria for the uniformly weak bounded of solutions of the system are obtained.
Explicit solutions of Fisher's equation with three zeros
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M. F. K. Abur-Robb
1990-01-01
Full Text Available Explicit traveling wave solutions of Fisher's equation with three simple zeros ut=uxx+u(1−u(u−a, a∈(0,1, are obtained for the wave speeds C=±2(12−a suggested by pure analytic considerations. Two types of solutions are obtained: one type is of a permanent wave form whereas the other is not.
Nonlinear MHD-equations: symmetries, solutions and conservation laws
International Nuclear Information System (INIS)
Samokhin, A.V.
1985-01-01
To investigate stability and nonlinear effects in a high-temperature plasma the system of two scalar nonlinear equations is considered. The algebra of classical symmetries of this system and a certain natural part of its conservation laws are described. It is shown that first, with symmetries one can derive invariant (self-similar) solutions, second, acting with symmetry on the known solution the latter can be included into parametric family
New numerical method for solving the solute transport equation
International Nuclear Information System (INIS)
Ross, B.; Koplik, C.M.
1978-01-01
The solute transport equation can be solved numerically by approximating the water flow field by a network of stream tubes and using a Green's function solution within each stream tube. Compared to previous methods, this approach permits greater computational efficiency and easier representation of small discontinuities, and the results are easier to interpret physically. The method has been used to study hypothetical sites for disposal of high-level radioactive waste
Numerical solution of the resistive magnetohydrodynamic boundary-layer equations
International Nuclear Information System (INIS)
Glasser, A.H.; Jardin, S.C.; Tesauro, G.
1983-10-01
Three different techniques are presented for numerical solution of the equations governing the boundary layer of resistive magnetohydrodynamic tearing and interchange instabilities in toroidal geometry. Excellent agreement among these methods and with analytical results provides confidence in the correctness of the results. Solutions obtained in regimes where analytical medthods fail indicate a new scaling for the tearing mode as well as the existence of a new regime of stability
International Nuclear Information System (INIS)
Inan, Ibrahim E.; Kaya, Dogan
2006-01-01
In this Letter by considering an improved tanh function method, we found some exact solutions of the potential Kadomtsev-Petviashvili equation. Some exact solutions of the system of the shallow water wave equation were also found
Solution of the reactor point kinetics equations by MATLAB computing
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Singh Sudhansu S.
2015-01-01
Full Text Available The numerical solution of the point kinetics equations in the presence of Newtonian temperature feedback has been a challenging issue for analyzing the reactor transients. Reactor point kinetics equations are a system of stiff ordinary differential equations which need special numerical treatments. Although a plethora of numerical intricacies have been introduced to solve the point kinetics equations over the years, some of the simple and straightforward methods still work very efficiently with extraordinary accuracy. As an example, it has been shown recently that the fundamental backward Euler finite difference algorithm with its simplicity has proven to be one of the most effective legacy methods. Complementing the back-ward Euler finite difference scheme, the present work demonstrates the application of ordinary differential equation suite available in the MATLAB software package to solve the stiff reactor point kinetics equations with Newtonian temperature feedback effects very effectively by analyzing various classic benchmark cases. Fair accuracy of the results implies the efficient application of MATLAB ordinary differential equation suite for solving the reactor point kinetics equations as an alternate method for future applications.
TRAVELING WAVE SOLUTIONS OF SOME FRACTIONAL DIFFERENTIAL EQUATIONS
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SERIFE MUGE EGE
2016-07-01
Full Text Available The modified Kudryashov method is powerful, efficient and can be used as an alternative to establish new solutions of different type of fractional differential equations applied in mathematical physics. In this article, we’ve constructed new traveling wave solutions including symmetrical Fibonacci function solutions, hyperbolic function solutions and rational solutions of the space-time fractional Cahn Hillihard equation D_t^α u − γD_x^α u − 6u(D_x^α u^2 − (3u^2 − 1D_x^α (D_x^α u + D_x^α(D_x^α(D_x^α(D_x^α u = 0 and the space-time fractional symmetric regularized long wave (SRLW equation D_t^α(D_t^α u + D_x^α(D_x^α u + uD_t^α(D_x^α u + D_x^α u D_t^α u + D_t^α(D_t^α(D_x^α(D_x^α u = 0 via modified Kudryashov method. In addition, some of the solutions are described in the figures with the help of Mathematica.
International Nuclear Information System (INIS)
Burde, G.I.
2002-01-01
A new approach to the use of the Lie group technique for partial and ordinary differential equations dependent on a small parameter is developed. In addition to determining approximate solutions to the perturbed equation, the approach allows constructing integrable equations that have solutions with (partially) prescribed features. Examples of application of the approach to partial differential equations are given
Multipermutation Solutions of the Yang-Baxter Equation
International Nuclear Information System (INIS)
Gateva-Ivanova, Tatiana; Cameron, Peter
2009-12-01
Set-theoretic solutions of the Yang-Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution consists of a set X and a function r : X x X → X x X which satisfies the braid relation. We examine solutions here mainly from the point of view of finite permutation groups: a solution gives rise to a map from X to the symmetric group Sym(X) on X satisfying certain conditions. Our results include many new constructions based on strong twisted union and wreath product, with an investigation of retracts and the multipermutation level and the solvable length of the groups defined by the solutions; and new results about decompositions and factorisations of the groups defined by invariant subsets of the solution. (author)
CMB spectral distortions as solutions to the Boltzmann equations
Energy Technology Data Exchange (ETDEWEB)
Ota, Atsuhisa, E-mail: a.ota@th.phys.titech.ac.jp [Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551 (Japan)
2017-01-01
We propose to re-interpret the cosmic microwave background spectral distortions as solutions to the Boltzmann equation. This approach makes it possible to solve the second order Boltzmann equation explicitly, with the spectral y distortion and the momentum independent second order temperature perturbation, while generation of μ distortion cannot be explained even at second order in this framework. We also extend our method to higher order Boltzmann equations systematically and find new type spectral distortions, assuming that the collision term is linear in the photon distribution functions, namely, in the Thomson scattering limit. As an example, we concretely construct solutions to the cubic order Boltzmann equation and show that the equations are closed with additional three parameters composed of a cubic order temperature perturbation and two cubic order spectral distortions. The linear Sunyaev-Zel'dovich effect whose momentum dependence is different from the usual y distortion is also discussed in the presence of the next leading order Kompaneets terms, and we show that higher order spectral distortions are also generated as a result of the diffusion process in a framework of higher order Boltzmann equations. The method may be applicable to a wider class of problems and has potential to give a general prescription to non-equilibrium physics.
Mild Solutions of Neutral Stochastic Partial Functional Differential Equations
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T. E. Govindan
2011-01-01
Full Text Available This paper studies the existence and uniqueness of a mild solution for a neutral stochastic partial functional differential equation using a local Lipschitz condition. When the neutral term is zero and even in the deterministic special case, the result obtained here appears to be new. An example is included to illustrate the theory.
Existence of solutions of nonlinear integrodifferential equations of ...
Indian Academy of Sciences (India)
The results are obtained by using semigroup theory and the Schauder fixed point ... The problem of existence of solutions of evolution equations with nonlocal ... we assume that there exists an operator E on DЕEЖИX given by the formula.
''Localized'' tachyonic wavelet-solutions of the wave equation
International Nuclear Information System (INIS)
Barut, A.O.; Chandola, H.C.
1993-05-01
Localized-nonspreading, wavelet-solutions of the wave equation □φ=0 with group velocity v>c and phase velocity u=c 2 /v< c are constructed explicitly by two different methods. Some recent experiments seem to find evidence for superluminal group velocities. (author). 7 refs, 2 figs
Numerical Solution of Hamilton-Jacobi Equations in High Dimension
2012-11-23
high dimension FA9550-10-1-0029 Maurizio Falcone Dipartimento di Matematica SAPIENZA-Universita di Roma P. Aldo Moro, 2 00185 ROMA AH930...solution of Hamilton-Jacobi equations in high dimension AFOSR contract n. FA9550-10-1-0029 Maurizio Falcone Dipartimento di Matematica SAPIENZA
Calculation of Volterra kernels for solutions of nonlinear differential equations
van Hemmen, JL; Kistler, WM; Thomas, EGF
2000-01-01
We consider vector-valued autonomous differential equations of the form x' = f(x) + phi with analytic f and investigate the nonanticipative solution operator phi bar right arrow A(phi) in terms of its Volterra series. We show that Volterra kernels of order > 1 occurring in the series expansion of
Asymptotic formulae for solutions of half-linear differential equations
Czech Academy of Sciences Publication Activity Database
Řehák, Pavel
2017-01-01
Roč. 292, January (2017), s. 165-177 ISSN 0096-3003 Institutional support: RVO:67985840 Keywords : half-linear differential equation * nonoscillatory solution * regular variation Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 1.738, year: 2016 http://www.sciencedirect.com/science/article/pii/S0096300316304581
Numerical Solution of Differential Algebraic Equations and Applications
DEFF Research Database (Denmark)
Thomsen, Per Grove
2005-01-01
These lecture notes have been written as part of a special course on the numerical solution of Differential Algebraic Equations and applications . The course was held at IMM in the spring of 2005. The authors of the different chapters have all taken part in the course and the chapters are written...
The Local Stability of Solutions for a Nonlinear Equation
Directory of Open Access Journals (Sweden)
Haibo Yan
2014-01-01
Full Text Available The approach of Kruzkov’s device of doubling the variables is applied to establish the local stability of strong solutions for a nonlinear partial differential equation in the space L1(R by assuming that the initial value only lies in the space L1(R∩L∞(R.
On nonnegative solutions of second order linear functional differential equations
Czech Academy of Sciences Publication Activity Database
Lomtatidze, Alexander; Vodstrčil, Petr
2004-01-01
Roč. 32, č. 1 (2004), s. 59-88 ISSN 1512-0015 Institutional research plan: CEZ:AV0Z1019905 Keywords : second order linear functional differential equations * nonnegative solution * two-point boundary value problem Subject RIV: BA - General Mathematics
Almost periodic solutions to systems of parabolic equations
Directory of Open Access Journals (Sweden)
Janpou Nee
1994-01-01
Full Text Available In this paper we show that the second-order differential solution is 2-almost periodic, provided it is 2-bounded, and the growth of the components of a non-linear function of a system of parabolic equation is bounded by any pair of con-secutive eigenvalues of the associated Dirichlet boundary value problems.
Radial solutions to semilinear elliptic equations via linearized operators
Directory of Open Access Journals (Sweden)
Phuong Le
2017-04-01
Full Text Available Let $u$ be a classical solution of semilinear elliptic equations in a ball or an annulus in $\\mathbb{R}^N$ with zero Dirichlet boundary condition where the nonlinearity has a convex first derivative. In this note, we prove that if the $N$-th eigenvalue of the linearized operator at $u$ is positive, then $u$ must be radially symmetric.
About Global Stable of Solutions of Logistic Equation with Delay
Kaschenko, S. A.; Loginov, D. O.
2017-12-01
The article is devoted to the definition of all the arguments for which all positive solutions of logistic equation with delay tend to zero for t → ∞. The authors have proved the acquainted Wright’s conjecture on evaluation of a multitude of such arguments. An approach that enables subsequent refinement of this evaluation has been developed.
Minimal solution of linear formed fuzzy matrix equations
Directory of Open Access Journals (Sweden)
Maryam Mosleh
2012-10-01
Full Text Available In this paper according to the structured element method, the $mimes n$ inconsistent fuzzy matrix equation $Ailde{X}=ilde{B},$ which are linear formed by fuzzy structured element, is investigated. The necessary and sufficient condition for the existence of a fuzzy solution is also discussed. some examples are presented to illustrate the proposed method.
Existence of Three Solutions for $p$-biharmonic Equation
Directory of Open Access Journals (Sweden)
Lin Li
2011-10-01
Full Text Available In this paper, the existence of at least three solutions to a Navier boundary problem involving the $p$-biharmonic equation, will be established. The technical approach is mainly based on the three critical points theorem of B. Ricceri.
Explosive solutions of elliptic equations with absorption and non ...
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
control theory and have been first studied by Lasry and Lions [8]. The corresponding parabolic equation was considered in Quittner [12]. In terms of the dynamic programming approach, an explosive solution of (1) corresponds to a value function (or Bellman function) associated to an infinite exit cost (see [8]). Bandle and ...
Review of Kaganove's solution for the reactor point kinetics equations
International Nuclear Information System (INIS)
Couto, R.T.; Santo, A.C.F. de.
1993-09-01
A review of Kaganove's method for the reactor point kinetics equations solution is performed. This was method chosen to calculate the power in ATR, a computer program for the analysis of reactivity transients. The reasons for this choice and the adaptation of the method to the purposes of ATR are presented. (author)
Analysis of Quadratic Diophantine Equations with Fibonacci Number Solutions
Leyendekkers, J. V.; Shannon, A. G.
2004-01-01
An analysis is made of the role of Fibonacci numbers in some quadratic Diophantine equations. A general solution is obtained for finding factors in sums of Fibonacci numbers. Interpretation of the results is facilitated by the use of a modular ring which also permits extension of the analysis.
Periodic solutions to second-order indefinite singular equations
Czech Academy of Sciences Publication Activity Database
Hakl, Robert; Zamora, M.
2017-01-01
Roč. 263, č. 1 (2017), s. 451-469 ISSN 0022-0396 Institutional support: RVO:67985840 Keywords : degree theory * indefinite singularity * periodic solution * singular differential equation Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 1.988, year: 2016 http://www.sciencedirect.com/science/article/pii/S0022039617301134
A perturbative solution to metadynamics ordinary differential equation.
Tiwary, Pratyush; Dama, James F; Parrinello, Michele
2015-12-21
Metadynamics is a popular enhanced sampling scheme wherein by periodic application of a repulsive bias, one can surmount high free energy barriers and explore complex landscapes. Recently, metadynamics was shown to be mathematically well founded, in the sense that the biasing procedure is guaranteed to converge to the true free energy surface in the long time limit irrespective of the precise choice of biasing parameters. A differential equation governing the post-transient convergence behavior of metadynamics was also derived. In this short communication, we revisit this differential equation, expressing it in a convenient and elegant Riccati-like form. A perturbative solution scheme is then developed for solving this differential equation, which is valid for any generic biasing kernel. The solution clearly demonstrates the robustness of metadynamics to choice of biasing parameters and gives further confidence in the widely used method.
A perturbative solution to metadynamics ordinary differential equation
Tiwary, Pratyush; Dama, James F.; Parrinello, Michele
2015-12-01
Metadynamics is a popular enhanced sampling scheme wherein by periodic application of a repulsive bias, one can surmount high free energy barriers and explore complex landscapes. Recently, metadynamics was shown to be mathematically well founded, in the sense that the biasing procedure is guaranteed to converge to the true free energy surface in the long time limit irrespective of the precise choice of biasing parameters. A differential equation governing the post-transient convergence behavior of metadynamics was also derived. In this short communication, we revisit this differential equation, expressing it in a convenient and elegant Riccati-like form. A perturbative solution scheme is then developed for solving this differential equation, which is valid for any generic biasing kernel. The solution clearly demonstrates the robustness of metadynamics to choice of biasing parameters and gives further confidence in the widely used method.
Invariant solutions of the supersymmetric sine-Gordon equation
International Nuclear Information System (INIS)
Grundland, A M; Hariton, A J; Snobl, L
2009-01-01
A comprehensive symmetry analysis of the N=1 supersymmetric sine-Gordon equation is performed. Two different forms of the supersymmetric system are considered. We begin by studying a system of partial differential equations corresponding to the coefficients of the various powers of the anticommuting independent variables. Next, we consider the super-sine-Gordon equation expressed in terms of a bosonic superfield involving anticommuting independent variables. In each case, a Lie (super)algebra of symmetries is determined and a classification of all subgroups having generic orbits of codimension 1 in the space of independent variables is performed. The method of symmetry reduction is systematically applied in order to derive invariant solutions of the supersymmetric model. Several types of algebraic, hyperbolic and doubly periodic solutions are obtained in explicit form.
Sharp bounds for periodic solutions of Lipschitzian differential equations
International Nuclear Information System (INIS)
Zevin, A A
2009-01-01
A general system of Lipschitzian differential equations, containing simultaneously terms without delay and with arbitrary constant and time-varying delays, is considered. For the autonomous case, a lower bound for the period of nonconstant periodic solutions, expressed in the respective supremum Lipschitz constants, is found. For nonautonomous periodic equations, explicit upper bounds for the amplitudes and maximum derivatives of periodic solutions are obtained. For all n ≥ 2, the bounds do not depend on n and, in general, are different from that for n = 1. All the bounds are sharp; they are attained in linear differential equations with piece-wise constant deviating arguments. A relation between the obtained bounds and the sharp bounds in other metrics is established
Asymptotic solutions and spectral theory of linear wave equations
International Nuclear Information System (INIS)
Adam, J.A.
1982-01-01
This review contains two closely related strands. Firstly the asymptotic solution of systems of linear partial differential equations is discussed, with particular reference to Lighthill's method for obtaining the asymptotic functional form of the solution of a scalar wave equation with constant coefficients. Many of the applications of this technique are highlighted. Secondly, the methods and applications of the theory of the reduced (one-dimensional) wave equation - particularly spectral theory - are discussed. While the breadth of application and power of the techniques is emphasised throughout, the opportunity is taken to present to a wider readership, developments of the methods which have occured in some aspects of astrophysical (particularly solar) and geophysical fluid dynamics. It is believed that the topics contained herein may be of relevance to the applied mathematician or theoretical physicist interest in problems of linear wave propagation in these areas. (orig./HSI)
International Nuclear Information System (INIS)
Xiao Yafeng; Xue Haili; Zhang Hongqing
2011-01-01
Based on Jacobi elliptic function and the Lame equation, the perturbation method is applied to get the multi-order envelope periodic solutions of the nonlinear Schrodinger equation and cubic nonlinear Schrodinger equation. These multi-order envelope periodic solutions can degenerate into the different envelope solitary solutions. (authors)
Numerical solution of plasma fluid equations using locally refined grids
International Nuclear Information System (INIS)
Colella, P.
1997-01-01
This paper describes a numerical method for the solution of plasma fluid equations on block-structured, locally refined grids. The plasma under consideration is typical of those used for the processing of semiconductors. The governing equations consist of a drift-diffusion model of the electrons and an isothermal model of the ions coupled by Poisson's equation. A discretization of the equations is given for a uniform spatial grid, and a time-split integration scheme is developed. The algorithm is then extended to accommodate locally refined grids. This extension involves the advancement of the discrete system on a hierarchy of levels, each of which represents a degree of refinement, together with synchronization steps to ensure consistency across levels. A brief discussion of a software implementation is followed by a presentation of numerical results
A comparison of certain variational solutions of neutron diffusion equation
International Nuclear Information System (INIS)
Altiparmakov, D.V.; Milgram, M.S.
1987-01-01
Using the R-function theory and the variational method of Kantorovich, an approximate solution of the neutron diffusion equation is constructed for a homogeneous spatial domain of arbitrary shape. Calculations have been carried out by five different types of trial functions for certain characteristic domains of polygonal shape (square, triangle, hexagon, rhombus nad L-shaped domain). In the case of non-convex polygons, the consequence of the R-function solution is very poor and a separate treatment of singularity seems to be necessary. Compared to the R-function solution, the singular function development is mathematically more complicated but superior in respect to convergence rate. (author)
Smooth solutions of the Navier-Stokes equations
International Nuclear Information System (INIS)
Pokhozhaev, S I
2014-01-01
We consider smooth solutions of the Cauchy problem for the Navier-Stokes equations on the scale of smooth functions which are periodic with respect to x∈R 3 . We obtain existence theorems for global (with respect to t>0) and local solutions of the Cauchy problem. The statements of these depend on the smoothness and the norm of the initial vector function. Upper bounds for the behaviour of solutions in both classes, which depend on t, are also obtained. Bibliography: 10 titles
Oscillatory solutions of the Cauchy problem for linear differential equations
Directory of Open Access Journals (Sweden)
Gro Hovhannisyan
2015-06-01
Full Text Available We consider the Cauchy problem for second and third order linear differential equations with constant complex coefficients. We describe necessary and sufficient conditions on the data for the existence of oscillatory solutions. It is known that in the case of real coefficients the oscillatory behavior of solutions does not depend on initial values, but we show that this is no longer true in the complex case: hence in practice it is possible to control oscillatory behavior by varying the initial conditions. Our Proofs are based on asymptotic analysis of the zeros of solutions, represented as linear combinations of exponential functions.
An Exact Solution of The Neutron Slowing Down Equation
Energy Technology Data Exchange (ETDEWEB)
Stefanovic, D [Boris Kidric Vinca Institute of Nuclear Sciences, Vinca, Belgrade (Yugoslavia)
1970-07-01
The slowing down equation for an infinite homogeneous monoatomic medium is solved exactly. The cross sections depend on neutron energy. The solution is given in analytical form within each of the lethargy intervals. This analytical form is the sum of probabilities which are given by the Green functions. The calculated collision density is compared with the one obtained by Bednarz and also with an approximate Wigner formula for the case of a resonance not wider than one collision interval. For the special case of hydrogen, the present solution reduces to Bethe's solution. (author)
Rational solutions to the KPI equation and multi rogue waves
Gaillard, Pierre
2016-04-01
We construct here rational solutions to the Kadomtsev-Petviashvili equation (KPI) as a quotient of two polynomials in x, y and t depending on several real parameters. This method provides an infinite hierarchy of rational solutions written in terms of polynomials of degrees 2 N(N + 1) in x, y and t depending on 2 N - 2 real parameters for each positive integer N. We give explicit expressions of the solutions in the simplest cases N = 1 and N = 2 and we study the patterns of their modulus in the (x , y) plane for different values of time t and parameters.
Iterative solution of nonlinear equations with strongly accretive operators
International Nuclear Information System (INIS)
Chidume, C.E.
1991-10-01
Let E be a real Banach space with a uniformly convex dual, and let K be a nonempty closed convex and bounded subset of E. Suppose T:K→K is a strongly accretive map such that for each f is an element of K the equation Tx=f has a solution in K. It is proved that each of the two well known fixed point iteration methods (the Mann and Ishikawa iteration methods) converges strongly to a solution of the equation Tx=f. Furthermore, our method shows that such a solution is necessarily unique. Explicit error estimates are given. Our results resolve in the affirmative two open problems (J. Math. Anal. Appl. Vol 151(2) (1990), p. 460) and generalize important known results. (author). 32 refs
Asymptotically Almost Periodic Solutions of Evolution Equations in Banach Spaces
Ruess, W. M.; Phong, V. Q.
Tile linear abstract evolution equation (∗) u'( t) = Au( t) + ƒ( t), t ∈ R, is considered, where A: D( A) ⊂ E → E is the generator of a strongly continuous semigroup of operators in the Banach space E. Starting from analogs of Kadets' and Loomis' Theorems for vector valued almost periodic Functions, we show that if σ( A) ∩ iR is countable and ƒ: R → E is [asymptotically] almost periodic, then every bounded and uniformly continuous solution u to (∗) is [asymptotically] almost periodic, provided e-λ tu( t) has uniformly convergent means for all λ ∈ σ( A) ∩ iR. Related results on Eberlein-weakly asymptotically almost periodic, periodic, asymptotically periodic and C 0-solutions of (∗), as well as on the discrete case of solutions of difference equations are included.
Periodic Solutions and S-Asymptotically Periodic Solutions to Fractional Evolution Equations
Directory of Open Access Journals (Sweden)
Jia Mu
2017-01-01
Full Text Available This paper deals with the existence and uniqueness of periodic solutions, S-asymptotically periodic solutions, and other types of bounded solutions for some fractional evolution equations with the Weyl-Liouville fractional derivative defined for periodic functions. Applying Fourier transform we give reasonable definitions of mild solutions. Then we accurately estimate the spectral radius of resolvent operator and obtain some existence and uniqueness results.
On numerical solution of Burgers' equation by homotopy analysis method
International Nuclear Information System (INIS)
Inc, Mustafa
2008-01-01
In this Letter, we present the Homotopy Analysis Method (shortly HAM) for obtaining the numerical solution of the one-dimensional nonlinear Burgers' equation. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Convergence of the solution and effects for the method is discussed. The comparison of the HAM results with the Homotopy Perturbation Method (HPM) and the results of [E.N. Aksan, Appl. Math. Comput. 174 (2006) 884; S. Kutluay, A. Esen, Int. J. Comput. Math. 81 (2004) 1433; S. Abbasbandy, M.T. Darvishi, Appl. Math. Comput. 163 (2005) 1265] are made. The results reveal that HAM is very simple and effective. The HAM contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region of solution series. The numerical solutions are compared with the known analytical and some numerical solutions
Polynomial solutions of the Monge-Ampère equation
Energy Technology Data Exchange (ETDEWEB)
Aminov, Yu A [B.Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Khar' kov (Ukraine)
2014-11-30
The question of the existence of polynomial solutions to the Monge-Ampère equation z{sub xx}z{sub yy}−z{sub xy}{sup 2}=f(x,y) is considered in the case when f(x,y) is a polynomial. It is proved that if f is a polynomial of the second degree, which is positive for all values of its arguments and has a positive squared part, then no polynomial solution exists. On the other hand, a solution which is not polynomial but is analytic in the whole of the x, y-plane is produced. Necessary and sufficient conditions for the existence of polynomial solutions of degree up to 4 are found and methods for the construction of such solutions are indicated. An approximation theorem is proved. Bibliography: 10 titles.
Singular solitons and other solutions to a couple of nonlinear wave equations
International Nuclear Information System (INIS)
Inc Mustafa; Ulutaş Esma; Biswas Anjan
2013-01-01
This paper addresses the extended (G'/G)-expansion method and applies it to a couple of nonlinear wave equations. These equations are modified the Benjamin—Bona—Mahoney equation and the Boussinesq equation. This extended method reveals several solutions to these equations. Additionally, the singular soliton solutions are revealed, for these two equations, with the aid of the ansatz method
New Numerical Solution of von Karman Equation of Lengthwise Rolling
Directory of Open Access Journals (Sweden)
Rudolf Pernis
2015-01-01
Full Text Available The calculation of average material contact pressure to rolls base on mathematical theory of rolling process given by Karman equation was solved by many authors. The solutions reported by authors are used simplifications for solution of Karman equation. The simplifications are based on two cases for approximation of the circular arch: (a by polygonal curve and (b by parabola. The contribution of the present paper for solution of two-dimensional differential equation of rolling is based on description of the circular arch by equation of a circle. The new term relative stress as nondimensional variable was defined. The result from derived mathematical models can be calculated following variables: normal contact stress distribution, front and back tensions, angle of neutral point, coefficient of the arm of rolling force, rolling force, and rolling torque during rolling process. Laboratory cold rolled experiment of CuZn30 brass material was performed. Work hardening during brass processing was calculated. Comparison of theoretical values of normal contact stress with values of normal contact stress obtained from cold rolling experiment was performed. The calculations were not concluded with roll flattening.
A stochastic solution of the advective transport equation with uncertainty
International Nuclear Information System (INIS)
Williams, M.M.R.
1991-01-01
A model has been developed for calculating the transport of water-borne radionuclides through layers of porous materials, such as rock or clay. The model is based upon a purely advective transport equation, in which the fluid velocity is a random variable, thereby simulating dispersion in a more realistic manner than the ad hoc introduction of a dispersivity. In addition to a random velocity field, which is an observable physical phenomenon, allowance is made for uncertainty in our knowledge of the parameters which enter the equation, e.g. the retardation coefficient. This too, is assumed to be a random variable and contributes to the stochasticity of the resulting partial differential equation of transport. The stochastic differential equation can be solved analytically and then ensemble averages taken over the associated probability distribution of velocity and retardation coefficient. A method based upon a novel form of the central limit theorem of statistics is employed to obtain tractable solutions of a system consisting of many serial legs of varying properties. One interesting conclusion is that the total flux out of a medium is significantly underestimated by using the deterministic solution with an average transit time compared with that from the stochastically averaged solution. The theory is illustrated numerically for a number of physically relevant cases. (author) 8 figs., 4 tabs., 7 refs
CSR Fields: Direct Numerical Solution of the Maxwell's Equation
International Nuclear Information System (INIS)
Novokhatski, Alexander
2011-01-01
We discuss the properties of the coherent electromagnetic fields of a very short, ultra-relativistic bunch in a rectangular vacuum chamber inside a bending magnet. The analysis is based on the results of a direct numerical solution of Maxwell's equations together with Newton's equations. We use a new dispersion-free time-domain algorithm which employs a more efficient use of finite element mesh techniques and hence produces self-consistent and stable solutions for very short bunches. We investigate the fine structure of the CSR fields including coherent edge radiation. This approach should be useful in the study of existing and future concepts of particle accelerators and ultrafast coherent light sources. The coherent synchrotron radiation (CSR) fields have a strong action on the beam dynamics of very short bunches, which are moving in the bends of all kinds of magnetic elements. They are responsible for additional energy loss and energy spread; micro bunching and beam emittance growth. These fields may bound the efficiency of damping rings, electron-positron colliders and ultrafast coherent light sources, where high peak currents and very short bunches are envisioned. This is relevant to most high-brightness beam applications. On the other hand these fields together with transition radiation fields can be used for beam diagnostics or even as a powerful resource of THz radiation. A history of the study of CSR and a good collection of references can be found in (1). Electromagnetic theory suggests several methods on how to calculate CSR fields. The most popular method is to use Lienard-Wiechert potentials. Other approach is to solve numerically the approximate equations, which are a Schrodinger type equation. These numerical methods are described in (2). We suggest that a direct solution of Maxwell's equations together with Newton's equations can describe the detailed structure of the CSR fields (3).
Classification of exact solutions to the generalized Kadomtsev-Petviashvili equation
International Nuclear Information System (INIS)
Pandir, Yusuf; Gurefe, Yusuf; Misirli, Emine
2013-01-01
In this paper, we study the Kadomtsev-Petviashvili equation with generalized evolution and derive some new results using the approach called the trial equation method. The obtained results can be expressed by the soliton solutions, rational function solutions, elliptic function solutions and Jacobi elliptic function solutions. In the discussion, we give a new version of the trial equation method for nonlinear differential equations.
Generalized nonlinear Proca equation and its free-particle solutions
Energy Technology Data Exchange (ETDEWEB)
Nobre, F.D. [Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex Systems, Rio de Janeiro, RJ (Brazil); Plastino, A.R. [Universidad Nacional Buenos Aires-Noreoeste, CeBio y Secretaria de Investigacion, Junin (Argentina)
2016-06-15
We introduce a nonlinear extension of Proca's field theory for massive vector (spin 1) bosons. The associated relativistic nonlinear wave equation is related to recently advanced nonlinear extensions of the Schroedinger, Dirac, and Klein-Gordon equations inspired on the non-extensive generalized thermostatistics. This is a theoretical framework that has been applied in recent years to several problems in nuclear and particle physics, gravitational physics, and quantum field theory. The nonlinear Proca equation investigated here has a power-law nonlinearity characterized by a real parameter q (formally corresponding to the Tsallis entropic parameter) in such a way that the standard linear Proca wave equation is recovered in the limit q → 1. We derive the nonlinear Proca equation from a Lagrangian, which, besides the usual vectorial field Ψ{sup μ}(vector x,t), involves an additional field Φ{sup μ}(vector x,t). We obtain exact time-dependent soliton-like solutions for these fields having the form of a q-plane wave, and we show that both field equations lead to the relativistic energy-momentum relation E{sup 2} = p{sup 2}c{sup 2} + m{sup 2}c{sup 4} for all values of q. This suggests that the present nonlinear theory constitutes a new field theoretical representation of particle dynamics. In the limit of massless particles the present q-generalized Proca theory reduces to Maxwell electromagnetism, and the q-plane waves yield localized, transverse solutions of Maxwell equations. Physical consequences and possible applications are discussed. (orig.)
Inverse scattering solution of the Chew-Low equation
International Nuclear Information System (INIS)
Nakano, K.
1985-01-01
Techniques for solving the inverse scattering problem are applied to the Chew-Low equation to obtain the nucleon form factor directly from the experimental phase shifts. A new dispersion relation is derived for the P 11 wave because of its sign-changing phase shift. A self-consistent solution for each channel is obtained, but the universality of form factor is not confirmed. Also, an iterative procedure based on Omnes' method is developed in order to solve coupled-channel, singular integral equations. (orig.)
Numerical solution of the Schroedinger equation with a polynomial potential
International Nuclear Information System (INIS)
Campoy, G.; Palma, A.
1986-01-01
A numerical method for solving the Schroedinger equation for a potential expressed as a polynomial is proposed. The basic assumption relies on the asymptotic properties of the solution of this equation. It is possible to obtain the energies and the stationary state functions simultaneously. They analyze, in particular, the cases of the quartic anharmonic oscillator and a hydrogen atom perturbed by a quadratic term, obtaining its energy eigenvalues for some values of the perturbation parameter. Together with the Hellmann-Feynman theorem, they use their algorithm to calculate expectation values of x'' for arbitrary positive values of n. 4 tables
Superdiffusions and positive solutions of nonlinear partial differential equations
Dynkin, E B
2004-01-01
This book is devoted to the applications of probability theory to the theory of nonlinear partial differential equations. More precisely, it is shown that all positive solutions for a class of nonlinear elliptic equations in a domain are described in terms of their traces on the boundary of the domain. The main probabilistic tool is the theory of superdiffusions, which describes a random evolution of a cloud of particles. A substantial enhancement of this theory is presented that can be of interest for everybody who works on applications of probabilistic methods to mathematical analysis.
Solution of the transport equation with account for inelastic collisions
International Nuclear Information System (INIS)
Kalashnikov, N.P.; Remizovich, V.S.; Ryazanov, M.I.
1980-01-01
The theory of charged particle scattering in a matter with account for inelastic collisions is considered. In ''directly-forward'' approximation the transport equation at the absence of elastic collisions is obtained. The solution of the transport equation is made without and with account for fluctuation of energy losses. Formulas for path-energy relation are given. Energy spectrum and distribution of fast charged particles with respect to paths are studied. The problem of quantum mechanical approach to the theory of multiple scattering of fast charged particles in a matter is discussed briefly
Global solutions of nonlinear Schrödinger equations
Bourgain, J
1999-01-01
This volume presents recent progress in the theory of nonlinear dispersive equations, primarily the nonlinear Schrödinger (NLS) equation. The Cauchy problem for defocusing NLS with critical nonlinearity is discussed. New techniques and results are described on global existence and properties of solutions with large Cauchy data. Current research in harmonic analysis around Strichartz's inequalities and its relevance to nonlinear PDE is presented. Several topics in NLS theory on bounded domains are reviewed. Using the NLS as an example, the book offers comprehensive insight on current research r
Analytical Solution of a Generalized Hirota-Satsuma Equation
Kassem, M.; Mabrouk, S.; Abd-el-Malek, M.
A modified version of generalized Hirota-Satsuma is here solved using a two parameter group transformation method. This problem in three dimensions was reduced by Estevez [1] to a two dimensional one through a Lie transformation method and left unsolved. In the present paper, through application of symmetry transformation the Lax pair has been reduced to a system of ordinary equations. Three transformations cases are investigated. The obtained analytical solutions are plotted and show a profile proper to deflagration processes, well described by Degasperis-Procesi equation.
On the Partial Analytical Solution of the Kirchhoff Equation
Michels, Dominik L.
2015-09-01
We derive a combined analytical and numerical scheme to solve the (1+1)-dimensional differential Kirchhoff system. Here the object is to obtain an accurate as well as an efficient solution process. Purely numerical algorithms typically have the disadvantage that the quality of solutions decreases enormously with increasing temporal step sizes, which results from the numerical stiffness of the underlying partial differential equations. To prevent that, we apply a differential Thomas decomposition and a Lie symmetry analysis to derive explicit analytical solutions to specific parts of the Kirchhoff system. These solutions are general and depend on arbitrary functions, which we set up according to the numerical solution of the remaining parts. In contrast to a purely numerical handling, this reduces the numerical solution space and prevents the system from becoming unstable. The differential Kirchhoff equation describes the dynamic equilibrium of one-dimensional continua, i.e. slender structures like fibers. We evaluate the advantage of our method by simulating a cilia carpet.
On the Partial Analytical Solution of the Kirchhoff Equation
Michels, Dominik L.; Lyakhov, Dmitry; Gerdt, Vladimir P.; Sobottka, Gerrit A.; Weber, Andreas G.
2015-01-01
We derive a combined analytical and numerical scheme to solve the (1+1)-dimensional differential Kirchhoff system. Here the object is to obtain an accurate as well as an efficient solution process. Purely numerical algorithms typically have the disadvantage that the quality of solutions decreases enormously with increasing temporal step sizes, which results from the numerical stiffness of the underlying partial differential equations. To prevent that, we apply a differential Thomas decomposition and a Lie symmetry analysis to derive explicit analytical solutions to specific parts of the Kirchhoff system. These solutions are general and depend on arbitrary functions, which we set up according to the numerical solution of the remaining parts. In contrast to a purely numerical handling, this reduces the numerical solution space and prevents the system from becoming unstable. The differential Kirchhoff equation describes the dynamic equilibrium of one-dimensional continua, i.e. slender structures like fibers. We evaluate the advantage of our method by simulating a cilia carpet.
Delay chemical master equation: direct and closed-form solutions.
Leier, Andre; Marquez-Lago, Tatiana T
2015-07-08
The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived.
An Algebraic Method for Constructing Exact Solutions to Difference-Differential Equations
International Nuclear Information System (INIS)
Wang Zhen; Zhang Hongqing
2006-01-01
In this paper, we present a method to solve difference differential equation(s). As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation(s).
International Nuclear Information System (INIS)
Chen Yong; Yan Zhenya
2005-01-01
In this paper (2 + 1)-dimensional Gardner equation is investigated using a sine-Gordon equation expansion method, which was presented via a generalized sine-Gordon reduction equation and a new transformation. As a consequence, it is shown that the method is more powerful to obtain many types of new doubly periodic solutions of (2 + 1)-dimensional Gardner equation. In particular, solitary wave solutions are also given as simple limits of doubly periodic solutions
Solution of 3-dimensional diffusion equation by finite Fourier transformation
International Nuclear Information System (INIS)
Krishnani, P.D.
1978-01-01
Three dimensional diffusion equation in Cartesian co-ordinates is solved by using the finite Fourier transformation. This method is different from the usual Fourier transformation method in the sense that the solutions are obtained without performing the inverse Fourier transformation. The advantage has been taken of the fact that the flux is finite and integrable in the finite region. By applying this condition, a two-dimensional integral equation, involving flux and its normal derivative at the boundary, is obtained. By solving this equation with given boundary conditions, all of the boundary values are determined. In order to calculate the flux inside the region, flux is expanded into three-dimensional Fourier series. The Fourier coefficients of the flux in the region are calculated from the boundary values. The advantage of this method is that the integrated flux is obtained without knowing the fluxes inside the region as in the case of finite difference method. (author)
Solutions of the KPI equation with smooth initial data
Boiti, M.; Pempinelli, F.; Pogrebkov, A.
1994-06-01
The solution $u(t,x,y)$ of the Kadomtsev--Petviashvili I (KPI) equation with given initial data $u(0,x,y)$ belonging to the Schwartz space is considered. No additional special constraints, usually considered in literature, as $\\int\\!dx\\,u(0,x,y)=0$ are required to be satisfied by the initial data. The problem is completely solved in the framework of the spectral transform theory and it is shown that $u(t,x,y)$ satisfies a special evolution version of the KPI equation and that, in general, $\\partial_t u(t,x,y)$ has different left and right limits at the initial time $t=0$. The conditions of the type $\\int\\!dx\\,u(t,x,y)=0$, $\\int\\!dx\\,xu_y(t,x,y)=0$ and so on (first, second, etc. `constraints') are dynamically generated by the evolution equation for $t\
Numerical Solution of Stochastic Nonlinear Fractional Differential Equations
El-Beltagy, Mohamed A.; Al-Juhani, Amnah
2015-01-01
Using Wiener-Hermite expansion (WHE) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. WHE is the only known expansion that handles the white/colored noise exactly. This work introduces a numerical estimation of the stochastic response of the Duffing oscillator with fractional or variable order damping and driven by white noise. The WHE technique is integrated with the Grunwald-Letnikov approximation in case of fractional order and with Coimbra approximation in case of variable-order damping. The numerical solver was tested with the analytic solution and with Monte-Carlo simulations. The developed mixed technique was shown to be efficient in simulating SPDEs.
Analytical solution of the PNP equations at AC applied voltage
International Nuclear Information System (INIS)
Golovnev, Anatoly; Trimper, Steffen
2012-01-01
A symmetric binary polymer electrolyte subjected to an AC voltage is considered. The analytical solution of the Poisson–Nernst–Planck equations (PNP) is found and analyzed for small applied voltages. Three distinct time regimes offering different behavior can be discriminated. The experimentally realized stationary behavior is discussed in detail. An expression for the external current is derived. Based on the theoretical result a simple method is suggested of measuring the ion mobility and their concentration separately. -- Highlights: ► Analytical solution of Poisson–Nernst–Planck equations. ► Binary polymer electrolyte subjected to an external AC voltage. ► Three well separated time scales exhibiting different behavior. ► The experimentally realized stationary behavior is discussed in detail. ► A method is proposed measuring the mobility and the concentration separately.
Limited-diffraction solutions to Maxwell and Schroedinger equations
International Nuclear Information System (INIS)
Lu, Jian-yu; Greenleaf, J.F.
1996-10-01
The authors have developed a new family of limited diffraction electromagnetic X-shaped waves based on the scalar X-shaped waves discovered previously. These waves are diffraction-free in theory and particle-like (wave packets), in that they maintain their shape as they propagate to an infinite distance. The 'X waves' possess (theoretically) infinitely extended 'arms' and - at least, the ones studied in this paper - have an infinite total energy: therefore, they are not physically realizable. However, they can be truncated in both space and time and 'approximated' by means of a finite aperture radiator so to get a large enough depth of interest (depth of field). In addition to the Maxwell equations, X wave solutions to the free Schroedinger equation are also obtained. Possible applications of these new waves are discussed. Finally, the authors discuss the appearance of the X-shaped solutions from the purely geometric point of view of the special relativity theory
Hipergeometric solutions to some nonhomogeneous equations of fractional order
Olivares, Jorge; Martin, Pablo; Maass, Fernando
2017-12-01
In this paper a study is performed to the solution of the linear non homogeneous fractional order alpha differential equation equal to I 0(x), where I 0(x) is the modified Bessel function of order zero, the initial condition is f(0)=0 and 0 definition for the fractional derivatives is considered. Fractional derivatives have become important in physical and chemical phenomena as visco-elasticity and visco-plasticity, anomalous diffusion and electric circuits. In particular in this work the values of alpha=1/2, 1/4 and 3/4. are explicitly considered . In these cases Laplace transform is applied, and later the inverse Laplace transform leads to the solutions of the differential equation, which become hypergeometric functions.
Numerical Solution of Stochastic Nonlinear Fractional Differential Equations
El-Beltagy, Mohamed A.
2015-01-07
Using Wiener-Hermite expansion (WHE) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. WHE is the only known expansion that handles the white/colored noise exactly. This work introduces a numerical estimation of the stochastic response of the Duffing oscillator with fractional or variable order damping and driven by white noise. The WHE technique is integrated with the Grunwald-Letnikov approximation in case of fractional order and with Coimbra approximation in case of variable-order damping. The numerical solver was tested with the analytic solution and with Monte-Carlo simulations. The developed mixed technique was shown to be efficient in simulating SPDEs.
Sn approach applied to the solution of transport equation
International Nuclear Information System (INIS)
Lopes, J.P.
1973-09-01
In this work the origin of the Transport Theory is considered and the Transport Equation for the movement of the neutron in a system is established in its more general form, using the laws of nuclear physics. This equation is used as the starting point for development, under adequate assumptions, of simpler models that render the problem suitable for numerical solution. Representation of this model in different geometries is presented. The different processes of nuclear physics are introduced briefly and discussed. In addition, the boundary conditions for the different cases and a general procedure for the application of the Conservation Law are stated. The last chapter deals specifically with the S n method, its development, definitions and generalities. Computational schemes for obtaining the S n solution in spherical and cylindrical geometry, and convergence acceleration methods are also developed. (author)
A Solution to the Fundamental Linear Fractional Order Differential Equation
Hartley, Tom T.; Lorenzo, Carl F.
1998-01-01
This paper provides a solution to the fundamental linear fractional order differential equation, namely, (sub c)d(sup q, sub t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the inductor terminated semi- infinite lossy line, is used to demonstrate the theory.
STRANGE ATTRACTORS ON PSEUDOSPECTRAL SOLUTIONS FOR DISSIPATIVE ZAKHAROV EQUATIONS
Institute of Scientific and Technical Information of China (English)
马书清; 常谦顺
2004-01-01
In this paper, the pseudospcctral method to solve the dissipative Zakharov equations is used. Its convergence is proved by priori estinates. The existence of the global attractors and the estimates of dimension are presented. A class of steady state solutions is also disscussed. The numerical results show that if the steady state solutions satisfy some special conditions, they become unstable and limit cycles and strange attractors will occur for very small perturbations.The largest Lyapunov exponent and analysis of the lincarized system are applied to explain these phenomena.
Operational Solution to the Nonlinear Klein-Gordon Equation
Bengochea, G.; Verde-Star, L.; Ortigueira, M.
2018-05-01
We obtain solutions of the nonlinear Klein-Gordon equation using a novel operational method combined with the Adomian polynomial expansion of nonlinear functions. Our operational method does not use any integral transforms nor integration processes. We illustrate the application of our method by solving several examples and present numerical results that show the accuracy of the truncated series approximations to the solutions. Supported by Grant SEP-CONACYT 220603, the first author was supported by SEP-PRODEP through the project UAM-PTC-630, the third author was supported by Portuguese National Funds through the FCT Foundation for Science and Technology under the project PEst-UID/EEA/00066/2013
On classical solutions of SU(3) gauge field equations
International Nuclear Information System (INIS)
Chakrabarti, A.
1975-01-01
Static classical solutions of SU(3) gauge field equations are studied. The roles of the O(3) subgroup and of the quadrupole generators are discussed systematically. The general form thus obtained leads, through-out, to a high degree of symmetry in the results. This brings in some simplifying features. An octet of scalar mesons is finally added. Certain classes of exact solutions are given that are singular at the origin. A generalized gauge condition is pointed out. The relation of the general form to known particular cases is discussed [fr
Entire solutions for bistable lattice differential equations with obstacles
Hoffman, Aaron; Vleck, E S Van
2018-01-01
The authors consider scalar lattice differential equations posed on square lattices in two space dimensions. Under certain natural conditions they show that wave-like solutions exist when obstacles (characterized by "holes") are present in the lattice. Their work generalizes to the discrete spatial setting the results obtained in Berestycki, Hamel, and Matuno (2009) for the propagation of waves around obstacles in continuous spatial domains. The analysis hinges upon the development of sub and super-solutions for a class of discrete bistable reaction-diffusion problems and on a generalization of a classical result due to Aronson and Weinberger that concerns the spreading of localized disturbances.
Exact solutions of the neutron slowing down equation
International Nuclear Information System (INIS)
Dawn, T.Y.; Yang, C.N.
1976-01-01
The problem of finding the exact analytic closed-form solution for the neutron slowing down equation in an infinite homogeneous medium is studied in some detail. The existence and unique properties of the solution of this equation for both the time-dependent and the time-independent cases are studied. A direct method is used to determine the solution of the stationary problem. The final result is given in terms of a sum of indefinite multiple integrals by which solutions of some special cases and the Placzek-type oscillation are examined. The same method can be applied to the time-dependent problem with the aid of the Laplace transformation technique, but the inverse transform is, in general, laborious. However, the solutions of two special cases are obtained explicitly. Results are compared with previously reported works in a variety of cases. The time moments for the positive integral n are evaluated, and the conditions for the existence of the negative moments are discussed
Ground state solutions for non-local fractional Schrodinger equations
Directory of Open Access Journals (Sweden)
Yang Pu
2015-08-01
Full Text Available In this article, we study a time-independent fractional Schrodinger equation with non-local (regional diffusion $$ (-\\Delta^{\\alpha}_{\\rho}u + V(xu = f(x,u \\quad \\text{in }\\mathbb{R}^{N}, $$ where $\\alpha \\in (0,1$, $N > 2\\alpha$. We establish the existence of a non-negative ground state solution by variational methods.
A general method for enclosing solutions of interval linear equations
Czech Academy of Sciences Publication Activity Database
Rohn, Jiří
2012-01-01
Roč. 6, č. 4 (2012), s. 709-717 ISSN 1862-4472 R&D Projects: GA ČR GA201/09/1957; GA ČR GC201/08/J020 Institutional research plan: CEZ:AV0Z10300504 Keywords : interval linear equations * solution set * enclosure * absolute value inequality Subject RIV: BA - General Mathematics Impact factor: 1.654, year: 2012
Lectures on the Numerical Solution of Partial Differential Equations.
1981-12-01
Mathematics Rockefeller University Professor F. Brezzi New York, New York 10021 Laboratorio di Analisi Numerica Universita di Pavia Professor Amiram Harten...equations and to control the spacing of the points sj. In the MFE process the grid points move with the solution and cluster atound areas of roughness...149-159. [28] Fichera, G.: Analisi essistenziale per le soluzioni die problemi al contorno misti relativi alle equazione ed ai sistemi di equazioni
Knitting Ansatz and solutions to Yang-Baxter equation
International Nuclear Information System (INIS)
Zhang Jun; Guo Hanying; Yan Hong
1997-01-01
The authors suggest a new method, named knitting Ansatz, to generate solutions to Yang-Baxter equation with lower dimensional representations of braid group. To support this Ansatz, the authors work out an example of a new 16 x 16 R-matrix constructed along this idea, with two 4 x 4 braid group representations of familiar 6-vertex type with different q-parameters
Global solutions to the equation of thermoelasticity with fading memory
Okada, Mari; Kawashima, Shuichi
2017-07-01
We consider the initial-history value problem for the one-dimensional equation of thermoelasticity with fading memory. It is proved that if the data are smooth and small, then a unique smooth solution exists globally in time and converges to the constant equilibrium state as time goes to infinity. Our proof is based on a technical energy method which makes use of the strict convexity of the entropy function and the properties of strongly positive definite kernels.
Methodology for the hybrid solution of systems of differential equations
International Nuclear Information System (INIS)
Larrinaga, E.F.; Lopez, M.A.
1993-01-01
This work shows a general methodology of solution to systems of differential equations in hybrid computers. Taking into account this methodology, a mathematical model was elaborated. It offers wide possibilities of recording and handling the results on the basis of using the hybrid system IBM-VIDAC 1224 which the ISCTN has. It also presents the results gained when simulating a simple model of a nuclear reactor, which was used in the validation of the results of the computational model
Exact solutions of nonlinear generalizations of the Klein Gordon and Schrodinger equations
International Nuclear Information System (INIS)
Burt, P.B.
1978-01-01
Exact solutions of sine Gordon and multiple sine Gordon equations are constructed in terms of solutions of a linear base equation, the Klein Gordon equation and also in terms of nonlinear base equations where the nonlinearity is polynomial in the dependent variable. Further, exact solutions of nonlinear generalizations of the Schrodinger equation and of additional nonlinear generalizations of the Klein Gordon equation are constructed in terms of solutions of linear base equations. Finally, solutions with spherical symmetry, of nonlinear Klein Gordon equations are given. 14 references
Periodic solutions for one dimensional wave equation with bounded nonlinearity
Ji, Shuguan
2018-05-01
This paper is concerned with the periodic solutions for the one dimensional nonlinear wave equation with either constant or variable coefficients. The constant coefficient model corresponds to the classical wave equation, while the variable coefficient model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For finding the periodic solutions of variable coefficient wave equation, it is usually required that the coefficient u (x) satisfies ess infηu (x) > 0 with ηu (x) = 1/2 u″/u - 1/4 (u‧/u)2, which actually excludes the classical constant coefficient model. For the case ηu (x) = 0, it is indicated to remain an open problem by Barbu and Pavel (1997) [6]. In this work, for the periods having the form T = 2p-1/q (p , q are positive integers) and some types of boundary value conditions, we find some fundamental properties for the wave operator with either constant or variable coefficients. Based on these properties, we obtain the existence of periodic solutions when the nonlinearity is monotone and bounded. Such nonlinearity may cross multiple eigenvalues of the corresponding wave operator. In particular, we do not require the condition ess infηu (x) > 0.
Cellular solutions for the Poisson equation in extended systems
International Nuclear Information System (INIS)
Zhang, X.; Butler, W.H.; MacLaren, J.M.; van Ek, J.
1994-01-01
The Poisson equation for the electrostatic potential in a solid is solved using three different cellular techniques. The relative merits of these different approaches are discussed for two test charge densities for which an analytic solution to the Poisson equation is known. The first approach uses full-cell multiple-scattering theory and results in the famililar structure constant and multipole moment expansion. This solution is shown to be valid everywhere inside the cell, although for points outside the muffin-tin sphere but inside the cell the sums must be performed in the correct order to yield meaningful results. A modification of the multiple-scattering-theory approach yields a second method, a Green-function cellular method, which only requires the solution of a nearest-neighbor linear system of equations. A third approach, a related variational cellular method, is also derived. The variational cellular approach is shown to be the most accurate and reliable, and to have the best convergence in angular momentum of the three methods. Coulomb energies accurate to within 10 -6 hartree are easily achieved with the variational cellular approach, demonstrating the practicality of the approach in electronic structure calculations
Solutions of the Wheeler-Feynman equations with discontinuous velocities.
de Souza, Daniel Câmara; De Luca, Jayme
2015-01-01
We generalize Wheeler-Feynman electrodynamics with a variational boundary value problem for continuous boundary segments that might include velocity discontinuity points. Critical-point orbits must satisfy the Euler-Lagrange equations of the action functional at most points, which are neutral differential delay equations (the Wheeler-Feynman equations of motion). At velocity discontinuity points, critical-point orbits must satisfy the Weierstrass-Erdmann continuity conditions for the partial momenta and the partial energies. We study a special setup having the shortest time-separation between the (infinite-dimensional) boundary segments, for which case the critical-point orbit can be found using a two-point boundary problem for an ordinary differential equation. For this simplest setup, we prove that orbits can have discontinuous velocities. We construct a numerical method to solve the Wheeler-Feynman equations together with the Weierstrass-Erdmann conditions and calculate some numerical orbits with discontinuous velocities. We also prove that the variational boundary value problem has a unique solution depending continuously on boundary data, if the continuous boundary segments have velocity discontinuities along a reduced local space.
Five-equation and robust three-equation methods for solution verification of large eddy simulation
Dutta, Rabijit; Xing, Tao
2018-02-01
This study evaluates the recently developed general framework for solution verification methods for large eddy simulation (LES) using implicitly filtered LES of periodic channel flows at friction Reynolds number of 395 on eight systematically refined grids. The seven-equation method shows that the coupling error based on Hypothesis I is much smaller as compared with the numerical and modeling errors and therefore can be neglected. The authors recommend five-equation method based on Hypothesis II, which shows a monotonic convergence behavior of the predicted numerical benchmark ( S C ), and provides realistic error estimates without the need of fixing the orders of accuracy for either numerical or modeling errors. Based on the results from seven-equation and five-equation methods, less expensive three and four-equation methods for practical LES applications were derived. It was found that the new three-equation method is robust as it can be applied to any convergence types and reasonably predict the error trends. It was also observed that the numerical and modeling errors usually have opposite signs, which suggests error cancellation play an essential role in LES. When Reynolds averaged Navier-Stokes (RANS) based error estimation method is applied, it shows significant error in the prediction of S C on coarse meshes. However, it predicts reasonable S C when the grids resolve at least 80% of the total turbulent kinetic energy.
Exact solutions of space-time fractional EW and modified EW equations
International Nuclear Information System (INIS)
Korkmaz, Alper
2017-01-01
The bright soliton solutions and singular solutions are constructed for the space-time fractional EW and the space-time fractional modified EW (MEW) equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform (FCT) and properties of modified Riemann–Liouville derivative. Then, various ansatz method are implemented to construct the solutions for both equations.
International Nuclear Information System (INIS)
Liu Chunping
2003-01-01
Using a direct algebraic method, more new exact solutions of the Kolmogorov-Petrovskii-Piskunov equation are presented by formula form. Then a theorem concerning the relation between the kink-type solution and the kink-bell-type solution of nonlinear evolution equations is given. Finally, the applications of the theorem to several well-known equations in physics are also discussed
Particular solutions of generalized Euler-Poisson-Darboux equation
Directory of Open Access Journals (Sweden)
Rakhila B. Seilkhanova
2015-01-01
Full Text Available In this article we consider the generalized Euler-Poisson-Darboux equation $$ {u}_{tt}+\\frac{2\\gamma }{t}{{u}_{t}}={u}_{xx}+{u}_{yy} +\\frac{2\\alpha }{x}{{u}_{x}}+\\frac{2\\beta }{y}{{u}_y},\\quad x>0,\\;y>0,\\;t>0. $$ We construct particular solutions in an explicit form expressed by the Lauricella hypergeometric function of three variables. Properties of each constructed solutions have been investigated in sections of surfaces of the characteristic cone. Precisely, we prove that found solutions have singularity $1/r$ at $r\\to 0$, where ${{r}^2}={{( x-{{x}_0}}^2}+{{( y-{{y}_0}}^2}-{{( t-{{t}_0}}^2}$.
On the General Analytical Solution of the Kinematic Cosserat Equations
Michels, Dominik L.
2016-09-01
Based on a Lie symmetry analysis, we construct a closed form solution to the kinematic part of the (partial differential) Cosserat equations describing the mechanical behavior of elastic rods. The solution depends on two arbitrary analytical vector functions and is analytical everywhere except a certain domain of the independent variables in which one of the arbitrary vector functions satisfies a simple explicitly given algebraic relation. As our main theoretical result, in addition to the construction of the solution, we proof its generality. Based on this observation, a hybrid semi-analytical solver for highly viscous two-way coupled fluid-rod problems is developed which allows for the interactive high-fidelity simulations of flagellated microswimmers as a result of a substantial reduction of the numerical stiffness.
Explosive attractor solutions to a universal cubic delay equation
Sanz-Orozco, D.; Berk, H. L.
2017-05-01
New explosive attractor solutions have been found in a universal cubic delay equation that has been studied in both the plasma and the fluid mechanics literature. Through computational simulations and analytic approximations, it is found that the oscillatory component of the explosive mode amplitude solutions are described through multi-frequency Fourier expansions with respect to a pseudo-time variable. The spectral dependence of these solutions as a function of a system parameter, ϕ , is studied. The mode amplitude that is described in the explosive regime has two main features: a well-known envelope ( t 0 - t ) - 5 / 2 , with t0 the blow-up time of the amplitude, and a spectrum of discrete oscillations with ever-increasing frequencies, which may give experimental information about the properties of a system's equilibrium.
Positive solutions for nonlocal dispersal equation with spatial degeneracy
Sun, Jian-Wen
2018-02-01
In this paper, we consider the positive solutions of the nonlocal dispersal equation \\int \\limits _{Ω }J(x,y)[u(y)-u(x)]dy=-λ m(x)u(x)+[c(x)+ɛ ]u^p(x) \\quad { in }\\bar{Ω }, where Ω \\subset R^N is a bounded domain, λ ,ɛ and p>1 are positive constants. The dispersal kernel J and the coefficient c( x) are nonnegative, but c( x) has a degeneracy in some subdomain of Ω . In order to study the influence of heterogeneous environment on the nonlocal system, we study the sharp spatial patterns of positive solutions as ɛ → 0. We obtain that the positive solutions always have blow-up asymptotic profiles in \\bar{Ω }. Meanwhile, we find that the profiles in degeneracy domain are different from the domain without degeneracy.
Continuum solutions of the Klein-Gordon equation
International Nuclear Information System (INIS)
Jansen, G.; Pusch, M.; Soff, G.
1987-10-01
We construct explicit solutions of the Klein-Gordon equation for continuum states. The role of the energy in the single-particle Klein-Gordon theory is elucidated. Special emphasis is laid on the determination of resonance states in the continuum for overcritical potentials. As examples for long-range interaction we depict solutions for the Coulomb potential of a point-like nucleus as an extended nucleus. The square-well potential and the exponential potential are treated to exemplify pecularities of short-range interactions. We also derive continuum solutions for a scalar interaction of square-well type. Finally we discuss the behaviour of a spin-0 particle in an external homogeneous magnetic field. (orig.)
On the General Analytical Solution of the Kinematic Cosserat Equations
Michels, Dominik L.; Lyakhov, Dmitry; Gerdt, Vladimir P.; Hossain, Zahid; Riedel-Kruse, Ingmar H.; Weber, Andreas G.
2016-01-01
Based on a Lie symmetry analysis, we construct a closed form solution to the kinematic part of the (partial differential) Cosserat equations describing the mechanical behavior of elastic rods. The solution depends on two arbitrary analytical vector functions and is analytical everywhere except a certain domain of the independent variables in which one of the arbitrary vector functions satisfies a simple explicitly given algebraic relation. As our main theoretical result, in addition to the construction of the solution, we proof its generality. Based on this observation, a hybrid semi-analytical solver for highly viscous two-way coupled fluid-rod problems is developed which allows for the interactive high-fidelity simulations of flagellated microswimmers as a result of a substantial reduction of the numerical stiffness.
Homoclinic and quasi-homoclinic solutions for damped differential equations
Directory of Open Access Journals (Sweden)
Chuan-Fang Zhang
2015-01-01
Full Text Available We study the existence and multiplicity of homoclinic solutions for the second-order damped differential equation $$ \\ddot{u}+c\\dot{u}-L(tu+W_u(t,u=0, $$ where L(t and W(t,u are neither autonomous nor periodic in t. Under certain assumptions on L and W, we obtain infinitely many homoclinic solutions when the nonlinearity W(t,u is sub-quadratic or super-quadratic by using critical point theorems. Some recent results in the literature are generalized, and the open problem proposed by Zhang and Yuan is solved. In addition, with the help of the Nehari manifold, we consider the case where W(t,u is indefinite and prove the existence of at least one nontrivial quasi-homoclinic solution.
Darboux Transformation and Explicit Solutions for Drinfel'd-Sokolov-Wilson Equation
International Nuclear Information System (INIS)
Geng Xianguo; Wu Lihua
2010-01-01
A generalized Drinfel'd-Sokolov-Wilson (DSW) equation and its Lax pair are proposed. A Darboux transformation for the generalized DSW equation is constructed with the help of the gauge transformation between spectral problems, from which a Darboux transformation for the DSW equation is obtained through a reduction technique. As an application of the Darboux transformations, we give some explicit solutions of the generalized DSW equation and DSW equation such as rational solutions, soliton solutions, periodic solutions. (general)
Soliton solutions of the (2 + 1)-dimensional Harry Dym equation via Darboux transformation
International Nuclear Information System (INIS)
Halim, A.A.
2008-01-01
This work introduces solitons solutions for the (2 + 1)-dimensional Harry Dym equation using Darboux transformation. The link between the (2 + 1)-dimensional Harry Dym equation and the linear system associated with the modified Kadomtzev-Patvishvili equation is used. Namely, soliton solutions for the linear system associated with the later equation are produced using Darboux transformation. These solutions are inserted in the mentioned link to produce soliton solutions for the (2 + 1)-dimensional Harry Dym equation
International Nuclear Information System (INIS)
Ofoedu, Eric U.; Malonza, David M.
2010-07-01
In this paper we study the hybrid iterative scheme to find a common element of a set of solutions of generalized mixed equilibrium problem, a set of common fixed points of finite family of weak relatively nonexpansive mapping, and null spaces of finite family of γ-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. Our results extend, improve and generalize the results of several authors which were announced recently. An application of our theorem to the solution of equations of Hammerstein-type is of independent interest. (author)
EXISTENCE AND UNIQUENESS OF SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATION WITH RANDOM COEFFICIENTS
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
This paper mainly deals with a stochastic differential equation (SDE) with random coefficients. Sufficient conditions which guarantee the existence and uniqueness of solutions to the equation are given.
New traveling wave solutions to AKNS and SKdV equations
International Nuclear Information System (INIS)
Ozer, Teoman
2009-01-01
We analyze the traveling wave solutions of Ablowitz-Kaup-Newell-Segur (AKNS) and Schwarz-Korteweg-de Vries (SKdV) equations. As the solution method for differential equations we consider the improved tanh approach. This approach provides to transform the partial differential equation into the ordinary differential equation and then obtain the new families of exact solutions based on the solutions of the Riccati equation. The different values of the coefficients of the Riccati equation allow us to obtain new type of traveling wave solutions to AKNS and SKdV equations.
On the numerical solution of the neutron fractional diffusion equation
International Nuclear Information System (INIS)
Maleki Moghaddam, Nader; Afarideh, Hossein; Espinosa-Paredes, Gilberto
2014-01-01
Highlights: • The new version of neutron diffusion equation which established on the fractional derivatives is presented. • The Neutron Fractional Diffusion Equation (NFDE) is solved in the finite differences frame. • NFDE is solved using shifted Grünwald-Letnikov definition of fractional operators. • The results show that “K eff ” strongly depends on the order of fractional derivative. - Abstract: In order to core calculation in the nuclear reactors there is a new version of neutron diffusion equation which is established on the fractional partial derivatives, named Neutron Fractional Diffusion Equation (NFDE). In the NFDE model, neutron flux in each zone depends directly on the all previous zones (not only on the nearest neighbors). Under this circumstance, it can be said that the NFDE has the space history. We have developed a one-dimension code, NFDE-1D, which can simulate the reactor core using arbitrary exponent of differential operators. In this work a numerical solution of the NFDE is presented using shifted Grünwald-Letnikov definition of fractional derivative in finite differences frame. The model is validated with some numerical experiments where different orders of fractional derivative are considered (e.g. 0.999, 0.98, 0.96, and 0.94). The results show that the effective multiplication factor (K eff ) depends strongly on the order of fractional derivative
Exact solutions of (3 + 1-dimensional generalized KP equation arising in physics
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Syed Tauseef Mohyud-Din
Full Text Available In this work, we have obtained some exact solutions to (3 + 1-dimensional generalized KP Equation. The improved tanϕ(ξ2-expansion method has been introduced to construct the exact solutions of nonlinear evolution equations. The obtained solutions include hyperbolic function solutions, trigonometric function solutions, exponential solutions, and rational solutions. Our study has added some new varieties of solutions to already available solutions. It is also worth mentioning that the computational work has been reduced significantly. Keywords: Improved tanϕ(ξ2-expansion method, Hyperbolic function solution, Trigonometric function solution, Rational solution, (3 + 1-dimensional generalized KP equation
On a direct approach to quasideterminant solutions of a noncommutative modified KP equation
International Nuclear Information System (INIS)
Gilson, C R; Nimmo, J J C; Sooman, C M
2008-01-01
A noncommutative version of the modified KP equation and a family of its solutions expressed as quasideterminants are discussed. The origin of these solutions is explained by means of Darboux transformations and the solutions are verified directly. We also verify directly an explicit connection between quasideterminant solutions of the noncommutative mKP equation and the noncommutative KP equation arising from the Miura transformation
Exact error estimation for solutions of nuclide chain equations
International Nuclear Information System (INIS)
Tachihara, Hidekazu; Sekimoto, Hiroshi
1999-01-01
The exact solution of nuclide chain equations within arbitrary figures is obtained for a linear chain by employing the Bateman method in the multiple-precision arithmetic. The exact error estimation of major calculation methods for a nuclide chain equation is done by using this exact solution as a standard. The Bateman, finite difference, Runge-Kutta and matrix exponential methods are investigated. The present study confirms the following. The original Bateman method has very low accuracy in some cases, because of large-scale cancellations. The revised Bateman method by Siewers reduces the occurrence of cancellations and thereby shows high accuracy. In the time difference method as the finite difference and Runge-Kutta methods, the solutions are mainly affected by the truncation errors in the early decay time, and afterward by the round-off errors. Even though the variable time mesh is employed to suppress the accumulation of round-off errors, it appears to be nonpractical. Judging from these estimations, the matrix exponential method is the best among all the methods except the Bateman method whose calculation process for a linear chain is not identical with that for a general one. (author)
Parallel solutions of the two-group neutron diffusion equations
International Nuclear Information System (INIS)
Zee, K.S.; Turinsky, P.J.
1987-01-01
Recent efforts to adapt various numerical solution algorithms to parallel computer architectures have addressed the possibility of substantially reducing the running time of few-group neutron diffusion calculations. The authors have developed an efficient iterative parallel algorithm and an associated computer code for the rapid solution of the finite difference method representation of the two-group neutron diffusion equations on the CRAY X/MP-48 supercomputer having multi-CPUs and vector pipelines. For realistic simulation of light water reactor cores, the code employees a macroscopic depletion model with trace capability for selected fission product transients and critical boron. In addition to this, moderator and fuel temperature feedback models are also incorporated into the code. The validity of the physics models used in the code were benchmarked against qualified codes and proved accurate. This work is an extension of previous work in that various feedback effects are accounted for in the system; the entire code is structured to accommodate extensive vectorization; and an additional parallelism by multitasking is achieved not only for the solution of the matrix equations associated with the inner iterations but also for the other segments of the code, e.g., outer iterations
Solution of generalized control system equations at steady state
International Nuclear Information System (INIS)
Vilim, R.B.
1987-01-01
Although a number of reactor systems codes feature generalized control system models, none of the models offer a steady-state solution finder. Indeed, if a transient is to begin from steady-state conditions, the user must provide estimates for the control system initial conditions and run a null transient until the plant converges to steady state. Several such transients may have to be run before values for control system demand signals are found that produce the desired plant steady state. The intent of this paper is (a) to present the control system equations assumed in the SASSYS reactor systems code and to identify the appropriate set of initial conditions, (b) to describe the generalized block diagram approach used to represent these equations, and (c) to describe a solution method and algorithm for computing these initial conditions from the block diagram. The algorithm has been installed in the SASSYS code for use with the code's generalized control system model. The solution finder greatly enhances the effectiveness of the code and the efficiency of the user in running it
Stochastic solutions to the Schrodinger equation for fermions
International Nuclear Information System (INIS)
Arnow, D.M.
1981-01-01
An exact stochastic method has been developed for generating the antisymmetric eigensolution of lowest index and its associated eigenvalue for the Schrodinger wave equation in 3N dimensions. The method is called the Green's function Monte Carlo method for fermions (FGFMC) because it is based on a Monte Carlo solution to the integral form of the Schrodinger equation (using Green's function) and because it is the fermion class of particles in physics which require antisymmetric solutions. The solution consists of two sets of 3N-dimensional points, [R/sub j/ + ] and [R/sub j/ - ], distributed by density functions psi + and psi - , whose difference, psi + -psi - , is proportional to the eigensolution, psi/sub F/. The FGFMC method is successfully applied to a one dimensional problem and a nine dimensional problem, the results of which are presented here. These results demonstrate that this method can be successfully applied to small physical problems on medium-scale computing machines. The key to this success was the transformation of the problem from exponential to linear cost as a function of accuracy. The strong dependence on dimensionality, however, currently results in an exponential cost as a function of problem size, and this, until overcome, imposes a severe barrier to calculations on large systems
Lower bounds for solutions of the Schroedinger equation
International Nuclear Information System (INIS)
Froese, R.G.
1983-01-01
For a large class of generalized N-body Hamiltonians H = -Δ + V the large absolute value x behavior of solutions to the Schroedinger equation H psi = H psi is studied. If E lies below the essential spectrum of H, then it is proved that lim R -1 log (absolute value psi/sub R/) = -α 0 R → infinity where absolute value psi/sub R/ 2 is the integral of absolute value psi 2 over a sphere of radius R and α 0 2 + E is a threshold or α 0 0. For E not necessarily below the essential spectrum of H, the same equation holds with absolute value psi/sub R/ 2 replaced by an integral of absolute value psi 2 over a spherical shell
Random ordinary differential equations and their numerical solution
Han, Xiaoying
2017-01-01
This book is intended to make recent results on the derivation of higher order numerical schemes for random ordinary differential equations (RODEs) available to a broader readership, and to familiarize readers with RODEs themselves as well as the closely associated theory of random dynamical systems. In addition, it demonstrates how RODEs are being used in the biological sciences, where non-Gaussian and bounded noise are often more realistic than the Gaussian white noise in stochastic differential equations (SODEs). RODEs are used in many important applications and play a fundamental role in the theory of random dynamical systems. They can be analyzed pathwise with deterministic calculus, but require further treatment beyond that of classical ODE theory due to the lack of smoothness in their time variable. Although classical numerical schemes for ODEs can be used pathwise for RODEs, they rarely attain their traditional order since the solutions of RODEs do not have sufficient smoothness to have Taylor ...
Analytical solution of Mori's equation with secant hyperbolic memory
International Nuclear Information System (INIS)
Tankeshwar, K.; Pathak, K.N.
1993-07-01
The equation of motion of the auto-correlation function has been solved analytically using a secant-hyperbolic form of the memory function. The analytical results obtained for the long time expansion together with the short time expansion provide a good description over the whole time domain as judged by their comparison with the numerical solution of Mori's equation of motion. We also find that the time evolution of the auto-correlation function is determined by a single parameter τ which is related to the frequency sum rules up to the fourth order. The auto-correlation function has been found to show simple decaying or oscillatory behaviour depending on whether the parameter τ is greater than or less than some critical values. Similarities as well as differences in time evolution of the auto-correlation have been discussed for exponential, secant-hyperbolic and Gaussian approaches of the memory function. (author). 16 refs, 5 figs
Approximate solution to neutron transport equation with linear anisotropic scattering
International Nuclear Information System (INIS)
Coppa, G.; Ravetto, P.; Sumini, M.
1983-01-01
A method to obtain an approximate solution to the transport equation, when both sources and collisions show a linearly anisotropic behavior, is outlined and the possible implications for numerical calculations in applied neutronics as well as shielding evaluations are investigated. The form of the differential system of equations taken by the method is quite handy and looks simpler and more manageable than any other today available technique. To go deeper into the efficiency of the method, some typical calculations concerning critical dimension of multiplying systems are then performed and the results are compared with the ones coming from the classical Ssub(N) approximations. The outcome of such calculations leads us to think of interesting developments of the method which could be quite useful in alternative to other today widespread approximate procedures, for any geometry, but especially for curved ones. (author)
A New Topology of Solutions of Chemical Equations
International Nuclear Information System (INIS)
Risteski, Ice B.
2013-01-01
In this work is induced a new topology of solutions of chemical equations by virtue of point-set topology in an abstract stoichiometrical space. Subgenerators of this topology are the coefficients of chemical reaction. Complex chemical reactions, as those of direct reduction of hematite with a carbon, often exhibit distinct properties which can be interpreted as higher level mathematical structures. Here we used a mathematical model that exploits the stoichiometric structure, which can be seen as a topology too, to derive an algebraic picture of chemical equations. This abstract expression suggests exploring the chemical meaning of topological concept. Topological models at different levels of realism can be used to generate a large number of reaction modifications, with a particular aim to determine their general properties. The more abstract the theory is, the stronger the cognitive power is
Numerical solution of three-dimensional magnetic differential equations
International Nuclear Information System (INIS)
Reiman, A.H.; Greenside, H.S.
1987-02-01
A computer code is described that solves differential equations of the form B . del f = h for a single-valued solution f, given a toroidal three-dimensional divergence-free field B and a single-valued function h. The code uses a new algorithm that Fourier decomposes a given function in a set of flux coordinates in which the field lines are straight. The algorithm automatically adjusts the required integration lengths to compensate for proximity to low order rational surfaces. Applying this algorithm to the Cartesian coordinates defines a transformation to magnetic coordinates, in which the magnetic differential equation can be accurately solved. Our method is illustrated by calculating the Pfirsch-Schlueter currents for a stellarator
Multiple solutions for the Schwarzian Korteweg-de Vries equation in (2 + 1) dimensions
International Nuclear Information System (INIS)
Ramirez, J.; Romero, J.L.; Bruzon, M.S.; Gandarias, M.L.
2007-01-01
In this paper we find new families of solutions for the (2 + 1)-dimensional integrable Schwarzian Korteweg-de Vries equation, that depend up to two arbitrary functions and a solution of a Riemann wave equation. Some of these solutions exhibit a rich dynamic, with a wide variety of qualitative behavior and structures that are exponentially localized. We have also found several families of overturning and intertwining solutions for the equation, that correspond to the nonconstant solutions of Riemann equations
Numerical solution of modified differential equations based on symmetry preservation.
Ozbenli, Ersin; Vedula, Prakash
2017-12-01
In this paper, we propose a method to construct invariant finite-difference schemes for solution of partial differential equations (PDEs) via consideration of modified forms of the underlying PDEs. The invariant schemes, which preserve Lie symmetries, are obtained based on the method of equivariant moving frames. While it is often difficult to construct invariant numerical schemes for PDEs due to complicated symmetry groups associated with cumbersome discrete variable transformations, we note that symmetries associated with more convenient transformations can often be obtained by appropriately modifying the original PDEs. In some cases, modifications to the original PDEs are also found to be useful in order to avoid trivial solutions that might arise from particular selections of moving frames. In our proposed method, modified forms of PDEs can be obtained either by addition of perturbation terms to the original PDEs or through defect correction procedures. These additional terms, whose primary purpose is to enable symmetries with more convenient transformations, are then removed from the system by considering moving frames for which these specific terms go to zero. Further, we explore selection of appropriate moving frames that result in improvement in accuracy of invariant numerical schemes based on modified PDEs. The proposed method is tested using the linear advection equation (in one- and two-dimensions) and the inviscid Burgers' equation. Results obtained for these tests cases indicate that numerical schemes derived from the proposed method perform significantly better than existing schemes not only by virtue of improvement in numerical accuracy but also due to preservation of qualitative properties or symmetries of the underlying differential equations.
Possible and impossible solutions of the Navier-Stokes equations
International Nuclear Information System (INIS)
Irmay, S.
1998-01-01
Flow of isochoric constant-viscosity fluids obeys continuity and the Navier-Stokes equations. They are difficult to solve being nonlinear with a nonslip boundary condition at solid walls. Berker presented many solutions, but some of them, e.g. irrotational velocity, contradict the repulsion condition. Radial flow, possible between two nonparallel planes, is shown to be impossible in a cone, though an approximate solution exists. Parallel (equidistant) streamlines are possible only if rectilinear, concentric or coaxial circles, or helices of equal inclination on coaxial cylinders. Two-way flows resemble ideal and Stokes flows. The author presents some spatial jets impacting on a fixed or parallelly moving boundary. A general unsteady spatial solution near a plane boundary is expressed as power series of z, distance from the wall, which shows most boundary layer solutions to be valid only up to z 2 terms. Uniform steady-state flow at a constant piezo metric gradient in the x-direction, between nonparallel planes, has a definite solution only up to second-order terms in (y,z), due to undefined boundary condition at ∞. Acceleration averaged over time gives insight into the properties of pseudoturbulent or chaotic (turbulent) flows. Turbulent shear is redefined and Reynolds (turbulent) stresses loose their meaning
Solutions for a diffusion equation with a backbone term
International Nuclear Information System (INIS)
Tateishi, A A; Lenzi, E K; Ribeiro, H V; Evangelista, L R; Mendes, R S; Da Silva, L R
2011-01-01
We investigate the diffusion equation ∂ t ρ=D y ∂ y 2 ρ+D x ∂ x 2 ρ+ D-bar x δ(y)∂ x μ ρ subjected to the boundary conditions ρ(±∞,y;t)=0 and ρ(x,±∞;t)=0, and the initial condition ρ(x,y;0)= ρ-hat (x,y). We obtain exact solutions in terms of the Green function approach and analyze the mean square displacement in the x and y directions. This analysis shows an anomalous spreading of the system which is characterized by different diffusive regimes connected to anomalous diffusion
Quantum effects from topological conditions in solutions of Einstein equations
Patiño, L
2003-01-01
In this paper it is shown that Dirac's approach to the quantization of the electric charge can be extended to gravitational configurations by defining a phase-like object related to the curvature of the space-time. Using this phase-like object, Dirac's argument is applied to the Kerr-Newmann and the Taub-NUT solutions to Einstein equations. As a result of this procedure we obtain that certain functions of the parameters entering the metric become quantized. Also, the phase acquired by an observer traveling along a loop around a curvature singularity is quantized. (Author)
Oscillation of solutions of some higher order linear differential equations
Directory of Open Access Journals (Sweden)
Hong-Yan Xu
2009-11-01
Full Text Available In this paper, we deal with the order of growth and the hyper order of solutions of higher order linear differential equations $$f^{(k}+B_{k-1}f^{(k-1}+\\cdots+B_1f'+B_0f=F$$ where $B_j(z (j=0,1,\\ldots,k-1$ and $F$ are entire functions or polynomials. Some results are obtained which improve and extend previous results given by Z.-X. Chen, J. Wang, T.-B. Cao and C.-H. Li.
Existence of infinitely many radial solutions for quasilinear Schrodinger equations
Directory of Open Access Journals (Sweden)
Gui Bao
2014-10-01
Full Text Available In this article we prove the existence of radial solutions with arbitrarily many sign changes for quasilinear Schrodinger equation $$ -\\sum_{i,j=1}^{N}\\partial_j(a_{ij}(u\\partial_iu +\\frac{1}{2}\\sum_{i,j=1}^{N}a'_{ij}(u\\partial_iu\\partial_ju+V(xu =|u|^{p-1}u,~x\\in\\mathbb{R}^N, $$ where $N\\geq3$, $p\\in(1,\\frac{3N+2}{N-2}$. The proof is accomplished by using minimization under a constraint.
Multigrid solution of diffusion equations on distributed memory multiprocessor systems
International Nuclear Information System (INIS)
Finnemann, H.
1988-01-01
The subject is the solution of partial differential equations for simulation of the reactor core on high-performance computers. The parallelization and implementation of nodal multigrid diffusion algorithms on array and ring configurations of the DIRMU multiprocessor system is outlined. The particular iteration scheme employed in the nodal expansion method appears similarly efficient in serial and parallel environments. The combination of modern multi-level techniques with innovative hardware (vector-multiprocessor systems) provides powerful tools needed for real time simulation of physical systems. The parallel efficiencies range from 70 to 90%. The same performance is estimated for large problems on large multiprocessor systems being designed at present. (orig.) [de
Numerical solutions of differential equations of an ionization chamber
International Nuclear Information System (INIS)
Novkovic, D.; Tomasevic, M.; Subotic, K.; Manic, S.
1998-01-01
A system of reduced differential equations generally valid for plane-parallel, cylindrical, and spherical ionization chambers filled with air, which is appropriate for numerical solution, has been derived. The system has been solved for all three geometries. The comparison of the calculated results of Armstrong and Tate, for plane-parallel ionization chambers, and Sprinkle and Tate, for spherical ionization chambers, with the present calculations has shown a good agreement. The calculated values for ionization chambers filled with CO 2 were also in good agreement with the experimental data of Moriuchi et al (author)
A new method for the solution of the Schroedinger equation
International Nuclear Information System (INIS)
Amore, Paolo; Aranda, Alfredo; De Pace, Arturo
2004-01-01
We present a new method for the solution of the Schroedinger equation applicable to problems of a non-perturbative nature. The method works by identifying three different scales in the problem, which then are treated independently: an asymptotic scale, which depends uniquely on the form of the potential at large distances; an intermediate scale, still characterized by an exponential decay of the wavefunction; and, finally, a short distance scale, in which the wavefunction is sizable. The notion of optimized perturbation is then used in the last two regimes. We apply the method to the quantum anharmonic oscillator and find it suitable to treat both energy eigenvalues and wavefunctions, even for strong couplings
On the solutions of fractional reaction-diffusion equations
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Jagdev Singh
2013-05-01
Full Text Available In this paper, we obtain the solution of a fractional reaction-diffusion equation associated with the generalized Riemann-Liouville fractional derivative as the time derivative and Riesz-Feller fractional derivative as the space-derivative. The results are derived by the application of the Laplace and Fourier transforms in compact and elegant form in terms of Mittag-Leffler function and H-function. The results obtained here are of general nature and include the results investigated earlier by many authors.
Adib, Arash; Poorveis, Davood; Mehraban, Farid
2018-03-01
In this research, two equations are considered as examples of hyperbolic and elliptic equations. In addition, two finite element methods are applied for solving of these equations. The purpose of this research is the selection of suitable method for solving each of two equations. Burgers' equation is a hyperbolic equation. This equation is a pure advection (without diffusion) equation. This equation is one-dimensional and unsteady. A sudden shock wave is introduced to the model. This wave moves without deformation. In addition, Laplace's equation is an elliptical equation. This equation is steady and two-dimensional. The solution of Laplace's equation in an earth dam is considered. By solution of Laplace's equation, head pressure and the value of seepage in the directions X and Y are calculated in different points of earth dam. At the end, water table is shown in the earth dam. For Burgers' equation, least-square method can show movement of wave with oscillation but Galerkin method can not show it correctly (the best method for solving of the Burgers' equation is discrete space by least-square finite element method and discrete time by forward difference.). For Laplace's equation, Galerkin and least square methods can show water table correctly in earth dam.
Sign-changing solutions for non-local elliptic equations
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Huxiao Luo
2017-07-01
Full Text Available This article concerns the existence of sign-changing solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions, $$\\displaylines{ -\\mathcal{L}_Ku=f(x,u,\\quad x\\in \\Omega, \\cr u=0,\\quad x\\in \\mathbb{R}^n\\setminus\\Omega, }$$ where $\\Omega\\subset\\mathbb{R}^n\\; (n\\geq2$ is a bounded, smooth domain and the nonlinear term f satisfies suitable growth assumptions. By using Brouwer's degree theory and Deformation Lemma and arguing as in [2], we prove that there exists a least energy sign-changing solution. Our results generalize and improve some results obtained in [27
Blowing up solutions for a biharmonic equation with critical nonlinearity
International Nuclear Information System (INIS)
El Mehdi, K.; Hammami, M.
2004-05-01
In this paper we consider the following biharmonic equation with critical exponent (P ε ):Δ 2 u=Ku (n+4)/(n-4)-ε , u>0 in Ω and u=Δu=0 on ∂Ω, where Ω is a smooth bounded domain in R n , n≥5, ε is a small positive parameter, and K is a smooth positive function in Ω-bar. We construct solutions of (P ε ) which blow up and concentrate at strict local maximum of K either at the boundary or in the interior of Ω. We also construct solutions of (P ε ) concentrating at an interior strict local minimum point of K. Finally, we prove a nonexistence result for the supercritical problem which is in sharp contrast to what happened for (P ε ). (author)
On the Solution of the Neutron Transport Equation
Energy Technology Data Exchange (ETDEWEB)
Depken, S
1962-12-15
The neutron transport equation has occupied the attention of many authors since Placzek, Wick and others made their first attempts to solve it, Even in the simple case of energy independent cross-sections, and disregarding the motion of the scattering nucleons, it is difficult to find a solution in an analytical form which is easily surveyable and fitted for numerical calculations. In Part I of this paper some new viewpoints will be introduced which enable the solution to be presented in its simplest possible form. Part II is devoted to an investigation of some functions introduced in Part I. In Part III the results are applied to the case of large energy lethargy, and the validity of derived formulas is discussed.
Intertwining solutions for magnetic relativistic Hartree type equations
Cingolani, Silvia; Secchi, Simone
2018-05-01
We consider the magnetic pseudo-relativistic Schrödinger equation where , m > 0, is an external continuous scalar potential, is a continuous vector potential and is a convolution kernel, is a constant, , . We assume that A and V are symmetric with respect to a closed subgroup G of the group of orthogonal linear transformations of . If for any , the cardinality of the G-orbit of x is infinite, then we prove the existence of infinitely many intertwining solutions assuming that is either linear in x or uniformly bounded. The results are proved by means of a new local realization of the square root of the magnetic laplacian to a local elliptic operator with Neumann boundary condition on a half-space. Moreover we derive an existence result of a ground state intertwining solution for bounded vector potentials, if G admits a finite orbit.
Exact Solutions to (2+1)-Dimensional Kaup-Kupershmidt Equation
International Nuclear Information System (INIS)
Lu Hailing; Liu Xiqiang
2009-01-01
In this paper, by using the symmetry method, the relationships between new explicit solutions and old ones of the (2+1)-dimensional Kaup-Kupershmidt (KK) equation are presented. We successfully obtain more general exact travelling wave solutions for (2+1)-dimensional KK equation by the symmetry method and the (G'/G)-expansion method. Consequently, we find some new solutions of (2+1)-dimensional KK equation, including similarity solutions, solitary wave solutions, and periodic solutions. (general)
Localized solutions for a nonlocal discrete NLS equation
Energy Technology Data Exchange (ETDEWEB)
Ben, Roberto I. [Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150, 1613 Los Polvorines (Argentina); Cisneros Ake, Luís [Department of Mathematics, ESFM, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos Edificio 9, 07738 México D.F. (Mexico); Minzoni, A.A. [Depto. Matemáticas y Mecánica, I.I.M.A.S.-U.N.A.M., Apdo. Postal 20-726, 01000 México D.F. (Mexico); Panayotaros, Panayotis, E-mail: panos@mym.iimas.unam.mx [Depto. Matemáticas y Mecánica, I.I.M.A.S.-U.N.A.M., Apdo. Postal 20-726, 01000 México D.F. (Mexico)
2015-09-04
We study spatially localized time-periodic solutions of breather type for a cubic discrete NLS equation with a nonlocal nonlinearity that models light propagation in a liquid crystal waveguide array. We show the existence of breather solutions in the limit where both linear and nonlinear intersite couplings vanish, and in the limit where the linear coupling vanishes with arbitrary nonlinear intersite coupling. Breathers of this nonlocal regime exhibit some interesting features that depart from what is seen in the NLS breathers with power nonlinearity. One property we see theoretically is the presence of higher amplitude at interfaces between sites with zero and nonzero amplitude in the vanishing linear coupling limit. A numerical study also suggests the presence of internal modes of orbitally stable localized modes. - Highlights: • Show existence of spatially localized solutions in nonlocal discrete NLS model. • Study spatial properties of localized solutions for arbitrary nonlinear nonlocal coupling. • Present numerical evidence that nonlocality leads to internal modes around stable breathers. • Present theoretical and numerical evidence for amplitude maxima at interfaces.
Localized solutions for a nonlocal discrete NLS equation
International Nuclear Information System (INIS)
Ben, Roberto I.; Cisneros Ake, Luís; Minzoni, A.A.; Panayotaros, Panayotis
2015-01-01
We study spatially localized time-periodic solutions of breather type for a cubic discrete NLS equation with a nonlocal nonlinearity that models light propagation in a liquid crystal waveguide array. We show the existence of breather solutions in the limit where both linear and nonlinear intersite couplings vanish, and in the limit where the linear coupling vanishes with arbitrary nonlinear intersite coupling. Breathers of this nonlocal regime exhibit some interesting features that depart from what is seen in the NLS breathers with power nonlinearity. One property we see theoretically is the presence of higher amplitude at interfaces between sites with zero and nonzero amplitude in the vanishing linear coupling limit. A numerical study also suggests the presence of internal modes of orbitally stable localized modes. - Highlights: • Show existence of spatially localized solutions in nonlocal discrete NLS model. • Study spatial properties of localized solutions for arbitrary nonlinear nonlocal coupling. • Present numerical evidence that nonlocality leads to internal modes around stable breathers. • Present theoretical and numerical evidence for amplitude maxima at interfaces
Homothetic and conformal symmetries of solutions to Einstein's equations
International Nuclear Information System (INIS)
Eardley, D.; Isenberg, J.; Marsden, J.; Moncrief, V.; Yale Univ., New Haven, CT
1986-01-01
We present several results about the nonexistence of solutions of Einstein's equations with homoethetic or conformal symmetry. We show that the only spatially compact, globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature, and also admitting an infinitesimal proper homothetic symmetry, are everywhere locally flat; this assumes that the matter fields either obey certain energy conditions, or are the Yang-Mills or massless Klein-Gordon fields. We find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions. We show that if the dominant energy condition is assumed, then Minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation. In other words, with the exceptions cited, homothetic or conformal Killing fields are in fact Killing in spatially compact or asymptotically flat spacetimes. In the conformal procedure for solving the initial value problem, we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry. (orig.)
Minimal parameter solution of the orthogonal matrix differential equation
Bar-Itzhack, Itzhack Y.; Markley, F. Landis
1990-01-01
As demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed emplying the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix.