Solving linear and nonlinear klein-gordon equations by new perturbation iteration transform method

We present an effective algorithm to solve the Linear and Nonlinear KleinGordon equation, which is based on the Perturbation Iteration Transform Method (PITM). The Klein-Gordon equation is the name given to the equation of motion of a quantum scalar or pseudo scalar field, a field whose quanta are spin-less particles. It describes the quantum amplitude for finding a point particle in various places, the relativistic wave function, but the particle propagates both forwards and backwards in time. The Perturbation Iteration Transform Method (PITM) is a combined form of the Laplace Transform Method and Perturbation Iteration Algorithm. The method provides the solution in the form of a rapidly convergent series. Some numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. The results show that the PITM is very efficient, simple and can be applied to other nonlinear problems.Publisher's Versio

A new analysis for Klein-Gordon model with local fractional derivative

Abstract This work adopts Yang's local fractional derivative to define the fractional Klein-Gordon equation in a fractal space or microgravity space. The variational principle of local fractional Klein-Gordon equation is successfully established via fractional semi-inverse transform method and the classical He's variational iteration method (HVIM) is used to obtain its approximate analytical solution

Analytical study of time-fractional order Klein–Gordon equation

AbstractIn this article, we study an approximate analytical solution of linear and nonlinear time-fractional order Klein–Gordon equations by using a recently developed semi analytical method referred as fractional reduced differential transform method with appropriate initial condition. In the study of fractional Klein–Gordon equation, fractional derivative is described in the Caputo sense. The validity and efficiency of the aforesaid method are illustrated by considering three computational examples. The solution profile behavior and effects of different fraction Brownian motion on solution profile of the three numerical examples are shown graphically

Matrix product states and variational methods applied to critical quantum field theory

We study the second-order quantum phase-transition of massive real scalar field theory with a quartic interaction ($\phi^4$ theory) in (1+1) dimensions on an infinite spatial lattice using matrix product states (MPS). We introduce and apply a naive variational conjugate gradient method, based on the time-dependent variational principle (TDVP) for imaginary time, to obtain approximate ground states, using a related ansatz for excitations to calculate the particle and soliton masses and to obtain the spectral density. We also estimate the central charge using finite-entanglement scaling. Our value for the critical parameter agrees well with recent Monte Carlo results, improving on an earlier study which used the related DMRG method, verifying that these techniques are well-suited to studying critical field systems. We also obtain critical exponents that agree, as expected, with those of the transverse Ising model. Additionally, we treat the special case of uniform product states (mean field theory) separately, showing that they may be used to investigate non-critical quantum field theories under certain conditions.Comment: 24 pages, 21 figures, with a minor improvement to the QFT sectio

PT-symmetric operators and metastable states of the 1D relativistic oscillators

We consider the one-dimensional Dirac equation for the harmonic oscillator and the associated second order separated operators giving the resonances of the problem by complex dilation. The same operators have unique extensions as closed PT-symmetric operators defining infinite positive energy levels converging to the Schroedinger ones as c tends to infinity. Such energy levels and their eigenfunctions give directly a definite choice of metastable states of the problem. Precise numerical computations shows that these levels coincide with the positions of the resonances up to the order of the width. Similar results are found for the Klein-Gordon oscillators, and in this case there is an infinite number of dynamics and the eigenvalues and eigenvectors of the PT-symmetric operators give metastable states for each dynamics.Comment: 13 pages, 2 figure