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时间分数阶Klein-Gordon型方程的解析近似解  [PDF]
郭鹏,王艺红,陶春兴,李常品
科技导报 , 2009,
Abstract: 采用Adomian分裂方法,给出在Caputo导数意义下的时间分数阶Klein-Gordon方程的解析近似解,并举例说明了Adomian分裂方法在求解上的高效性,通过4个表给出的近似解和精确解的误差,可以看出Adomian分裂方法在求解时间分数阶Klein-Gordon方程时能得到很高的精度。
Explicit Solution For Klein-Gordon Equation, in Four Dimensions, For any Arbitrary potential. A New Approach  [PDF]
Saeed Otarod
Physics , 2003,
Abstract: Klein-Gordon Equation has been solved in four dimension. The potential has been chosen to be any arbitrary field Potential.
Casimir Effect Associated with Fractional Klein-Gordon Field  [PDF]
S. C. Lim,L. P. Teo
Physics , 2011,
Abstract: This paper gives a brief review on the recent work on fractional Klein-Gordon fields, in particular on the Casimir effect associated to fractional Klein-Gordon fields in various geometries and boundary conditions. New results on Casimir piston due to a fractional Klein-Gordon massive field are given.
Fractional Klein-Gordon equation for linear dispersive phenomena: analytical methods and applications  [PDF]
Roberto Garra,Enzo Orsingher,Federico Polito
Mathematics , 2013, DOI: 10.1109/ICFDA.2014.6967381
Abstract: In this paper we discuss some exact results related to the fractional Klein--Gordon equation involving fractional powers of the D'Alembert operator. By means of a space-time transformation, we reduce the fractional Klein--Gordon equation to a fractional hyper-Bessel-type equation. We find an exact analytic solution by using the McBride theory of fractional powers of hyper-Bessel operators. A discussion of these results within the framework of linear dispersive wave equations is provided. We also present exact solutions of the fractional Klein-Gordon equation in the higher dimensional cases. Finally, we suggest a method of finding travelling wave solutions of the nonlinear fractional Klein-Gordon equation with power law nonlinearities.
New explicit exact solution of one type of the sine-Gordon equation with self-consistent source
一类带自相容源的sine-Gordon方程新的显式精确解

Su Jun,Xu Wei,Duan Dong-Hai,Xu Gen-Jiu,
苏军
,徐伟,段东海,徐根玖

物理学报 , 2011,
Abstract: This paper deals with one type of sine-Gordon with self-consistent source (SGESCS). The explicit exact solution of the equation is investigated using a generalized binary Darboux transformation. The complexiton solution for the equation is finally obtained.
Fractional Klein-Gordon equations and related stochastic processes  [PDF]
Roberto Garra,Enzo Orsingher,Federico Polito
Mathematics , 2013, DOI: 10.1007/s10955-014-0976-0
Abstract: This paper presents finite-velocity random motions driven by fractional Klein-Gordon equations of order $\alpha \in (0,1]$. A key tool in the analysis is played by the McBride's theory which converts fractional hyper-Bessel operators into Erdelyi-Kober integral operators. Special attention is payed to the fractional telegraph process whose space-dependent distribution solves a non-homogeneous fractional Klein-Gordon equation. The distribution of the fractional telegraph process for $\alpha = 1$ coincides with that of the classical telegraph process and its driving equation converts into the homogeneous Klein-Gordon equation. Fractional planar random motions at finite velocity are also investigated, the corresponding distributions obtained as well as the explicit form of the governing equations. Fractionality is reflected into the underlying random motion because in each time interval a binomial number of deviations $B(n,\alpha)$ (with uniformly-distributed orientation) are considered. The parameter $n$ of $B(n,\alpha)$ is itself a random variable with fractional Poisson distribution, so that fractionality acts as a subsampling of the changes of directions. Finally the behaviour of each coordinate of the planar motion is examined and the corresponding densities obtained. Extensions to $N$-dimensional fractional random flights are envisaged as well as the fractional counterpart of the Euler-Poisson-Darboux equation to which our theory applies.
A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein-Gordon Equation of Arbitrary (Fractional) Orders
A. M. A. El-Sayed,A. Elsaid,D. Hammad
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/581481
Abstract: The reliable treatment of homotopy perturbation method (HPM) is applied to solve the Klein-Gordon partial differential equation of arbitrary (fractional) orders. This algorithm overcomes the difficulty that arises in calculating complicated integrals when solving nonlinear equations. Some numerical examples are presented to illustrate the efficiency of this technique.
Difficulties with the Klein-Gordon Equation  [PDF]
E. Comay
Physics , 2003,
Abstract: Relying on the variational principle, it is proved that new contradictions emerge from an analysis of the Lagrangian density of the Klein-Gordon field: normalization problems arise and interaction with external electromagnetic fields cannot take place. By contrast, the Dirac equation is free of these problems. Other inconsistencies arise if the Klein-Gordon field is regarded as a classical field.
Asymptotic properties of solutions of the Maxwell Klein Gordon equation with small data  [PDF]
Lydia Bieri,Shuang Miao,Sohrab Shahshahani
Mathematics , 2014,
Abstract: We prove peeling estimates for the small data solutions of the Maxwell Klein Gordon equations with non-zero charge and with a non-compactly supported scalar field, in $(3+1)$ dimensions. We obtain the same decay rates as in an earlier work by Lindblad and Sterbenz, but giving a simpler proof. In particular we dispense with the fractional Morawetz estimates for the electromagnetic field, as well as certain space-time estimates. In the case that the scalar field is compactly supported we can avoid fractional Morawetz estimates for the scalar field as well. All of our estimates are carried out using the double null foliation and in a gauge invariant manner.
Notes On The Klein-Gordon Equation  [PDF]
Fredrick Michael
Physics , 2010,
Abstract: In this article, we derive the scalar parametrized Klein-Gordon equation from the formal information theory framework. The least biased probability distribution is obtained, and the scalar equation is recast in terms of a Fokker-Planck equation in terms of the imaginary time, or a Schroedinger equation for the proper time. This method yields the Green's function parametrized by an evolution parameter. The derivation can then allow the use of potentials as constraints along with the Hamiltonian or moments of the evolution. The information theoretic, analogously the maximum entropy method, also allows one to examine the possibility of utilizing generalized and non-extensive statistics in the derivation. This approach yields non-linear evolution in the parametrized Klein-Gordon partial differential equations. Furthermore, we examine the Klein-Gordon equation in curved space-time, and we compare our results to the results of Schwinger and Dewitt obtained from path integral approaches.
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