Spectral Analysis of Fractional Hyperbolic Diffusion Equations with Random Data

Journal of Statistical Physics - The paper studies the fundamental solutions to fractional in time hyperbolic diffusion equation or telegraph equations and their properties. Then it derives the...     

On the theory of classic mesodiffusion

A simple model of the classical random walk of particles with a constant speed and anisotropic angular distribution is used to study the characteristic features of mesodiffusion, that is, of an intermediate stage between the ballistic regime (short times) and ordinary diffusion (long times). In the extreme case of anisotropy, namely, walking along a straight line, the process can be described by the telegraph equation, whose solution contains δ-functions accounting for the ballistic component. As the anisotropy becomes less pronounced, the δ-singularity transforms into a frontal burst (the quasi-ballistic component), beyond which the distribution can be satisfactorily described by the telegraph approximation. In the other extreme case of isotropic walking, the frontal burst disappears and the telegraph approximation, contrary to general belief, proves to be cruder than the diffusion approximation.     

A stochastic model of Maxwell's equations in 1+1 dimensions

We show that the random walk model due to Mark Kac which underlies the telegraph equations may be modified to produce Maxwell's field equations in 1+1 dimensions. This provides the field equations with a representation in terms of classical particles. It also establishes the Kac model as a strong conceptual link between the diffusion, telegraph, and Maxwell equations, and suggests that recent simulations of the Schrödinger and Dirac equations are analogous to Maxwell's equation in terms of interpretation.     

Autowaves in double-wire lines with the exponential-type nonlinear active element

The regimes corresponding to the appearance of localized excitation pulses in a nonlinear double-wire line with an exponential-type active element similar to that occurring in the distributed p-n junctions and nerve fibers are studied on the basis of exact solutions. It is shown that the line of this type is described by the nonlinear telegraph equation if there is a running inductance and by the one-dimensional nonlinear diffusion equation if it is absent. The main properties of the excitation waves and conditions for their appearance are examined.     

Exact Analytic Solution of the Simplified Telegraph Model of Propagation and Dissipation of Excitation Fronts

The telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion in several branches of sciences (E. Ahmed, H. A. Abdusalam, and E. S. Fahmy, 2001, Int. J. Mod. Phys. C 12(5), 717; E. Ahmed and H. A. Abdusalam, 2004, Chaos, Solitons and Fractals 22, 583; H. A. Abdusalam and E. S. Fahmy, 2003, Chaos, Solitons and Fractals 18, 259; Abdusalam, 2004 Appl. Math. Comp. 157, 515). An excitation wave in cardiac tissue fails to propagate if the transmembrane voltage at its front rises too slow and does not excite the tissue ahead of it. Then the sharp voltage profile of the front will dissipate, and subsequent spread of voltage will be purely diffusive. This mechanism is impossible in FitzHugh–Nagumo type system (V. N. Biktashev, 2003, Int. J. Bifarcation and Chaos 13(12), 3605). Biktashev suggested a simplified mathematical model for this mechanism and in the present work we generalize this model to telegraph system. Our generalized telegraph model has exact traveling front solutions and we show the effect of the time delay on the velocity and we show that, the post-front voltage depends on two parameters in which one of them is the time delay.     

Adomian decomposition method for solving the telegraph equation in charged particle transport

In this paper, the analysis for the telegraph equation in case of isotropic small angle scattering from the Boltzmann transport equation for charged particle is presented. The Adomian decomposition is used to solve the telegraph equation. By means of MAPLE the Adomian polynomials of obtained series (ADM) solution have been calculated. The behaviour of the distribution function are shown graphically. The results reported in this article provide further evidence of the usefulness of Adomain decomposition for obtaining solution of linear and nonlinear problems.     

频率受随机过程调制的体系中的光谱扩散

通过在Bloch方程中增加描述对频率微扰的随机过程项{ω(t)},我们讨论了系统在窄谱带的光激发后的光谱扩散。特别是以下两种情况:1)如果ω以相同的概率变化为系综中的任一ω’,光谱是一个指数式衰减的定域峰和指数式增长的动态非均匀背底的叠加。2)如果ω'到ω的概率仅与|ω’-ω|有关,非定域峰随时间变宽。作为一个例子,我们讨论了频率受多个独立随机电报过程调制的系统中的光谱扩散。     

A Harmonic Solution for the Hyperbolic Heat Conduction Equation and Its Relationship to the Guyer–Krumhansl Equation

A particular solution of the hyperbolic heat-conduction equation was constructed using the method of operators. The evolution of a harmonic solution is studied, which simulates the propagation of electric signals in long wire transmission lines. The structures of the solutions of the telegraph equation and of the Guyer–Krumhansl equation are compared. The influence of the phonon heat-transfer mechanism in the environment is considered from the point of view of heat conductivity. The fulfillment of the maximum principle for the obtained solutions is considered. The frequency dependences of heat conductivity in the telegraph equation and in an equation of the Guyer–Krumhansl type are studied and compared with each other. The influence of the Knudsen number on heat conductivity in the model of thin films is studied.     

Approximate solution of two-term fractional-order diffusion,wave-diffusion,and telegraph models arising in mathematical physics using optimal homotopy asymptotic method

This paper deals with the investigation of the analytical approximate solutions for two-term fractional-order diffusion, wave-diffusion, and telegraph equations. The fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], (1,2), and [1,2], respectively. In this paper, we extended optimal homotopy asymptotic method (OHAM) for two-term fractional-order wave-diffusion equations. Highly approximate solution is obtained in series form using this extended method. Approximate solution obtained by OHAM is compared with the exact solution. It is observed that OHAM is a prevailing and convergent method for the solutions of nonlinear-fractional-order time-dependent partial differential problems. The numerical results rendering that the applied method is explicit, effective, and easy to use, for handling more general fractional-order wave diffusion, diffusion, and telegraph problems.     

Series expansion solutions for the multi-term time and space fractional partial differential equations in two- and three-dimensions

Fractional partial differential equations with more than one fractional derivative in time describe some important physical phenomena, such as the telegraph equation, the power law wave equation, or the Szabo wave equation. In this paper, we consider two- and three-dimensional multi-term time and space fractional partial differential equations. The multi-term time-fractional derivative is defined in the Caputo sense, whose order belongs to the interval (1,2],(2,3],(3,4] or (0,m], and the space-fractional derivative is referred to as the fractional Laplacian form. We derive series expansion solutions based on a spectral representation of the Laplacian operator on a bounded region. Some applications are given for the two- and three-dimensional telegraph equation, power law wave equation and Szabo wave equation.     

Random Evolutions Are Driven by the Hyperparabolic Operators

We resolve the long-standing problem of describing the multidimensional random evolutions by means of the telegraph equations. This problem was posed by Mark Kac more than 50 years ago and has become the subject of intense discussion among researchers on whether the multidimensional random flights could be described by the telegraph equations similarly to the one-dimensional case. We give the exhaustive answer to this question and show that the multidimensional random evolutions are driven by the hyperparabolic operators composed of the telegraph operators and their integer powers. The only exception is the 2D random flight whose transition density is the fundamental solution to the two-dimensional telegraph equation. The reason of the exceptionality of the 2D-case is explained. We also show that, under the standard Kac’s condition, the governing hyperparabolic operator turns into the generator of the Brownian motion.     

Velocity of propagation in diffusional quantum theory

An equation of diffusional quantum theory which takes into account the finite velocity of propagation is derived from Kelvin's telegraph equation and Fürth's relation. The equation is then used to derive the ground state of quantum systems and to derive the Sommerfeld-Dirac expression for the ionization potential of hydrogen-like ions.     

Exact travelling wave solutionsof some nonlinear evolutionary equations

A direct algebraic method of obtaining exact solutions to nonlinear PDE's is applied to certain set of nonlinear nonlocal evolutionary equations, including nonlinear telegraph equation, hyperbolic generalization of Burgers equation and some spatially nonlocal hydrodynamic-type model. Special attention is paid to the construction of the kink-like and soliton-like solutions.     

基于区间的传输线感应电压不确定性分析方法

为分析多导体传输线耦合情况下线缆结构参数的不确定性对终端电压的影响,引入了一种基于区间分析的切比雪夫(Chebyshev)多项式逼近方法。该方法首先将传输线电报方程转换为常微分方程求解;其次采用Chebyshev多项式求得电报方程的扩张函数,进而获得终端电压的波动范围。相比于混沌多项式方法和蒙特卡罗（MC）法,此方法只需要输入随机参数的波动范围。针对电磁脉冲辐照下高度和间距随机变动的多导体线束进行仿真,仿真结果表明,间距基本不影响终端电压,终端电压对高度更为敏感。在计算结果基本一致的情况下,Chebyshev多项式逼近方法的计算耗时远小于MC方法。     

频率受随机电报过程调制的系统的相干瞬态过程

由随机电报过程的性质出发,给出了频率受一个随机电报过程调制系统的Bloch方程,得到了这种系统中的自由感应衰减和光子回波衰减.     

The renewal equation for persistent diffusion

Persistent diffusion in one dimension, in which the velocity of the diffusing particle is a dichotomic Markov process, is considered. The flow is non-Markovian, but the position and the velocity together constitute a Markovian diffusion process. We solve the coupled forward Kolmogorov equations and the coupled backward Kolmogorov equations with appropriate initial conditions, to establish a generalized (matrix) form of the renewal equation connecting the probability densities and first passage time distributions for persistent diffusion.     

Two-dimensional theory of elf electromagnetic wave propagation in the earth-ionosphere waveguide channel

We present a surface theory of the 2-D telegraph equation and describe the methods for obtaining parameters L and C of this theory from the electromagnetic field of the dominant normal wave and its propagation constant together with the first two azimuthal derivatives. To excite the waveguide by vertical and horizontal electric dipoles, we determine the external sources of the 2-D telegraph equation, which are the 2-D point external voltage and the oriented point external specific voltage, respectively. The relation between the effective and physical sources is practically independent of the ionospheric conditions. The effective source of the horizontal dipole is proportional to the earth 's surface impedance at the source location.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 39, No. 9, pp. 1103–1113, September, 1996.This work was supported by grant 01.067 of the Competition Center at St. Petersburg State University.     

Two-Dimensional Legendre Wavelets for Solving Time-Fractional Telegraph Equation

In this paper, we develop an accurate and efficient Legendre wavelets method for numerical solution of the well known time-fractional telegraph equation. In the proposed method we have employed both of the operational matrices of fractional integration and differentiation to get numerical solution of the time-telegraph equation. The power of this manageable method is confirmed. Moreover, the use of Legendre wavelet is found to be accurate, simple and fast.     

On conformable double Laplace transform

In this study authors introduce the conformable double Laplace transform which can be used to solve fractional partial differential equations that represents many physical and engineering models. In these models the derivatives and integrals are in the sense of newly defined conformable type. Then some properties of conformable double Laplace transform are expressed. Finally fractional heat equation and fractional telegraph equation which is used in various applications in science and engineering investigated as an application of this new transform.     

Fractional Klein–Gordon Equations and Related Stochastic Processes

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 Erdélyi–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 direction. 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.