Fractional telegraph equations (original) (raw)

Fractional Diffusion--Telegraph Equations and Their Associated Stochastic Solutions

Theory of Probability & Its Applications, 2018

We present the stochastic solution to a generalized fractional partial differential equation involving a regularized operator related to the so-called Prabhakar operator and admitting, amongst others, as specific cases the fractional diffusion equation and the fractional telegraph equation. The stochastic solution is expressed as a Lévy process time-changed with the inverse process to a linear combination of (possibly subordinated) independent stable subordinators of different indices. Furthermore a related SDE is derived and discussed.

Fractional and nonlinear diffusion equation: additional results

Physica A: Statistical Mechanics and its Applications, 2004

We investigate the solutions of a generalized di usion equation which extends some known equations such as the fractional di usion equation and the porous medium equation. We start our study by considering the linear case and the nonlinear case afterward. The linear case is analyzed taking fractional time and spatial derivatives into account. In this context, we also discuss the modiÿcations that emerge by considering a di usion coe cient given by D(x)˙|x| −Â. For the nonlinear case accomplishing the fractional time derivative, we discuss scaling behavior of the time and the asymptotic for the solution of the nonlinear fractional di usion equation. In this case, the connection between the asymptotic solution found here and the nonextensive Tsallis statistics is performed.

Time-Changed Processes Governed by Space-Time Fractional Telegraph Equations

Stochastic Analysis and Applications, 2014

In this work we construct compositions of processes of the form S 2β n c 2 L ν (t) , t > 0, ν ∈ 0, 1 2 , β ∈ (0, 1], n ∈ N, whose distribution is related to space-time fractional n-dimensional telegraph equations. We present within a unifying framework the pde connections of n-dimensional isotropic stable processes S 2β n whose random time is represented by the inverse L ν (t), t > 0, of the superposition of independent positively-skewed stable processes, H ν (t) = H 2ν 1 (t) + (2λ) 1 ν H ν 2 (t), t > 0, (H 2ν 1 , H ν 2 , independent stable subordinators). As special cases for n = 1, ν = 1 2 and β = 1 we examine the telegraph process T at Brownian time B (Orsingher and Beghin [20]) and establish the equality in distribution B c 2 L 1 2 (t) law = T (|B(t)|), t > 0. Furthermore the iterated Brownian motion (Allouba and Zheng [2]) and the two-dimensional motion at finite velocity with a random time are investigated. For all these processes we present their counterparts as Brownian motion at delayed stable-distributed time. The last section of the paper is devoted to the interplay between timefractional hyperbolic equations and processes defined on the n-dimensional Poincaré half-space. Contents 1 ν H ν 2 (t) 2.1. The inverse process L ν (t) 3. n-dimensional stable laws and fractional Laplacian 4. Space-time fractional telegraph equation 4.1. The case ν = 1 2 , subordinator with drift 4.2. The case ν = 1 3 , convolutions of Airy functions 4.3. The planar case 5. Hyperbolic fractional telegraph equations References

Analysis of Solutions of Time Fractional Telegraph Equation

Journal of the Korea Society for Industrial and Applied Mathematics, 2014

In this paper, the solution of time fractional telegraph equation is obtained by using Adomain decomposition method and compared with various other method to determine the efficiency of Adomain decomposition method. These methods are used to obtain the series solutions. Finally, results are analysed by plotting the solutions for various fractional orders.

Time-fractional telegraph equations and telegraph processes with brownian time

Probability Theory and Related Fields, 2004

We study the fundamental solutions to time-fractional telegraph equations of order 2α. We are able to obtain the Fourier transform of the solutions for any α and to give a representation of their inverse, in terms of stable densities. For the special case α=1/2, we can show that the fundamental solution is the distribution of a telegraph process with Brownian time. In a special case, this becomes the density of the iterated Brownian motion, which is therefore the fundamental solution to a fractional diffusion equation of order 1/2 with respect to time.

Analytical study of fractional equations describing anomalous diffusion of energetic particles

Journal of Physics: Conference Series, 2017

To present the main influence of anomalous diffusion on the energetic particle propagation, the fractional derivative model of transport is developed by deriving the fractional modified Telegraph and Rayleigh equations. Analytical solutions of the fractional modified Telegraph and the fractional Rayleigh equations, which are defined in terms of Caputo fractional derivatives, are obtained by using the Laplace transform and the Mittag-Leffler function method. The solutions of these fractional equations are given in terms of special functions like Fox's H, Mittag-Leffler, Hermite and Hyper-geometric functions. The predicted travelling pulse solutions are discussed in each case for different values of fractional order.

An Approximate-Analytical Solution to Analyze Fractional View of Telegraph Equations

IEEE Access, 2020

In the present research article, a modified analytical method is applied to solve time-fractional telegraph equations. The Caputo-operator is used to express the derivative of fractional-order. The present method is the combination of two well-known methods namely Mohan transformation method and Adomian decomposition method. The validity of the proposed technique is confirmed through illustrative examples. It is observed that the obtained solutions have strong contact with the exact solution of the examples. Moreover, it is investigated that the present method has the desired degree of accuracy and provided the graphs closed form solutions of all targeted examples. The graphs have verified the convergence analysis of fractional-order solutions to integer-order solution. In conclusion, the suggested method is simple, straightforward and an effective technique to solve fractional-order partial differential equations. INDEX TERMS Mohand transformation, telegraph equations, Adomian decomposition method, Caputo operator.