Fractional taylor series and fractional initial conditions have been introduced. The fractional derivatives are taken in the caputo sense. In this paper, we consider the numerical solution of the time. In particular, fractional order telegraph equations have been considered and solved for many researchers, using di erent methods.
The equations come from oliver heaviside who developed the transmission line model in the 1880s. An efficient new perturbative laplace method for spacetime. Solving timefractional order telegraph equation via sinc. In order to show the efficiency of the proposed method, we present. The space and time fractional derivatives are considered in caputo sense and the solution is obtained in terms of mittagleffler functions.
Generally, telegraph equations are used within wave propagation of electric signals in a cable transmission line as well as wave phenomena. This method is based on perturbation theory and the laplace transformation. Analytic and approximate solutions of spacetime fractional. Numerical solution of nonlinear spacetime fractionalorder. In this paper, a method of separating variables is effectively implemented for solving a timefractional telegraph equation tfte. The telegraphers equations or just telegraph equations are a pair of coupled, linear partial differential equations that describe the voltage and current on an electrical transmission line with distance and time. In the first step the laplace transform is applied to the time fractional telegraph equation, which reduces the problem to a finite set of elliptic equations which are solved with the help of local radial basis functions method in parallel. Analytical solution for the timefractional telegraph equation. We present a new method to solve time fractional order telegraph equation tfote by using bernstein polynomials. Orsingher and beghin 27 discussed the timefractional telegraph equation. The equations come from oliver heaviside who developed the transmission line model starting with an august 1876 paper, on the extra current 6667 the model demonstrates that the.
Pdf generalized differential transform method to spacetime. The caputo fractional formula is used in the discretization of the time fractional derivative. 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. Timefractional telegraph equation and telegraph processes with brownian time article pdf available in probability theory and related fields 1281.
Consider the timefractional telegraph equation of order. On the approximation of timefractional telegraph equations using. Pdf generalized differential transform method to space. Timefractional telegraph equations and telegraph processes with brownian time 149 in the last step we have applied formula 2. Space time fractional telegraph equations 5 beghin 20. Twodimensional legendre wavelets for solving time fractional telegraph equation volume 6 issue 2 m. The organization of the rest of this article is as follows. We are able to solve the model equations in a simple analytic form by a generalized separating variable method in some particular interesting cases. Xu, existence and uniqueness of the weak solution of the spacetime fractional diffusion equation and a spectral method approximation, commun. In this paper, we are going to study fast high order techniques for the fractional telegraph equation.
A compact scheme is then presented to improve the convergence order in space. The salient feature of the article is the graphical exhibition of the movement of solute concentration for different particular cases due to the presence and absence of reaction term when the proposed scheme is applied to the considered nonlinear fractionalorder spacetime advectionreactiondiffusion model. However, all the previous works avoid the term of fractional initial conditions in the space time telegraph equations. The fractional derivative is expressed in the caputo definition of fractional derivatives. Mar 12, 2019 in this paper, we consider the numerical solution of the time.
First, we present a derivation of the tft equation from the balance of energy equation and the fractional mc equation. Pdf timefractional telegraph equation and telegraph processes. In table 1, we compare the obtained numerical results with the method of interpolating scaling functions operational matrix 15 isf, for. The method depends basically on an expansion of approximated solution in a series of sinc function and shifted legendre polynomials. Research article double laplace transform method for solving. Twodimensional legendre wavelets for solving timefractional.
Original article timefractional telegraph equation for hydrogen di. Stability and convergence of the numerical solution were presented in details. Rdtm solution of caputo time fractionalorder hyperbolic. In this section, we consider the fractional order telegraph equation which is a generalized version of the integer order equation developed in the preceding section. The main objective of this work is to present and analyze a new method for approximating the time fractional telegraph equation based on the application of spectral legendregalerkin method. In this paper, an operational matrix of integration based on haar wavelets hw is introduced, and a procedure for applying the matrix to solve space and time fractional telegraph equations is formulated. The second case is related to the choice of a powerlaw relaxation function and leads to nonlinear fractional wavetype equation. Apr 27, 2020 the salient feature of the article is the graphical exhibition of the movement of solute concentration for different particular cases due to the presence and absence of reaction term when the proposed scheme is applied to the considered nonlinear fractionalorder spacetime advectionreactiondiffusion model. Orsingher and beghin 4 discussed the time fractional telegraph equation. In the present work, a hybrid transformbased localized meshless method is constructed for the solution of time fractional telegraph equations. On the approximation of timefractional telegraph equations. Beghin and orsingher 19 considered the fractional telegraph equation with partial fractional derivatives of rational order. A novel barycentric lagrange interpolation collocation method is developed to solve this equation.
Fast high order difference schemes for the time fractional. The generalized equation is called timefractional variable order telegraph equation. Spacetime fractional telegraph equations 5 beghin 20. Generalized differential transform method to space time fractional telegraph equation. Pdf we study the fundamental solutions to timefractional telegraph equations of order 2. The fractional telegraph equation has recently been considered by many authors. The governing equations have been solved by a recent reliable semianalytic method known as the reduced differential transformation method rdtm. In this paper we derived the solution of two homogeneous spacetime fractional telegraph. The hptm is combined in the form of laplace transform and homotopy perturbation method. Orsingher and beghin 27 discussed the time fractional telegraph equation. A flexible numerical method for the time fractional telegraph equation is proposed and analyzed in this paper. This technique has been applied to deal with fractional telegraph differential equation defined by caputo fractional derivative for fractional orders. Timefractional telegraph equations and telegraph processes.
We study the fundamental solutions to timefractional telegraph equations of order 2 we are able to obtain the fourier transform of the solutions for any. Zhao, the spacefractional telegraph equation and the related fractional telegraph process, chinese annals of mathematics series b 24, 1. Timefractional telegraph equation for hydrogen diffusion. One of the most popular ways is the finite difference method. Numerical solution for fractional model of telegraph equation. In this study, a mathematical model has been developed for the second order hyperbolic onedimensional time fractional telegraph equation tfte. Solution of some fractional order telegraph equations. Fourierlaplace transform to solve time fractional telegraph equation. Sep 19, 2016 in this paper, we introduce a numerical method for solving time fractional order telegraph equation. Pdf available in probability theory and related fields 1281. The fractional derivative has been described in the caputo sense. This new equation was solved numerically via the cranknicholson scheme. In this paper, we consider a fractional model of telegraph equation in terms of voltage and current. Twodimensional legendre wavelets for solving timefractional telegraph equation volume 6 issue 2 m.
The general representations of solutions is obtained and the theorems of solution existence and uniquiness for boundary value problem is proved. Timefractional telegraph equations and telegraph processes with brownian time. A steady barycentric lagrange interpolation method for the 2d. Finite difference approximation of a generalized time. Nov 12, 2003 we study the fundamental solutions to timefractional telegraph equations of order 2 we are able to obtain the fourier transform of the solutions for any. Pdf wavelet method for a class of space and time fractional. Nov 27, 2012 in this paper, we propose a new technique for solving space time fractional telegraph equations. We discuss and derive the analytical solution for three basic problems of the so called timefractional telegraph equation. The numerical algorithm based on the homotopy perturbation transform method hptm is applied to obtain analytic and approximate solutions of the space time fractional telegraph equations. In this paper we derived the solution of two homogeneous space time fractional telegraph. Fractional telegraph equation and its solution by natural. A steady barycentric lagrange interpolation method for the. We use generalized differential transform method gdtm to derive the solution of spacetime fractional telegraph equation in closed form. The expansion coefficients are then determined by reducing the time.
Numerical solution of nonlinear spacetime fractional. Analytical solution of fractionalorder hyperbolic telegraph. Variational iteration method is proposed by he and used by sevimlican for solving space and time fractional telegraph equations. A flexible numerical method for the timefractional telegraph equation is proposed and analyzed in this paper. Xu, a spacetime spectral method for the time fractional diffusion equation, siam j. This is shown by the timefractional telegraph equation 1. Analytical solution for the timefractional telegraph. Double laplace transform method for solving space and time fractional. By implementation of bernstein polynomials operational matrices of fractional differential on tfote, we reduce the original problem to a linear system of algebraic equations. A nh, analytical solution for the timefractional telegraph equation by the method of separating variables, journal of mathematical analysis and applications 338, 2 2008, 6477.
The main objective of the present paper is to extend the application of the variational iteration method vim to obtain approximate solution of the space and time fractional telegraph equations. We use generalized differential transform method gdtm to derive the solution of space time fractional telegraph equation in closed form. Analytical solution for the timefractional telegraph equation by the. Pdf we derive and discuss the analytical solution for the generalized time fractional telegraph equation with the help of the sumudu and fourier. Numerical solution of timefractional order telegraph. Siam journal on scientific computing siam society for. Numerical methods for partial differential equations 36. It is worth to point out that, the method based on using the operational matrix for solving fractional order di erential equations is computer oriented. In this paper, a fast high order difference scheme is first proposed to solve the time fractional telegraph equation based on the 2. An approximation to solution of space and time fractional. The one dimensional spacetime fractional telegraph equation is given by. We are able to obtain the fourier transform of the solutions. The solution is discretized with a new finite difference scheme in time, and a local discontinuous galerkin ldg method in space. We note that the space and time fractional derivatives are considered in caputo sense in this paper.
Pdf timefractional telegraph equation and telegraph. Pdf timefractional telegraph equations and telegraph. Generalized differential transform method to spacetime fractional telegraph equation. Analytical solution for the generalized timefractional.
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