Time-fractional Telegraph Equation Research Articles

LOCAL TIME FRACTIONAL REDUCED DIFFERENTIAL TRANSFORM METHOD FOR SOLVING LOCAL TIME FRACTIONAL TELEGRAPH EQUATIONS

In this paper, we seek to find solutions of the local time fractional Telegraph equation (LTFTE) by employing the local time fractional reduced differential transform method (LTFRDTM). This method produces a numerical approximate solution having the form of an infinite series that converges to a closed form solution in many cases. We apply LTFRDTM on four different LTFTEs to examine the efficiency of the proposed method. The yielded results established the effectiveness of LTFRDTM as a reliable and solid approach for obtaining solutions of LTFTEs. The solutions coincided with the exact solution in the ordinary case when [Formula: see text]. It also required minimal amount of computational work and saved a lot of time.

Genocchi Wavelet Method for the Solution of Time-Fractional Telegraph Equations with Dirichlet Boundary Conditions

Genocchi Wavelet Method for the Solution of Time-Fractional Telegraph Equations with Dirichlet Boundary Conditions

A second-order fitted scheme combined with time two-grid technique for two-dimensional nonlinear time fractional telegraph equations involving initial singularity

In this paper, we derive the improved regularity for two-dimensional nonlinear time fractional telegraph equations by virtue of the technic of decomposition at first. Then, the famous L2-1σ formula is adopted to approximate the Caputo derivative and the central finite difference method is used for spatial discretization. The convergence accuracy of the proposed method is second order in both temporal and spatial direction. Meanwhile, for the sake of reducing the time of solving the high dimensional nonlinear problems, an efficient time two-grid algorithm is proposed. Furthermore, stability analysis of the proposed scheme is studied by the energy method. At last, numerical experiments are presented to verify the validity of the theoretical statements.

A least squares finite element method for time fractional telegraph equation with Vieta-Lucas basis functions

We proposed a Least Squares Finite Element Method (LSFEM) for the approximate solution of Time fractional telegraph equation (TFTE). The method implemented the Vieta-Lucas polynomials expressed in the Caputo fractional sense as basis functions. The LSFEM reduces the TFTE into algebraic system with element matrices and vectors, while the Conjugate Gradient Method (CGM) solved the system for the unknowns with the aid of MAPLE 18. The results generated implies that the proposed method converges favorably to the theoretical solution when the number of elements is large with a large space-time step used for the parameter t.

A high-order space-time spectral method for the distributed-order time-fractional telegraph equation

A high-order space-time spectral method for the distributed-order time-fractional telegraph equation

A High Order Numerical Scheme for Time-Fractional Telegraph Equation Via Cubic Spline in Tension

A High Order Numerical Scheme for Time-Fractional Telegraph Equation Via Cubic Spline in Tension

Numerical solution of time-fractional telegraph equations using wavelet transform

Numerical solution of time-fractional telegraph equations using wavelet transform

A novel method based on fractional order Gegenbauer wavelet operational matrix for the solutions of the multi-term time-fractional telegraph equation of distributed order

In this article, we propose an effective scheme based on a combination of the Tau method and fractional-order Gegenbauer wavelets for solving the multi-term time-fractional differential equations of distributed order. First, we define fractional order Gegenbauer wavelets and then obtain operational matrices of these orthogonal functions. Applying the Legendre–Gauss quadrature for the integral term, we use function approximations obtained by the presented wavelets and the Tau method for the solution of the distributed-order multi-term time-fractional telegraph equation. The proposed method reduces the numerical solution of multi order time-fractional equations to a system of algebraic equations. Then, the convergence analysis and error bounds of the proposed scheme are studied. Three illustrative examples are solved to justify the effectiveness of the proposed method compared with some previously published results.

A Semi-Discretization Method Based on Finite Difference and Differential Transform Methods to Solve the Time-Fractional Telegraph Equation

The telegraph equation is a hyperbolic partial differential equation that has many applications in symmetric and asymmetric problems. In this paper, the solution of the time-fractional telegraph equation is obtained using a hybrid method. The numerical simulation is performed based on a combination of the finite difference and differential transform methods, such that at first, the equation is semi-discretized along the spatial ordinate, and then the resulting system of ordinary differential equations is solved using the fractional differential transform method. This hybrid technique is tested for some prominent linear and nonlinear examples. It is very simple and has a very small computation time; also, the obtained results demonstrate that the exact solutions are exactly symmetric with approximate solutions. The results of our scheme are compared with the two-dimensional differential transform method. The numerical results show that the proposed method is more accurate and effective than the two-dimensional fractional differential transform technique. Also, the implementation process of this method is very simple, so its computer programming is very fast.

Romanovski–Jacobi polynomials for the numerical solution of multi-dimensional multi-order time fractional telegraph equations

In this paper, the one- and two-dimensional multi-order time fractional telegraph equations are introduced. Two collocation methods based on the one- and two-dimensional Romanovski–Jacobi polynomials are proposed to solve them. In this way, some operational matrices related to the classical and fractional derivatives of these polynomials are achieved at first. Then, the solution of the problem under consideration is approximated by a finite expansion of these polynomials with unknown coefficients. Next, an algebraic system of equations is created in terms of the expansion coefficients by applying the derived operational matrices and the collocation technique. Eventually, by solving this system and determining the expressed coefficients, a solution for the multi-order time fractional telegraph equation under consideration is obtained. The correctness of the developed schemes are checked by solving some examples.

Invariant Solutions and Conservation Laws of the Time-Fractional Telegraph Equation

In this study, the Lie symmetry analysis is given for the time-fractional telegraph equation with the Riemann–Liouville derivative. This equation is useable to describe the physical processes of models possessing memory. By applying classical and nonclassical Lie symmetry analysis for the telegraph equation with α , β time-fractional derivatives and some technical computations, new infinitesimal generators are obtained. The actual methods give some classical symmetries while the nonclassical approach will bring back other symmetries to these equations. The similarity reduction and conservation laws to the fractional telegraph equation are found.

A fully spectral tau method for a class of linear and nonlinear variable-order time-fractional partial differential equations in multi-dimensions

We present a linear and nonlinear comprehensive class of variable-order (VO) time-fractional partial differential equations in multi-dimensions with Caputo VO fractional operator. This class comprises, as special cases, several linear and nonlinear equations of important applications in diverse fields, such as, the VO time-fractional telegraph equation, the VO time-fractional diffusion-wave equation and the VO time-fractional Klein–Gordon equation. A novel and accurate spectral tau method based on the shifted Gegenbauer polynomials (SGPs) is presented for the numerical treatment of the aforementioned class of equations. Actually, applying the tau method for solving this class of equations is very difficult, especially in the presence of the nonlinear term of these equations and the nature of the VO derivatives. To overcome this difficulty, new operational matrices for the VO fractional derivative and the approximation of the multiplication of the space–time Kronecker product vectors in multi-dimension are derived. These new matrices have the ability to remove the aforementioned difficulties that facing for applying the tau method. Several linear and nonlinear numerical examples are examined and compared in multi-dimensions with other methods to verify the validity, applicability and accuracy of the proposed methodology.

A Stable and Convergent Hybridized Discontinuous Galerkin Method for Time-Fractional Telegraph Equations

In this paper, we extend the application of the hybridized discontinuous Galerkin (HDG) method to solve time-fractional telegraph equations. In fact, we use an HDG method for space discretization and L1 and L2 finite difference schemes using non-uniform meshes for time discretization. Thanks to a special kind of discrete Gronwall inequality, we prove that the HDG method is unconditionally stable and it is convergent with the optimal spatial order of convergence. Two numerical experiments are tested to confirm the theoretical results.

Semi-analytical approximation of time-fractional telegraph equation via natural transform in Caputo derivative

Abstract In the present research study, time-fractional hyperbolic telegraph equations are solved iteratively using natural transform in one, two, and three dimensions. The fractional derivative is considered in the Caputo sense. These equations serve as a model for the wave theory process of signal processing and transmission of electric impulses. To evaluate the validity and effectiveness of the suggested strategy, a graphical comparison of approximated and exact findings is performed. Convergence analysis of the approximations utilising L ∞ {L}_{\infty } has been done using tables. The suggested approach may successfully and without errors solve a wide variety of ordinary differential equations, partial differential equations (PDEs), fractional PDEs, and fractional hyperbolic telegraph equations.

An Error Analysis of Implicit Finite Difference Method with Mamadu-Njoseh Basis Functions for Time Fractional Telegraph Equation

In this paper, we proposed and analyzed the error estimate of an implicit finite difference method with Mamadu-Njoseh as basis functions for time fractional telegraph equation. To enhance the efficacy of the method we first transform the Caputo type fractional derivative into Riemann-Liouville derivatives. The error analysis of the method is stated and proven. Also, the optimal results for scalars unknown in norm were derived for the two-dimensional case. Numerical illustrations are presented to test the reliability of the analytical and computed results. The resulting numerical evidence shows that the proposed method convergences more rapidly than the standard finite difference method. MAPLE 18 is used for all mathematical procedures in this paper.

Convergence Analysis of Space Discretization of Time Fractional Telegraph Equation

Convergence Analysis of Space Discretization of Time Fractional Telegraph Equation

New exact solutions to space–time fractional telegraph equation with conformable derivative

In this paper, we extract variety of new exact traveling wave solutions of space–time fractional nonlinear Telegraph equation for transmission lines by using improved generalized Riccati equation mapping (IGREM) method. The aforementioned equation has been solved for the first time using conformable fractional derivative. The nonlinear Telegraph equation plays crucial role in the mathematical study of voltage and current on an electrical transmission line. In an effort to motivate the researchers, we have used very efficacious technique called IGREM method to find a variety of soliton solutions, in which appropriate conformable fractional wave transformation converts nonlinear partial fractional differential equation into nonlinear ordinary equation which is then solved to find the exact solutions of equation. Hence, we succeed in constructing rich solutions that include periodic wave, singular periodic wave solutions, dark, combined dark–bright and rational solutions. To investigate wave profiles of various solutions, 2D and 3D graphs have been explained. The originality of this work has been proved by comparing the obtained results with previously present results in literature. This method has substantial advantage over other methods that it can be used to solve higher order nonlinear PDEs, and this method allows to solve these equations more conveniently with the aid of mathematical software.

A convergent exponential B-spline collocation method for a time-fractional telegraph equation

A convergent exponential B-spline collocation method for a time-fractional telegraph equation

On the solution of generalized time-fractional telegraphic equation

In this article, we have introduced a nonlinear extension of the generalized time-fractional telegraph equation (TFTE). Further, an analytical solution to this equation has been obtained by using the Laplace homotopy perturbation method (LHPM). Some known results are also discussed as special cases which supports the strength and viability of the work.

A novel adaptive meshless method for solving the nonlinear time fractional telegraph equations on arbitrary domains

A novel adaptive meshless method for solving the nonlinear time fractional telegraph equations on arbitrary domains