Time-fractional Telegraph Research Articles

AN INNOVATIVE METHOD FOR SOLVING LINEAR AND NONLINEAR FRACTIONAL TELEGRAPH EQUATIONS

This work investigates and solves the time-fractional telegraph equations (TFTEs) occurring in electromagnetism, which serve as mathematical models in several practically significant applied research domains. Elzaki transform (ET) is employed in this process. Caputo sense describes fractional derivatives. Solutions of TFTEs were found in an easy-to-understand, step-by-step way using ET. In addition, instances are given to show how the phrase can be applied and how valid it is for the problem-solving form. The exact solutions and the analytical solutions accord well for the tested problems. This work also discusses the convergence of the ET technique to the exact solution of TFTEs. Several examples of linear and nonlinear TFTEs are used to demonstrate the suggested methodology. The novel technique’s results show that it is an effective way to solve TFTEs, and it makes the procedure easier.

Super-convergence and Stability Analysis of the Finite Element Orthogonal Collocation Method for Time-Fractional Telegraph Equation

Super-convergence and Stability Analysis of the Finite Element Orthogonal Collocation Method for Time-Fractional Telegraph Equation

Efficient Solutions to Time-Fractional Telegraph Equations with Chebyshev Neural Networks

Abstract The objective of this study is to employ artificial neural networks (ANNs) as a new method for solving time fractional telegraph equations (TFTEs), which are typically tackled using the Caputo fractional derivative in scientific investigations. The utilisation of Chebyshev polynomials in place of a hidden layer serves to improve computing performance and expand the range of input patterns. A feed-forward neural network (NN) model is utilised, using Adam optimisation, to further refine network parameters and minimise mistakes. Two problems are selected, and the proposed method is employed to obtain their numerical solutions in order to showcase the effectiveness of the methodology. The efficiency of the proposed method is demonstrated by a comparison with the analytical solutions.

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 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.

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.

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.

Improved Gegenbauer spectral tau algorithms for distributed-order time-fractional telegraph models in multi-dimensions

The distributed-order fractional telegraph models are commonly used to describe the phenomenas of diffusion and wave-like anomalous, which can model processes without a power-law scale across the entire temporal domain. To increase the range of implementation of distributed-order fractional telegraph models, there is a need to present effective and accurate numerical algorithms to solve these models, as these models are hard to solve analytically. In this work, a novel matrix representation of the distributed-order fractional derivative based on shifted Gegenbauer (SG) polynomials has been derived. Also, two efficient algorithms based on the aforementioned operatonal matrix and the spectral tau method have been constructed for solving the one- and two-dimensional (1D and 2D) distributed-order time-fractional telegraph models with spatial variable coefficients. Also, a new operational matrix of the multiplication of space vectors has been built to have the ability in applying the tau method in the 2D case. The convergence and error bound analysis of the presented techniques are investigated. Moreover, the presented algorithms are applied on four miscellaneous test examples to illustrate the robustness and effectiveness of these algorithms.

A New analytical Modeling for Fractional Telegraph Equation Arising in Electromagnetic

In this article, the He’s variation iteration method (VIM) and Elzaki integral transform are proposed to analyze the time-fractional telegraph equations arising in electromagnetics. The Caputo sense is used to describe fractional derivatives. One of the advantages of this technique is that there is neither need to compute the Lagrange multiplier by calculating the integration in recurrence relations or via taking the convolution theorem. Further, to decrease nonlinear computational terms, the Adomian polynomial is identified with the homotopy perturbation method (HPM). The proposed method is applied to some examples of linear and nonlinear fractional telegraph equations. The solutions obtained by the new computational technique indicate that this method is efficient and facilitates the process of solving time fractional differential equations.

A stability analysis for multi-term fractional delay differential equations with higher order

As a widely used tool modeling some processes and systems in a variety of fields, fractional delay differential equations (FDDEs) with higher order have attracted much attention of the scientific community for years. Motivated by Yao et al. (2022), in which a single term has been done, we are much more interested in the stability analysis for multi-term FDDEs. In addition to the widely used Laplace transform method and decoupling technique for the characteristic equation, a region embedding technique is first introduced to handle the multiple fractional exponents. The existing results are generalized to multi-term FDDEs with higher order and the damping term of the classical integer-order delay differential equation is extended to fractional calculus. Numerical simulations for FDDEs and time-fractional telegraph equations with time delay are presented to illustrate the efficiency and validity of our results.

Hybrid techniques for approximate analytical solution of space- and time-fractional telegraph equations

Hybrid techniques for approximate analytical solution of space- and time-fractional telegraph equations

Novel and accurate Gegenbauer spectral tau algorithms for distributed order nonlinear time-fractional telegraph models in multi-dimensions

This paper sheds light on the numerical solutions of the multi-dimensional distributed-order (DO) nonlinear time-fractional telegraph equations (TFTEs) models. We present novel and accurate shifted Gegenbauer spectral tau schemes for solving the aforementioned models. In the proposed schemes, the known and unknown functions are approximated in terms of the shifted Gegenbauer polynomials (SGPs) by using the Kronecker product structure in time and space. The matrix representation of the DO fractional derivative of SGPs in the Caputo sense has been derived. Also, novel operational matrices of the multiplication of shifted Gegenbauer functions in multi-dimension have been constructed to remove the obstacle that facing the use of the tau method in the presence of nonlinear term or variable coefficients or both of them. The proposed method takes full advantage of the nonlocal nature of DO fractional differential operators. The presented method is applied on six miscellaneous test examples to illustrate its robustness and effectiveness for smooth and non smooth solutions.