Finite Difference Approximation of a Generalized Time-Fractional Telegraph Equation (original) (raw)
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Advances in Difference Equations
We discuss the numerical solution of the time-fractional telegraph equation. The main purpose of this work is to construct and analyze stable and high-order scheme for solving the time-fractional telegraph equation efficiently. The proposed method is based on a generalized finite difference scheme in time and Legendre spectral Galerkin method in space. Stability and convergence of the method are established rigorously. We prove that the temporal discretization scheme is unconditionally stable and the numerical solution converges to the exact one with order O(τ 2-α + N 1-ω), where τ , N, and ω are the time step size, polynomial degree, and regularity of the exact solution, respectively. Numerical experiments are carried out to verify the theoretical claims.
High order approximation on non-uniform meshes for generalized time-fractional telegraph equation
MethodsX
This paper presents a high order approximation scheme to solve the generalized fractional telegraph equation (GFTE) involving the generalized fractional derivative (GFD). The GFD is characterized by a scale function σ (t) and a weight function ω(t). Thus, we study the solution behavior of the GFTE for different σ (t) and ω(t). The scale function either stretches or contracts the solution while the weight function dramatically shifts the numerical solution of the GFTE. The time fractional GFTE is approximated using quadratic scheme in the temporal direction and the compact finite difference scheme in the spatial direction. To improve the numerical scheme's accuracy, we use the non-uniform mesh. The convergence order of the whole discretized scheme is, O (τ 2 α−3 , h 4) , where τ and h are the temporal and spatial step sizes respectively. The outcomes of the work are as follows: • The error estimate for approximation of the GFD on non-uniform meshes is established. • The numerical scheme's stability and convergence are examined. • Numerical results for four examples are compared with those obtained using other method. The study shows that the developed scheme achieves higher accuracy than the scheme discussed in literature.
Numerical Approximation of a Class of Time-Fractional Differential Equations
Computational Mathematics and Variational Analysis, 2020
We consider a class of linear fractional partial differential equations containing two time-fractional derivatives of orders α, β ∈ (0, 2) and elliptic operator on space variable. Three main types of such equations with α and β in the corresponding subintervals were determined. The existence of weak solutions of the corresponding initial-boundary value problems has been proved. Some finite difference schemes approximating these problems are proposed and their stability is proved. Estimates of their convergence rates, in special discrete energetic Sobolev's norms, are obtained. The theoretical results are confirmed by numerical examples.
Convergence Analysis of Space Discretization of Time Fractional Telegraph Equation
Mathematics and Statistics, 2023
The role of fractional differential equations in the advancement of science and technology cannot be overemphasized. The time fractional telegraph equation (TFTE) is a hyperbolic partial differential equation (HPDE) with applications in frequency transmission lines such as the telegraph wire, radio frequency, wire radio antenna, telephone lines, and among others. Consequently, numerical procedures (such as finite element method, H1 – Galerkin mixed finite element method, finite difference method, and among others) have become essential tools for obtaining approximate solutions for these HPDEs. It is also essential for these numerical techniques to converge to a given analytic solution to certain rate. The Ritz projection is often used in the analysis of stability, error estimation, convergence and superconvergence of many mathematical procedures. Hence, this paper offers a rigorous and comprehensive analysis of convergence of the space discretized time-fractional telegraph equation. To this effect, we define a temporal mesh on with a finite element space in Mamadu-Njoseh polynomial space, , of degree An interpolation operator (also of a polynomial space) was introduced along the fractional Ritz projection to prove the convergence theorem. Basically, we have employed both the fractional Ritz projection and interpolation technique as superclose estimate in - norm between them to avoid a difficult Ritz operator construction to achieve the convergence of the method.
The fractional telegraph partial differential equation with fractional Atangana-Baleanu-Caputo (ABC) derivative is studied. Laplace method is used to find the exact solution of this equation. Stability inequalities are proved for the exact solution. Difference schemes for the implicit finite method are constructed. The implicit finite method is used to deal with modelling the fractional telegraph differential equation defined by Caputo fractional of Atangana-Baleanu (AB) derivative for different interval. Stability of difference schemes for this problem is proved by the matrix method. Numerical results with respect to the exact solution confirm the accuracy and effectiveness of the proposed method.
Fast High Order Difference Schemes for the Time Fractional Telegraph Equation
John Wiley & Sons, Ltd, 2020
In this paper, a fast high order difference scheme is first proposed to solve the time fractional telegraph equation based on the 2-1 formula for the Caputo fractional derivative, which reduces the storage and computational cost for calculation. A compact scheme is then presented to improve the convergence order in space. The unconditional stability and convergence in maximum norm are proved for both schemes, with the accuracy order (2 +h 2) and (2 +h 4), respectively. Difficulty arising from the two Caputo fractional derivatives is overcome by some detailed analysis. Finally, we carry out numerical experiments to show the efficiency and accuracy, by comparing with the 2-1 method.
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.
Analytic and approximate solutions of the space- and time-fractional telegraph equations
Applied Mathematics and Computation, 2005
The Adomian decomposition method is used to obtain analytic and approximate solutions of the space-and time-fractional telegraph equations. The space-and timefractional derivatives are considered in the Caputo sense. The analytic solutions are calculated in the form of series with easily computable terms. Some examples are given. The results reveal that the Adomian method is very effective and convenient.
Finite difference method for the fractional order pseudo telegraph integro-differential equation
Journal of Applied Mathematics and Computational Mechanics, 2022
The main goal of this paper is to investigate the numerical solution of the fractional order pseudo telegraph integro-differential equation. We establish the first order finite difference scheme. Then for the stability analysis of the constructed difference scheme, we give theoretical statements and prove them. We also support our theoretical statements by performing numerical experiments for some fractions of order α.