Application of polynomial scaling functions for numerical solution of telegraph equation

Rashidinia, Jalil; Jokar, Mahmood

doi:10.1080/00036811.2014.998654

Cited by 22 publications

(12 citation statements)

References 24 publications

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“…In addition, Table displays the maximum absolute errors

M A E = | u (x, t) - u_{N} (x, t) |

for N = 16, obtained by our method (GOMM), the PSFM in , and the interpolating scaling functions method (ISFM) in . These two tables show that our method (GOMM) is more accurate than those in .…”

Section: Illustrative Examplesmentioning

confidence: 98%

“…Example (). Consider the nonlinear telegraph equation subject to the initial conditions (33) and the nonhomogeneous boundary conditions (29), that is,

\begin{matrix} left \\ α = 2, γ = 1, ℓ = τ = 1, p_{0} (x) = e^{x}, p_{1} (x) = - 2 e^{x}, q_{0} (t) = e^{- 2 t}, q_{1} (t) = e^{1 - 2 t}, and \\ f (x, t, u) = - {(u (x, t))}^{2} + e^{2 x - 4 t} - e^{x - 2 t} . \end{matrix}

The exact solution is

u (x, t) = e^{x - 2 t} .

Figures and display the absolute errors

| u (x, t) - u_{N} (x, t) |

for

N = 5 and 7

, obtained by our method COMM and the differential transform method (DTM) in , thus showing the wide applicability and efficiency of our method for solving nonlinear problems. Example (). Consider the following telegraph equation

\partial_{t t} u (x, t) + 2 α \partial_{t} u (x, t) + β^{2} <...>

…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…Consider the following telegraph equation

\partial_{t t} u (x, t) + 2 α \partial_{t} u (x, t) + β^{2} u (x, t) = \partial_{x x} u (x, t) + f (x, t), (x, t) \in [0, 1] \times [0, 1], α > β \geq 0

with the following initial and boundary conditions

\begin{matrix} left \\ u (x, 0) = \sin x, & \partial_{t} u (x, 0) = 0, & x \in [0, 1], \\ u (0, t) = 0, & u (1, t) = \sin 1 \cos t, & t \in [0, 1], \end{matrix}

where the source term f(x, t) is chosen to be consistent with the exact solution

u (x, t) = \cos t \sin x

. Table displays the maximum absolute errors

M A E = | u (x, t) - u_{N} (x, t) |

for N = 8, obtained by our method (GOMM) and the polynomial scaling functions method (PSFM) in . In addition, Table displays the maximum absolute errors

M A E = | u (x, t) - u_{N} (x, t

…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…Numerical Methods for Partial Differential Equations DOI 10.1002/num Table II displays the maximum absolute errors MAE = |u(x, t)−u N (x, t)| for N = 8, obtained by our method (GOMM) and the polynomial scaling functions method (PSFM) in [37]. In addition, Table III displays the maximum absolute errors MAE = |u(x, t) − u N (x, t)| for N = 16, obtained by our method (GOMM), the PSFM in [37], and the interpolating scaling functions method (ISFM) in [10]. These two tables show that our method (GOMM) is more accurate than those in [10,37].…”

Section: Example 3 ([5]mentioning

confidence: 99%

“…Comparison between the absolute error |u − u N | for Example 3, N = 7, obtained by our method (COMM) and DTM in [5]. [Color figure can be viewed at wileyonlinelibrary.com]Figures 2 and 3display the absolute errors |u(x, t) − u N (x, t)| for N = 5 and 7, obtained by our method COMM and the differential transform method (DTM) in[5], thus showing the wide applicability and efficiency of our method for solving nonlinear problems.Example 4 ([10,37]). Consider the following telegraph equation∂ tt u(x, t) + 2α ∂ t u(x, t) + β 2 u(x, t) = ∂ xx u(x, t) + f (x, t), (x, t) ∈ [0, 1] × [0, 1], α > β ≥ 0(38)with the following initial and boundary conditions u(x, 0) = sin x, ∂ t u(x, 0) = 0, x ∈ [0, 1], u(0, t) = 0, u(1, t) = sin 1 cos t, t ∈ [0, 1],…”

mentioning

confidence: 99%

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New Tchebyshev‐Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations

Abd-Elhameed

Doha

Youssri

et al. 2016

Numerical Methods Partial

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The telegraph equation describes various phenomena in many applied sciences. We propose two new efficient spectral algorithms for handling this equation. The principal idea behind these algorithms is to convert the linear/nonlinear telegraph problems (with their initial and boundary conditions) into a system of linear/nonlinear equations in the expansion coefficients, which can be efficiently solved. The main advantage of our algorithm in the linear case is that the resulting linear systems have special structures that reduce the computational effort required for solving them. The numerical algorithms are supported by a careful convergence analysis for the suggested Chebyshev expansion. Some illustrative examples are given to demonstrate the wide applicability and high accuracy of the proposed algorithms. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1553–1571, 2016

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“…In addition, Table displays the maximum absolute errors

M A E = | u (x, t) - u_{N} (x, t) |

for N = 16, obtained by our method (GOMM), the PSFM in , and the interpolating scaling functions method (ISFM) in . These two tables show that our method (GOMM) is more accurate than those in .…”

Section: Illustrative Examplesmentioning

confidence: 98%

“…Example (). Consider the nonlinear telegraph equation subject to the initial conditions (33) and the nonhomogeneous boundary conditions (29), that is,

\begin{matrix} left \\ α = 2, γ = 1, ℓ = τ = 1, p_{0} (x) = e^{x}, p_{1} (x) = - 2 e^{x}, q_{0} (t) = e^{- 2 t}, q_{1} (t) = e^{1 - 2 t}, and \\ f (x, t, u) = - {(u (x, t))}^{2} + e^{2 x - 4 t} - e^{x - 2 t} . \end{matrix}

The exact solution is

u (x, t) = e^{x - 2 t} .

Figures and display the absolute errors

| u (x, t) - u_{N} (x, t) |

for

N = 5 and 7

\partial_{t t} u (x, t) + 2 α \partial_{t} u (x, t) + β^{2} <...>

…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…Consider the following telegraph equation

\partial_{t t} u (x, t) + 2 α \partial_{t} u (x, t) + β^{2} u (x, t) = \partial_{x x} u (x, t) + f (x, t), (x, t) \in [0, 1] \times [0, 1], α > β \geq 0

with the following initial and boundary conditions

\begin{matrix} left \\ u (x, 0) = \sin x, & \partial_{t} u (x, 0) = 0, & x \in [0, 1], \\ u (0, t) = 0, & u (1, t) = \sin 1 \cos t, & t \in [0, 1], \end{matrix}

where the source term f(x, t) is chosen to be consistent with the exact solution

u (x, t) = \cos t \sin x

. Table displays the maximum absolute errors

M A E = | u (x, t) - u_{N} (x, t) |

for N = 8, obtained by our method (GOMM) and the polynomial scaling functions method (PSFM) in . In addition, Table displays the maximum absolute errors

M A E = | u (x, t) - u_{N} (x, t

…”

Section: Illustrative Examplesmentioning

confidence: 99%

Section: Example 3 ([5]mentioning

confidence: 99%

mentioning

confidence: 99%

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New Tchebyshev‐Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations

Abd-Elhameed

Doha

Youssri

et al. 2016

Numerical Methods Partial

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Numerical Solution of the Rosenau‐KdV‐RLW equation via combination of a polynomial scaling function collocation and finite difference method

Oruç,

Esen,

Bulut

2024

Math Methods in App Sciences

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In this paper, we established a polynomial scaling method to investigate the numerical solution of Rosenau–Korteweg De Vries‐regularized long wave (Rosenau‐KdV‐RLW) equation. We start with discretization of the time variable of the equation using a finite difference approach equipped with a linearization. After the time discretization, we have used polynomial scaling functions for the discretization of the spatial variable. These two discretizations give us the desired discrete system of equations to obtain numerical solutions. We further derive an error estimate for the proposed method. We have applied the proposed method to Rosenau‐KdV, Rosenau‐RLW, and Rosenau‐KdV‐RLW equations and used error norms to examine the accuracy and reliability of the presented method. Also, to enhance accuracy of the results, we utilize Richardson extrapolation. The comparisons with the analytical solution and earlier studies that use different methods indicate that the proposed method is accurate and reliable.

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New unconditionally stable scheme for telegraph equation based on weighted Laguerre polynomials

Zhang

Miao

2017

Numerical Methods Partial

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This article proposes a new unconditionally stable scheme to solve one‐dimensional telegraph equation using weighted Laguerre polynomials. Unlike other numerical schemes, the time derivatives in the equation can be expanded analytically based on the Laguerre polynomials and basis functions. By applying a Galerkin temporal testing procedure and using the orthogonal property of weighted Laguerre polynomials, the time variable can be eliminated from computations, which results in an implicit equation. After solving the equation recursively one can obtain the numerical results of telegraph equation by using the expanded coefficients. Some numerical examples are considered to validate the accuracy and stability of this proposed scheme, and the results are compared with some existing numerical schemes.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1603–1615, 2017

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Application of polynomial scaling functions for numerical solution of telegraph equation

Cited by 22 publications

References 24 publications

New Tchebyshev‐Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations

New Tchebyshev‐Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations

Numerical Solution of the Rosenau‐KdV‐RLW equation via combination of a polynomial scaling function collocation and finite difference method

New unconditionally stable scheme for telegraph equation based on weighted Laguerre polynomials

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