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Numerical algorithm based on Haar-Sinc collocation method for solving the hyperbolic PDEs.

Pirkhedri A, Javadi HH, Navidi HR - ScientificWorldJournal (2014)

Bottom Line: The advantages of this technique are that not only is the convergence rate of Sinc approximation exponential but the computational speed also is high due to the use of the Haar operational matrices.This technique is used to convert the problem to the solution of linear algebraic equations via expanding the required approximation based on the elements of Sinc functions in space and Haar functions in time with unknown coefficients.To analyze the efficiency, precision, and performance of the proposed method, we presented four examples through which our claim was confirmed.

View Article: PubMed Central - PubMed

Affiliation: Department of Computer Engineering, Islamic Azad University, Science and Research Branch, Tehran, Iran.

ABSTRACT
The present study investigates the Haar-Sinc collocation method for the solution of the hyperbolic partial telegraph equations. The advantages of this technique are that not only is the convergence rate of Sinc approximation exponential but the computational speed also is high due to the use of the Haar operational matrices. This technique is used to convert the problem to the solution of linear algebraic equations via expanding the required approximation based on the elements of Sinc functions in space and Haar functions in time with unknown coefficients. To analyze the efficiency, precision, and performance of the proposed method, we presented four examples through which our claim was confirmed.

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Plot of the absolute error e(x, t) = /y(x, t) − yn,k(x, t)/ with n = 3,  k = 32 for Example 3.
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fig8: Plot of the absolute error e(x, t) = /y(x, t) − yn,k(x, t)/ with n = 3,  k = 32 for Example 3.

Mentions: Consider the nonlinear telegraph equation [5]:(52)∂2y∂x2=∂2y∂t2+2∂y∂t+y2−e2x−4t+ex−2t,with the following initial and boundary conditions:(53)yx,0=ex,  ytx,0=−2ex,y(0,t)=e−2t,  y(1,t)=e1−2t.The exact solution is given by(54)y(x,t)=ex−2t.Figures 5, 6, 7, and 8 show the absolute error function /yexact(x, t) − yn,k(x, t)/ obtained by the present method with n = 3 and k = 4,8, 16 and 32. We can see clearly that better accuracy can be achieved by increasing the number of Haar collocation points and using the arbitrary precision ability of Mathematica software, we are able to establish more accurate results.


Numerical algorithm based on Haar-Sinc collocation method for solving the hyperbolic PDEs.

Pirkhedri A, Javadi HH, Navidi HR - ScientificWorldJournal (2014)

Plot of the absolute error e(x, t) = /y(x, t) − yn,k(x, t)/ with n = 3,  k = 32 for Example 3.
© Copyright Policy - open-access
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4248428&req=5

fig8: Plot of the absolute error e(x, t) = /y(x, t) − yn,k(x, t)/ with n = 3,  k = 32 for Example 3.
Mentions: Consider the nonlinear telegraph equation [5]:(52)∂2y∂x2=∂2y∂t2+2∂y∂t+y2−e2x−4t+ex−2t,with the following initial and boundary conditions:(53)yx,0=ex,  ytx,0=−2ex,y(0,t)=e−2t,  y(1,t)=e1−2t.The exact solution is given by(54)y(x,t)=ex−2t.Figures 5, 6, 7, and 8 show the absolute error function /yexact(x, t) − yn,k(x, t)/ obtained by the present method with n = 3 and k = 4,8, 16 and 32. We can see clearly that better accuracy can be achieved by increasing the number of Haar collocation points and using the arbitrary precision ability of Mathematica software, we are able to establish more accurate results.

Bottom Line: The advantages of this technique are that not only is the convergence rate of Sinc approximation exponential but the computational speed also is high due to the use of the Haar operational matrices.This technique is used to convert the problem to the solution of linear algebraic equations via expanding the required approximation based on the elements of Sinc functions in space and Haar functions in time with unknown coefficients.To analyze the efficiency, precision, and performance of the proposed method, we presented four examples through which our claim was confirmed.

View Article: PubMed Central - PubMed

Affiliation: Department of Computer Engineering, Islamic Azad University, Science and Research Branch, Tehran, Iran.

ABSTRACT
The present study investigates the Haar-Sinc collocation method for the solution of the hyperbolic partial telegraph equations. The advantages of this technique are that not only is the convergence rate of Sinc approximation exponential but the computational speed also is high due to the use of the Haar operational matrices. This technique is used to convert the problem to the solution of linear algebraic equations via expanding the required approximation based on the elements of Sinc functions in space and Haar functions in time with unknown coefficients. To analyze the efficiency, precision, and performance of the proposed method, we presented four examples through which our claim was confirmed.

Show MeSH