TY - JOUR

T1 - Theory and Computations for the Nonlinear Burgers' Equation via the Use of Sinc-Galerkin Method

AU - Al-Momani, Anwar

AU - Al-Khaled, Kamel

N1 - Publisher Copyright:
© 2022 Anwar Al-Momani and Kamel Al-Khaled.
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PY - 2022/4/12

Y1 - 2022/4/12

N2 - In this research, Burgers' equation, which is well known to be nonlinear partial differential equation, has many applications for studying some physical phenomena in the disciplines we mention, water waves, plasma waves, and ion acoustic plasma waves. The paper demonstrates a comprehensive performance assessment for the use of sinc methodology to solve Burgers' equation; the method used in this research depends mainly on the use of the sinc function as a basis, where the equation under consideration was converted into an integral equation of Volterra type. Then, the x-derivatives were approximated and replaced by their corresponding sinc matrices and the resulting integral equation was also treated by approximating the integral via the use of definite integral formula for sinc functions. The method used is the sinc method, taking into account the domain on the x-axis and the time axis tt, where transformation functions intervened to make the problem applicable in their domains. The solution has been converted into an algebraic equation that is easy to deal with through any iterative method. As for the proof, the convergence of the resulting solution has been proven using the fixed-point method. It turns out that the approximate solution to the problem under consideration converges to the exact solution with an exponential order. Some examples were presented, and the effectiveness and accuracy of the method were shown by displaying the results in tables and graphs, which showed the efficacy and ease of the sinc method.

AB - In this research, Burgers' equation, which is well known to be nonlinear partial differential equation, has many applications for studying some physical phenomena in the disciplines we mention, water waves, plasma waves, and ion acoustic plasma waves. The paper demonstrates a comprehensive performance assessment for the use of sinc methodology to solve Burgers' equation; the method used in this research depends mainly on the use of the sinc function as a basis, where the equation under consideration was converted into an integral equation of Volterra type. Then, the x-derivatives were approximated and replaced by their corresponding sinc matrices and the resulting integral equation was also treated by approximating the integral via the use of definite integral formula for sinc functions. The method used is the sinc method, taking into account the domain on the x-axis and the time axis tt, where transformation functions intervened to make the problem applicable in their domains. The solution has been converted into an algebraic equation that is easy to deal with through any iterative method. As for the proof, the convergence of the resulting solution has been proven using the fixed-point method. It turns out that the approximate solution to the problem under consideration converges to the exact solution with an exponential order. Some examples were presented, and the effectiveness and accuracy of the method were shown by displaying the results in tables and graphs, which showed the efficacy and ease of the sinc method.

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U2 - 10.1155/2022/6073471

DO - 10.1155/2022/6073471

M3 - Article

AN - SCOPUS:85129116053

SN - 2090-0147

VL - 2022

SP - 6073471:1-6073471:10

JO - Journal of Electrical and Computer Engineering

JF - Journal of Electrical and Computer Engineering

M1 - 6073471

ER -