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Traveling and Shock Wave Simulations in A Viscous Burgers’ Equation with Periodic Boundary Conditions

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Abstract

A fourth order numerical method based on cubic B-spline functions has been proposed to solve the periodic Burgers’ equation. The method is capable of capturing the physical behavior of the Burgers’ equation very efficiently. Obtained solutions confirm that the Burgers’ equation sets a balance between nonlinearity and diffusion. The term \(u u_{x}\) produces a shocking up effect that causes waves to break while the diffusion term \(u_{xx}\) helps to smooth out shocks. As the diffusion term decreases, the smooth viscous solutions tend to become discontinuous shock waves. Approximate solutions are in good agreement with the earlier studies. Four well known problems of the periodic Burgers’ equation have been considered to test the efficiency of the method. With small modifications, method can be employed to solve different kind of partial differential equations arising in various areas of science and engineering.

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Authors and Affiliations

  1. Department of Mathematics, Jaypee Institute of Information Technology, Noida, India

    R. C. Mittal

  2. Department of Mathematics, The North Cap University, Gurugram, Haryana, India

    Rajni Rohila

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  1. R. C. Mittal
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  2. Rajni Rohila
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Correspondence to Rajni Rohila.

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Mittal, R.C., Rohila, R. Traveling and Shock Wave Simulations in A Viscous Burgers’ Equation with Periodic Boundary Conditions. Int. J. Appl. Comput. Math 4, 150 (2018). https://doi.org/10.1007/s40819-018-0582-y

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  • DOI: https://doi.org/10.1007/s40819-018-0582-y

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