Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Similarity solutions for the generalized equation of steady transonic gas flow with a singular source


Authors: Hamid Bellout, Kuppalapalle Vajravelu and Robert A. Van Gorder
Journal: Quart. Appl. Math. 73 (2015), 379-389
MSC (2010): Primary 35Q53, 37K10, 35D30, 34E05
DOI: https://doi.org/10.1090/qam/1381
Published electronically: March 31, 2015
MathSciNet review: 3357500
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Abstract: In this brief paper, we consider the generalized equation of steady transonic gas flow with the addition of a singular source term. While the addition of a source term often destroys self-similarity of such flows, we demonstrate that a self-similar solution can still exist in the case of a singular source. We first reduce the governing nonlinear partial differential equation into an ordinary differential equation for a class of similarity solutions. Then, we study the existence of solutions for this similarity equation. After that, several explicit solution forms are given. In constructing exact solutions analytically, we demonstrate that dual solution branches may exist for some parameter regimes. For those parameter regimes where exact or analytical solutions are not possible, we obtain numerical solutions. The results demonstrate interesting properties of the solutions which warrant further study.


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Additional Information

Hamid Bellout
Affiliation: Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115
Email: bellout@math.niu.edu

Kuppalapalle Vajravelu
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: Kuppalapalle.Vajravelu@ucf.edu

Robert A. Van Gorder
Affiliation: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
Email: Robert.VanGorder@maths.ox.ac.uk

DOI: https://doi.org/10.1090/qam/1381
Keywords: Transonic gas flow, singular source, analytical solution, exact solution, existence result
Received by editor(s): July 22, 2013
Published electronically: March 31, 2015
Additional Notes: The second author was supported in part by NSF grant #1144246.
Article copyright: © Copyright 2015 Brown University

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