Existence of Solutions for a Class of Impulsive Burgers Equation

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system’s guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.46698/x1302-5604-8948-x Existence of Solutions for a Class of Impulsive Burgers Equation Georgiev, S. G. , Hakem, A. Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 2. Abstract: We study a class of impulsive Burgers equations. A new topological approach is applied to prove the existence of at least one and at least two nonnegative classical solutions. The arguments are based on recent theoretical results. Here we focus our attention on a class of Burgers equations and we investigate it for the existence of classical solutions. The Burgers equation can be used for modeling both traveling and standing nonlinear plane waves. The simplest model equation can describe the second-order nonlinear effects connected with the propagation of high-amplitude (finite-amplitude waves) plane waves and, in addition, the dissipative effects in real fluids. There are several approximate solutions to the Burgers equation. These solutions are always fixed to areas before and after the shock formation. For an area where the shock wave is forming no approximate solution has yet been found. Therefore, it is therefore necessary to solve the Burgers equation numerically in this area. Keywords: Burgers equation, impulsive Burgers equation, positive solution, fixed point, cone, sum of operators Language: English Download the full text For citation: Georgiev, S. G. and Hakem, A. Existence of Solutions for a Class of Impulsive Burgers Equation, Vladikavkaz Math. J., 2024, vol. 26, no. 2, pp. 26-38. DOI 10.46698/x1302-5604-8948-x + References 1. Konicek, P., Bednarik, M. and Cervenka, M. Finite-Amplitude Acoustic Waves in a Liquid-Filled Tube, Hydroacoustics (Editors: E. Kozaczka, G. Grelowska), Gdynia, Naval Academy, 2000, pp. 85-88. 2. Mitome, H. An Exact Solution for Finite-Amplitude Plane Sound Waves in a Dissipative Fluid, Journal of the Acoustical Society of America, 1989, vol. 86, no. 6, pp. 2334-2338. 3. Ochmann, M. Nonlinear Resonant Oscilllations in Closed Tubes - An Application of the Averaging Method, Journal of the Acoustical Society of America, 1985, vol. 77, no. 1, pp. 61-66. DOI: 10.1121/1.391901. 4. Agarwal, R. P., Meehan, M. and O'Regan, D. Fixed Point Theory and Applications, Cambridge Tracts in Mathematics, Vol. 141, Cambridge University Press, 2001. DOI: 10.1017/CBO9780511543005. 5. Djebali, S. and Mebarki, K. Fixed Point Index for Expansive Perturbation of \(k\)-Set Contraction Mappings, Topological Methods in Nonlinear Analysis, 2019, vol. 54, no. 2, pp. 613-640. DOI: 10.12775/TMNA.2019.055. 6. Mouhous, M., Georgiev, S. and Mebarki, K. Existence of Solutions for a Class of First Order Boundary Value Problems, Archivum Mathematicum, 2022, vol. 58, no. 3, pp. 141-158. DOI: 10.5817/AM2022-3-141. 7. Polyanin, A. and Manzhirov, A. Handbook of Integral Equations, CRC Press, 1998. ← Contents of issue


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