Skip to main content
SpringerLink
Log in

A systematic literature review of Burgers’ equation with recent advances

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

Even if numerical simulation of the Burgers’ equation is well documented in the literature, a detailed literature survey indicates that gaps still exist for comparative discussion regarding the physical and mathematical significance of the Burgers’ equation. Recently, an increasing interest has been developed within the scientific community, for studying non-linear convective–diffusive partial differential equations partly due to the tremendous improvement in computational capacity. Burgers’ equation whose exact solution is well known, is one of the famous non-linear partial differential equations which is suitable for the analysis of various important areas. A brief historical review of not only the mathematical, but also the physical significance of the solution of Burgers’ equation is presented, emphasising current research strategies, and the challenges that remain regarding the accuracy, stability and convergence of various schemes are discussed. One of the objectives of this paper is to discuss the recent developments in mathematical modelling of Burgers’ equation and thus open doors for improvement. No claim is made that the content of the paper is new. However, it is a sincere effort to outline the physical and mathematical importance of Burgers’ equation in the most simplified ways. We throw some light on the plethora of challenges which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a numerical simulation of ordinary / partial differential equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. H Bateman, Mon. Weather Rev. 43, 163 (1915)

    Article  ADS  Google Scholar 

  2. J M Burgers, Trans. R. Netherlands Acad. Sci. Amsterdam 17, 1 (1939)

    Google Scholar 

  3. J M Burgers, Adv. Appl. Mech. 1, 171 (1948)

    Article  Google Scholar 

  4. F T Nieuwstadt and J A Steketee, Selected papers of J M Burgers (Springer Science+Business Media, BV, Dordrecht, 1995)

    Book  Google Scholar 

  5. E Hopf, Commun. Pure Appl. Math. 3, 201 (1950)

    Article  Google Scholar 

  6. J D Cole, Quart. Appl. Math. 9, 225 (1951)

    Article  MathSciNet  Google Scholar 

  7. J D Murray, J. Fluid Mech. 59, 263 (1973)

    Article  ADS  Google Scholar 

  8. J Baker, A Armaou and P D Christofides, J. Math. Anal. Appl. 252, 230 (2000)

    Article  MathSciNet  Google Scholar 

  9. R Iturriaga and K Khanin, Commun. Math. Phys. 232, 377 (2003)

    Article  ADS  Google Scholar 

  10. I A Bogaevsky, arXiv:math-ph/0407073v1, 1–12 (2004)

  11. J Bec and K Khanin, Phys. Rep. 447, 1 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  12. A Chekhlov and V Yakhot, Phys. Rev. E 51, R2739 (1995)

    Article  ADS  Google Scholar 

  13. A Chekhlov and V Yakhot, Phys. Rev. E 52, 5681 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  14. V Gurarie and A Migdal, Phys. Rev. E 54, 4908 (1996)

    Article  ADS  Google Scholar 

  15. V Yakhot and A Chekhlov, Phys. Rev. Lett. 77, 3118 (1996)

    Article  ADS  Google Scholar 

  16. E Balkovsky, G Falkovich, I Kolokolov and V Lebedev, Phys. Rev. Lett. 78, 1452 (1997)

    Article  ADS  Google Scholar 

  17. E Weinan, K Khanin, A Mazel and Y Sinai, Phys. Rev. Lett. 78, 1904 (1997)

    Article  ADS  Google Scholar 

  18. E Weinan, K Khanin, A Mazel and Y Sinai, Ann. Math. 151, 877 (2000)

    Article  MathSciNet  Google Scholar 

  19. J Bec, U Frisch and K Khanin, J. Fluid Mech. 416, 239267 (2000)

    Article  Google Scholar 

  20. J Bec, R Iturriaga and K Khanin, Phys. Rev. Lett. 89, 024501 (2002)

    Article  ADS  Google Scholar 

  21. J Bec and K Khanin, J. Stat. Phys. 113, 741 (2003)

    Article  Google Scholar 

  22. D Gomes, R Iturriaga, K Khanin and P Padilla, Moscow Math. J. 5, 613 (2005)

    MathSciNet  Google Scholar 

  23. G Kreiss and H O Kreiss, Appl. Numer. Math. 2, 161 (1986)

    Article  MathSciNet  Google Scholar 

  24. H Yang and A Przekwas, J. Comput. Phys. 102, 139 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  25. L G Reyna and M J Ward, Commun. Pure Appl. Math. XLVIII, 79 (1995)

    Article  Google Scholar 

  26. R S Hirsh, J. Comput. Phys. 19, 90 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  27. D E Panayotounakos and D Drikakis, ZAMM Z. Angew Math. Mech. 75, 437 (1995)

    Article  Google Scholar 

  28. P Peebles, The large scale structure of the Universe (Princeton University Press, Princeton, 1980)

    Google Scholar 

  29. S N Gurbatov and A I Saichev, Radiofizika 27, 456 (1984)

    ADS  Google Scholar 

  30. S N Gurbatov, A I Saichev and S F Shandarin, Sov. Phys. Dokl. 38, 321 (1984)

    Google Scholar 

  31. M Vergassola, B Dubrulle, U Frisch and A Nullez, Astron. Astrophys. 289, 325 (1994)

    ADS  Google Scholar 

  32. S Albeverio, S A Molchanov and D Surgailis, Probab. Theory Relat. Fields 100, 457 (1994)

    Article  Google Scholar 

  33. P Peebles, Principles of physical cosmology (Princeton University Press, Princeton, 1993)

    Google Scholar 

  34. S A Molchanov, D Surgailis and W A Woyczynski, Ann. Appl. Prob. 7, 200 (1997)

    Article  Google Scholar 

  35. B Greenshields, Proc. High. Res. Board 14, 448 (1935)

    Google Scholar 

  36. T Musha and H Higuchi, Jpn J. Appl. Phys. 15, 1271 (1976)

    Article  ADS  Google Scholar 

  37. T Musha and H Higuchi, Jpn J. Appl. Phys. 17, 811 (1978)

    Article  ADS  Google Scholar 

  38. M Leibig, Phys. Rev. E 49, 184 (1994)

    Article  ADS  Google Scholar 

  39. R Kuhne, N Gartner, C Messner and A J Rathi (eds), TRB Spe. Rep. 165, 2nd edn, Traffic flow theory (1998)

  40. D Chowdhury, L Santen and A Schadschneider, Phys. Rep. 329, 199 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  41. A Aw and M Rascle, SIAM J. Appl. Math. 60, 916 (2000)

    Article  MathSciNet  Google Scholar 

  42. X Yu, Chin. Phys. 11(11), 1128 (2002)

    Article  ADS  Google Scholar 

  43. J A Atwell and B B King, Proceedings of the American Control Conference (2000) pp. 2745–2749

  44. E Moreau and O Vallee, Proceedings of the American Control Conference, arXiv:physics/0501019 (2005)

  45. E Moreau and O Vallee, Phys. Rev. E 73, Article ID016112, 4 pages (2006)

  46. J Yepez, arXiv:quant-ph/0210092

  47. J Yepez, J. Stat. Phys. 107, 203 (2002)

    Article  Google Scholar 

  48. J Yepez, Phys. Rev. A 74, 042322 (2006)

    Article  ADS  Google Scholar 

  49. Z Chen, J Yepez and D G Cory, Phys. Rev. A 74, 042321 (2006)

    Article  ADS  Google Scholar 

  50. R Krause and D Ruprecht, Domain decomposition methods in science and engineering XXI (Springer, 2014) pp. 647–655

  51. K Huang, V Kindratenko and N Rizwan-Uddin, Trans. Am. Nucl. Soc. 100, 319 (2009)

    Google Scholar 

  52. S Satake and H Yoshimori, ICHMT Dig. Lib. Online (Begel House Inc., 2012)

  53. S H Kuo, C W Hsieh, R K Lin and W H Sheu, Algorithms and architectures for parallel processing (Springer, 2010) pp. 297–307

  54. F Puffer, R Tetzlaff and D Wolf, in: Proceedings of 1995 URSI Int. Symposium on Signals, Systems, and Electronics (1995) pp. 501–504

  55. M Hayati and B Karami, J. Appl. Sci. 7, 2812 (2007)

    Article  ADS  Google Scholar 

  56. J Zueco, Math. Comput. Model. 47, 401 (2008)

    Article  MathSciNet  Google Scholar 

  57. G Hetmanczyk and K Ochs, in: 51st Midwest Symposium on Circuits and Systems (2008) pp. 161–164

  58. S E Esipov, Phys. Rev. E 52, 3711 (1995)

    Article  ADS  Google Scholar 

  59. Y-M Tsai, H-C Kuo, Y-C Chang and Y-H Tseng WOS:000308786700006 (2012)

  60. R Menard, Kalman filtering of Burgers’ equation and its application to atmospheric data assimilation, Ph.D. thesis (Department of Atmospheric and Oceanic Sciences, McGiIl University, Montral)

  61. P D Düben and S I Dolaptchiev, Theor. Comput. Fluid Dyn. 29, 311 (2015)

    Article  Google Scholar 

  62. P Menounou and A N Athanasiadis, J. Acoust. Soc. Am. 126, 983 (2009)

    Article  ADS  Google Scholar 

  63. K L Gee, V W Sparrow, M M James, J M Downing, C M Hobbs, T B Gabrielson and A A Atchley, J. Acoust. Soc. Am. 123, 4082 (2008)

    Article  ADS  Google Scholar 

  64. K L Gee, Prediction of nonlinear jet noise propagation, Ph.D. thesis (The Pennsylvania State University, 2005)

  65. S Lee, P J Morris and K S Brentner, AIAA Paper 1384 (2010)

  66. K Shah and T Singh, Eur. J. Pure Appl. Math. 9, 114 (2016)

    MathSciNet  Google Scholar 

  67. W-J Liu and M Krstić, Int. J. Adapt. Control Signal. Process. 15, 745 (2001)

    Article  Google Scholar 

  68. H Park, M Lee and Y Jang, Comput. Methods Appl. Mech. Eng. 166, 289 (1998)

    Article  ADS  Google Scholar 

  69. I Kucuk and I Sadek, Appl. Math. Comput. 210, 126 (2009)

    MathSciNet  Google Scholar 

  70. H Park and Y Jang, Int. J. Eng. Sci. 38, 785 (2000)

    Article  Google Scholar 

  71. J A Burns and S Kang, Nonlin. Dyn. 2, 235 (1991)

    Article  Google Scholar 

  72. S Volkwein. Comput. Opt. Appl. 18, 115 (2001)

    Article  MathSciNet  Google Scholar 

  73. N Smaoui, J. Comput. Appl. Math. 182, 91 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  74. O Glass and S Guerrero, SIAM J. Control Opt. 46, 1211 (2007)

    Article  MathSciNet  Google Scholar 

  75. E Y Rodin, J. Math. Anal. Appl. 30, 401 (1970)

    Article  MathSciNet  Google Scholar 

  76. E R Benton and G W Platzman, Quart. Appl. Math. 30, 195 (1972)

    Article  MathSciNet  Google Scholar 

  77. K B Wolf, L Hlavatỳ and S Steinberg, J. Math. Anal. Appl. 114, 340 (1986)

    Article  MathSciNet  Google Scholar 

  78. S Nerney, E J Schmahl and Z Musielak, Quart. Appl. Math. LIV, 63 (1996)

  79. A Kudryavtsev and O Sapozhnikov, Acoust. Phys. 57, 311 (2011)

    Article  ADS  Google Scholar 

  80. V K Srivastava and M K Awasthi, Ain Shams Eng. J. 5, 533 (2014)

    Article  Google Scholar 

  81. T M Elzaki and H Kim, Appl. Math. Sci. 8, 2931 (2014)

    Google Scholar 

  82. W Wood, Commun. Numer. Meth. Eng. 22, 797 (2006)

    Article  Google Scholar 

  83. M M Cecchi, R Nociforo and P P Grego, Le Mate-matiche 51, 43 (1997)

    Google Scholar 

  84. I Christie, D F Griffiths, A R Mitchell and J M Sanz-Serna, IMA J. Numer. Anal. 1, 253 (1981)

    Article  MathSciNet  Google Scholar 

  85. H Nguyen and J Reynen, Numer. Methods Nonlin. Problems 2, 718 (1982)

    Google Scholar 

  86. A H Salas, Appl. Math. Comput. 216, 18 (2010)

    MathSciNet  Google Scholar 

  87. C A J Fletcher, Int. J. Numer. Methods Fluids 3, 213 (1983)

    Article  ADS  Google Scholar 

  88. H Aref and P Daripa, SIAM J. Sci. Stat. Comput. 5, 856 (1984)

    Article  Google Scholar 

  89. S Kutluay, A Bahadir and A Ozdes, J. Comput. Appl. Math. 103, 251 (1999)

    Article  MathSciNet  Google Scholar 

  90. A R Bahadir, Appl. Math. Comput. 137, 131 (2003)

    MathSciNet  Google Scholar 

  91. I Hassanien, A Salama and H Hosham, Appl. Math. Comput. 170, 781 (2005)

    MathSciNet  Google Scholar 

  92. M K Kadalbajoo, K K Sharma and A Awasthi, Appl. Math. Comput. 170, 1365 (2005)

    MathSciNet  Google Scholar 

  93. M K Kadalbajoo and A Awasthi, Appl. Math. Comput. 182, 1430 (2006)

    MathSciNet  Google Scholar 

  94. W Liao, Appl. Math. Comput. 206, 755 (2008)

    MathSciNet  Google Scholar 

  95. P P Xu and Z Sun, Numer. Meth. Part. Diff. Eqn 25, 172 (2009)

    Article  Google Scholar 

  96. K Pandey, L Verma and A K Verma, Appl. Math. Comput. 215, 2206 (2009)

    MathSciNet  Google Scholar 

  97. P Huang and A Abduwali, Comput. Math. Appl. 59, 2452 (2010)

    Article  MathSciNet  Google Scholar 

  98. W Liao, Int. J. Numer. Meth. Fluids 64, 565 (2010)

    Article  Google Scholar 

  99. M Tamsir and V K Srivastava, Int. J. Sci. Eng. Res. 2, 1 (2011)

    Google Scholar 

  100. M Mousa, A Abadeer and M Abbas, Int. J. Pure Appl. Math. 75, 169 (2012)

    Google Scholar 

  101. M C Kweyu, W A Manyonge, A Koross and V Ssemaganda, Appl. Math. Sci. 6, 5603 (2012)

    MathSciNet  Google Scholar 

  102. S S Wani and S H Thakar, Int. J. Appl. Phys. Math. 3, 324 (2013)

    Article  Google Scholar 

  103. V K Srivastava, M K Awasthi and S Singh, AIP Adv. 3, 122 (2013)

    Google Scholar 

  104. V K Srivastava, S Singh and M K Awasthi, AIP Adv. 3, 122105-1 (2013)

    Google Scholar 

  105. E Rothe, Math. Annal. 102, 650 (1930)

    Article  Google Scholar 

  106. L A Kurtz, R E Smith, C L Parks and L R Boney, Comput. Fluids  6, 49 (1977)

    Article  ADS  Google Scholar 

  107. L F Shampine, Numer. Methods Part. Diff. Eqn 10, 739 (1994)

    Article  Google Scholar 

  108. O Oymak and N Selcuk, Int. J. Numer. Methods Fluids 23, 455 (1996)

    Article  ADS  Google Scholar 

  109. N Selcuk, T Tarhan and S Tanrikulu, Int. J. Numer. Methods Engng 53, 1615 (2002)

    Article  Google Scholar 

  110. M Javidi, Int. Math. Forum 1, 1565 (2006)

    Article  MathSciNet  Google Scholar 

  111. H O Bakodah, IJRRAS 8, 328 (2011)

    MathSciNet  Google Scholar 

  112. I J Schoenberg, Quart. Appl. Math. 4, 112 (1946)

    Article  MathSciNet  Google Scholar 

  113. P Jain and D Holla, Int. J. Nonlin. Mech. 13, 213 (1978)

    Article  Google Scholar 

  114. P Arminjon and C Beauchamp, Int. J. Numer. Methods Engng 12, 415 (1978)

    Article  Google Scholar 

  115. E Varoglu and W D L Finn, Int. J. Numer. Methods Engng 16, 171 (1980)

    Article  Google Scholar 

  116. J Caldwell, P Wanless and A Cook, Appl. Math. Model. 5, 189 (1981)

    Article  Google Scholar 

  117. J Caldwell and P Smith, Appl. Math. Model. 6, 381 (1982)

    Article  Google Scholar 

  118. R Gelinas, S Doss and K Miller, J. Comput. Phys. 40, 202 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  119. J Caldwell, P Wanless and A Cook, Appl. Math. Model. 11, 211 (1987)

    Article  Google Scholar 

  120. A H A Ali and G A Gardner, Comput. Meth. Appl. Mech. Engng 100, 325 (1992)

    Article  ADS  Google Scholar 

  121. R Shankar, T V Singh and A A Bassaif, Int. J. Numer. Methods Fluids 20, 1263 (1995)

    Article  ADS  Google Scholar 

  122. T Öziş, E Aksan and A Özdeş, Appl. Math. Comput. 139, 417 (2003)

    MathSciNet  Google Scholar 

  123. A Dogan, Appl. Math. Comput. 157, 331 (2004)

    ADS  MathSciNet  Google Scholar 

  124. S Kutluay, A Esen and I Dag, J. Comput. Appl. Math. 167, 21 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  125. E Aksan and A Özdeş, Appl. Math. Comput. 156, 395 (2004)

    MathSciNet  Google Scholar 

  126. T Özis, A Esen and S Kutluay, Appl. Math. Comput. 65, 237 (2005)

    Google Scholar 

  127. E Aksan, Appl. Math. Comput. 170, 895 (2005)

    MathSciNet  Google Scholar 

  128. M A Ramadan, T S El-Danaf and F E A Alaal, Chaos Solitons Fractals 26, 1249 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  129. İ Dağ, B Saka and A Boz, Appl. Math. Comput. 166, 506 (2005)

    MathSciNet  Google Scholar 

  130. I Dağ, D Irk and B Saka, Appl. Math. Comput. 163, 199 (2005)

    MathSciNet  Google Scholar 

  131. B R Kumar and M Mehra, BIT Numer. Math. 45, 543 (2005)

    Google Scholar 

  132. M A Ramadan, T S El-Danaf and F E Alaal, Open Appl. Math. J. 1, 15 (2007)

    Article  MathSciNet  Google Scholar 

  133. B Saka, İ Dağ and D Irk, The ANZIAM J. 49, 389 (2008)

    Article  MathSciNet  Google Scholar 

  134. Z Jiang and R Wang, J. Inform. Comput. Sci, 7, 1013 (2010)

    Google Scholar 

  135. G Zhao, X Yu and R Zhang, Comput. Math. Appl. 62, 3279 (2011)

    Article  MathSciNet  Google Scholar 

  136. R Mittal and R Jain, Appl. Math. Comput. 218, 7839 (2012)

    MathSciNet  Google Scholar 

  137. R Kannan and Z J Wang, Int. J. Numer. Methods Fluids 69, 781 (2012)

    Article  ADS  Google Scholar 

  138. R Jiwari, R C Mittal and K K Sharma, Appl. Math. Comput. 219, 6680 (2013)

    MathSciNet  Google Scholar 

  139. Q Yang, Abstract Appl. Anal. 2013, 1 (2013)

    Google Scholar 

  140. G Arora and B K Singh, Appl. Math. Comput. 224, 166 (2013)

    MathSciNet  Google Scholar 

  141. A J Chorin, J. Comput. Phys. 2, 12 (1967)

    Article  ADS  Google Scholar 

  142. A J Chorin, Math. Comput. 22, 745 (1968)

    Article  Google Scholar 

  143. S V Patankar, Numerical heat transfer and fluid flow (Hemisphere, Washington, DC)

  144. R I Issa, J. Comput. Phys. 62, 40 (1985)

    Article  ADS  Google Scholar 

  145. R C Mittal and P Singhal, Commun. Numer. Methods Engng 9, 397 (1993)

    Article  ADS  Google Scholar 

  146. B Elton, Quart. Appl. Math. 30, 195 (1972)

    Article  MathSciNet  Google Scholar 

  147. D Zhang, G Wei, D Kouri and D Hoffman, Phys. Fluids 9, 1853 (1997)

    Article  ADS  Google Scholar 

  148. J Burns, A Balogh, D Gilliam and V Shubol, J. Math. Syst. Estimat. Control 8, 1 (1998)

    Google Scholar 

  149. O Brander and J Hedenfalk, Wave Motion 28, 319 (1998)

    Article  MathSciNet  Google Scholar 

  150. Y Hon and X Mao, Appl. Math. Comput. 95, 37 (1998)

    MathSciNet  Google Scholar 

  151. M B Abd-el Malek and S M El-Mansi, J. Comput. Appl. Math. 115, 1 (2000)

    Article  MathSciNet  Google Scholar 

  152. S Abbasbandy and M Darvishi, Appl. Math. Comput. 170, 95 (2005)

    MathSciNet  Google Scholar 

  153. S Abbasbandy and M Darvishi, Appl. Math. Comput. 163, 1265 (2005)

    MathSciNet  Google Scholar 

  154. M Abdou and A Soliman, J. Comput. Appl. Math. 181, 245 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  155. S F Radwan, J. Comput. Appl. Math. 174, 383 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  156. K Sakai and I Kimura, J. Comput. Appl. Math. 173, 39 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  157. M Darvishi and M Javidi, Appl. Math. Comput. 173, 421 (2006)

    MathSciNet  Google Scholar 

  158. M Gulsu, Appl. Math. Comput. 175, 1245 (2006)

    MathSciNet  Google Scholar 

  159. A Gorguis, Appl. Math. Comput. 173, 126 (2006)

    MathSciNet  Google Scholar 

  160. M Basto, V Semiao and F L Calheiros, J. Comput. Appl. Math. 206, 927 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  161. X Wu and J Zhang, Comp. Math. Appl. 56, 242 (2008)

    Article  Google Scholar 

  162. X H Zhang, J Ouyang and L Zhang, Eng. Anal. Boundary Elements 33, 356 (2009)

    Article  MathSciNet  Google Scholar 

  163. J Biazar and H Aminikhah, Math. Comput. Model. 49, 1394 (2009)

    Article  Google Scholar 

  164. L Zhang, J Ouyang, X Wang and X Zhang, J. Comput. Phys. 229, 7147 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  165. H Zhu, H Shu and M Ding, Comput. Math. Appl. 60, 840 (2010)

    Article  MathSciNet  Google Scholar 

  166. A Asaithambi, Appl. Math. Comput. 216, 2700 (2010)

    MathSciNet  Google Scholar 

  167. F Liu and W Shi, Commun. Nonlinear Sci. Numer. Simul. 16, 150 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  168. C Allery, A Hamdouni, D Ryckelynck and N Verdon, Appl. Math. Comput. 217, 6671 (2011)

    MathSciNet  Google Scholar 

  169. M Sarboland and A Aminataei, ISRN Appl. Math, Article ID 254086 (2012)

    Google Scholar 

  170. H Aminikhah, Appl. Math. Model. 37, 5979 (2013)

    Article  MathSciNet  Google Scholar 

  171. H Xie and D Li, Appl. Math. Model. 37, 9215 (2013)

    Article  MathSciNet  Google Scholar 

  172. S-L Mei, J. Appl. Math. 2013, 1 (2013)

    Article  ADS  Google Scholar 

  173. X Ruyi, Math. Prob. Engng 2013, 1 (2013)

    Article  Google Scholar 

  174. Y Gao, L H Le and B C Shi, Appl. Math. Comput. 219, 7685 (2013)

    MathSciNet  Google Scholar 

  175. P J Olver, Applications of Lie groups to differential equations (Springer-Verlag, New York, 1986)

    Book  MATH  Google Scholar 

  176. H Liu, J Li and Q Zhang, J. Comput. Appl. Math. 228, 1 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  177. B M Boghosian, P Love and J Yepez, Philos. Trans.: Math. Phys. Engng Sci. 362, 1691 (2004)

    Article  ADS  Google Scholar 

  178. B Inan and A R Bahadir, Pramana – J. Phys. 81, 547 (2013)

    Article  ADS  Google Scholar 

  179. J Biazar, Z Ayati and S Shahbazi, Am. J. Numer. Anal. 2, 1 (2014)

    Google Scholar 

  180. O Diyer, O Chakrone and D Sbibih, Bol. da Soc. Par. de Mate. 33, 109 (2014)

    Google Scholar 

  181. M Sarboland and A Aminataei, J. Appl. Math. 2014, 1 (2014)

    Article  Google Scholar 

  182. I Ganaie and V Kukreja, Appl. Math. Comput. 237, 571 (2014)

    MathSciNet  Google Scholar 

  183. J Talwar and R Mohanty, Int. J. Model. Simul. Sci. Comput. 5, 1350029 (2014)

    Article  Google Scholar 

  184. X Liu, Y Zhou, L Zhang and J Wang, Sci. China Technol. Sci. 57, 1285 (2014)

    Article  Google Scholar 

  185. F Gao and C Chi, Appl. Math. Comput. 229, 414 (2014)

    MathSciNet  Google Scholar 

  186. F Ylmaz and B Karasözen, J. Comput. Appl. Math. 259, 771 (2014)

    Article  MathSciNet  Google Scholar 

  187. J Chung, Y J Kim and M Slemrod, J. Diff. Eqns 257, 2520 (2014)

    Article  ADS  Google Scholar 

  188. X Zhang, H Tian and W Chen, Comput. Math. Appl. 66, 2425 (2014)

    Article  MathSciNet  Google Scholar 

  189. A C Velivelli and K M Bryden, Adv. Engng Softw. 70, 104 (2014)

    Article  Google Scholar 

  190. V K Srivastava, M Tamsir, M K Awasthi and S Singh, AIP Adv. 4, 037119 (2014)

    Google Scholar 

  191. K Goyal and M Mehra, Comput. Math. Appl. 68, 568 (2014)

    Article  MathSciNet  Google Scholar 

  192. M Khan, Alex. Engng J. 53, 485 (2014)

    Article  Google Scholar 

  193. M Kumar and S Pandit, Comput. Phys. Commun. 185, 809 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  194. B M Vaganan and E E Priya, Pramana – J. Phys. 85, 861 (2015)

    Article  ADS  Google Scholar 

  195. M Sarboland and A Aminataei, Appl. Math. Inform. Sci. 9, 245 (2015)

    Article  Google Scholar 

  196. V Mukundan and A Awasthi, Appl. Math. Comput. 262, 282 (2015)

    MathSciNet  Google Scholar 

  197. A Prakash, M Kumar and K K Sharma, Appl. Math. Comput. 260, 314 (2015)

    MathSciNet  Google Scholar 

  198. E G Saleeby, J. Math. Anal. Appl. 425, 508 (2015)

    Article  MathSciNet  Google Scholar 

  199. R Jiwari, Comput. Phys. Commun. 188, 59 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  200. R Jiwari, Comput. Phys. Commun. 183, 2413 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  201. T Zhanlav, O Chuluunbaatar and V Ulziibayar, Appl. Math. Comput. 250, 701 (2015)

    MathSciNet  Google Scholar 

  202. R C Mittal and A Tripathi, Engng Comput. 32, 1275 (2015)

    Article  Google Scholar 

  203. R Mittal, H Kaur and V Mishra, Int. J. Comput. Math. 92, 1643 (2015)

    Article  MathSciNet  Google Scholar 

  204. R K Mohanty, W Dai and F Han, Appl. Math. Comput. 256, 381 (2015)

    MathSciNet  Google Scholar 

  205. M A Abdulwahhab, Commun. Nonlin. Sci. Numer. Simul. 20, 98 (2015)

    Article  MathSciNet  Google Scholar 

  206. H Bhatt and A Khaliq, Comput. Phys. Commun. C 200, 117 (2016)

    Article  ADS  Google Scholar 

  207. M Bonkile, A Awasthi and S Jayaraj, Int. J. Converg. Comput. 2, 54 (2016)

    Article  Google Scholar 

  208. M Seydaolu, U Erdoan and T Özi, J. Comput. Appl. Math. 291, 410 (2016)

    Article  MathSciNet  Google Scholar 

  209. Y Guo, Y Shi and Y Li, Appl. Math. Comput. 281, 172 (2016)

    MathSciNet  Google Scholar 

  210. R Sinuvasan, K M Tamizhmani and P G L Leach, Pramana – J. Phys. 88, 74 (2017)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous referees for their valuable time, effort and extensive comments which help to improve the quality of this paper.

Author information

Authors and Affiliations

  1. Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai,  400 076, India

    Mayur P Bonkile

  2. Department of Mathematics, National Institute of Technology Calicut, Kozhikode, 673 601, India

    Ashish Awasthi, C Lakshmi, Vijitha Mukundan & V S Aswin

Authors
  1. Mayur P Bonkile
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ashish Awasthi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C Lakshmi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vijitha Mukundan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. V S Aswin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ashish Awasthi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bonkile, M.P., Awasthi, A., Lakshmi, C. et al. A systematic literature review of Burgers’ equation with recent advances. Pramana - J Phys 90, 69 (2018). https://doi.org/10.1007/s12043-018-1559-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12043-018-1559-4

Keywords

PACS Nos

Search

Navigation