New exact solutions for a generalised Burgers-Fisher equation

New travelling wave solutions for a generalised Burgers-Fisher (GBF) equation are obtained. They arise from the solutions of nonlinear second-order equations that can be linearised by a generalised Sundman transformation. The reconstruction problem involves a one-parameter family of first-order equa...

Full description

Autores:
Mendoza, J.
Muriel, C.
Tipo de recurso:
http://purl.org/coar/resource_type/c_816b
Fecha de publicación:
2021
Institución:
Corporación Universidad de la Costa
Repositorio:
REDICUC - Repositorio CUC
Idioma:
eng
OAI Identifier:
oai:repositorio.cuc.edu.co:11323/8794
Acceso en línea:
https://hdl.handle.net/11323/8794
https://doi.org/10.1016/j.chaos.2021.111360
https://repositorio.cuc.edu.co/
Palabra clave:
Generalised sundman transformation
λ−Symmetries
Generalised
Burgers-Fisher equations
Travelling wave solutions
Rights
openAccess
License
CC0 1.0 Universal
id RCUC2_1c3df7682ad9922c3e33c65f333f8cbd
oai_identifier_str oai:repositorio.cuc.edu.co:11323/8794
network_acronym_str RCUC2
network_name_str REDICUC - Repositorio CUC
repository_id_str
dc.title.spa.fl_str_mv New exact solutions for a generalised Burgers-Fisher equation
title New exact solutions for a generalised Burgers-Fisher equation
spellingShingle New exact solutions for a generalised Burgers-Fisher equation
Generalised sundman transformation
λ−Symmetries
Generalised
Burgers-Fisher equations
Travelling wave solutions
title_short New exact solutions for a generalised Burgers-Fisher equation
title_full New exact solutions for a generalised Burgers-Fisher equation
title_fullStr New exact solutions for a generalised Burgers-Fisher equation
title_full_unstemmed New exact solutions for a generalised Burgers-Fisher equation
title_sort New exact solutions for a generalised Burgers-Fisher equation
dc.creator.fl_str_mv Mendoza, J.
Muriel, C.
dc.contributor.author.spa.fl_str_mv Mendoza, J.
Muriel, C.
dc.subject.spa.fl_str_mv Generalised sundman transformation
λ−Symmetries
Generalised
Burgers-Fisher equations
Travelling wave solutions
topic Generalised sundman transformation
λ−Symmetries
Generalised
Burgers-Fisher equations
Travelling wave solutions
description New travelling wave solutions for a generalised Burgers-Fisher (GBF) equation are obtained. They arise from the solutions of nonlinear second-order equations that can be linearised by a generalised Sundman transformation. The reconstruction problem involves a one-parameter family of first-order equations of Chini type. Firstly we obtain a unified expression of a one-parameter family of exact solutions, few of which have been reported in the recent literature by using hitherto not interrelated procedures, such as the tanh method, the modified tanh-coth method, the Exp-function method, the first integral method, or the improved expansion method. Upon certain condition on the coefficients of the GBF equation, the procedure successes in finding all the possible travelling wave solutions, given through a single expression depending on two arbitrary parameters, and expressed in terms of the Lerch Transcendent function. Finally, the case is completely solved, classifying all the admitted travelling wave solutions into either a one-parameter family of exponential solutions, or into a two-parameter family of solutions that involve Bessel functions and modified Bessel functions. For particular subclasses of the GBF equation new families of solutions, depending on one or two arbitrary parameters and given in terms of the exponential, trigonometric, and hyperbolic functions, are also reported.
publishDate 2021
dc.date.accessioned.none.fl_str_mv 2021-10-21T13:48:12Z
dc.date.available.none.fl_str_mv 2021-10-21T13:48:12Z
dc.date.issued.none.fl_str_mv 2021
dc.date.embargoEnd.none.fl_str_mv 2023
dc.type.spa.fl_str_mv Pre-Publicación
dc.type.coar.spa.fl_str_mv http://purl.org/coar/resource_type/c_816b
dc.type.content.spa.fl_str_mv Text
dc.type.driver.spa.fl_str_mv info:eu-repo/semantics/preprint
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dc.type.version.spa.fl_str_mv info:eu-repo/semantics/acceptedVersion
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status_str acceptedVersion
dc.identifier.issn.spa.fl_str_mv 0960-0779
dc.identifier.uri.spa.fl_str_mv https://hdl.handle.net/11323/8794
dc.identifier.doi.spa.fl_str_mv https://doi.org/10.1016/j.chaos.2021.111360
dc.identifier.instname.spa.fl_str_mv Corporación Universidad de la Costa
dc.identifier.reponame.spa.fl_str_mv REDICUC - Repositorio CUC
dc.identifier.repourl.spa.fl_str_mv https://repositorio.cuc.edu.co/
identifier_str_mv 0960-0779
Corporación Universidad de la Costa
REDICUC - Repositorio CUC
url https://hdl.handle.net/11323/8794
https://doi.org/10.1016/j.chaos.2021.111360
https://repositorio.cuc.edu.co/
dc.language.iso.none.fl_str_mv eng
language eng
dc.relation.references.spa.fl_str_mv [1] H. Bateman Some recent researches on the motion of fluids Mon Weather Rev, 43 (4) (1915), pp. 163-170
[2] J.M. Burgers A mathematical model illustrating the theory of turbulence Advances in Applied Mechanics, 1, Elsevier (1948), pp. 171-199
[3] J. Murray On Burgers’ model equations for turbulence J Fluid Mech, 59 (2) (1973), pp. 263-279
[4] J. Yepez An efficient quantum algorithm for the one-dimensional burgers equation arXiv preprint quant-ph/0210092 (2002)
[5] R.S. Hirsh Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique J Comput Phys, 19 (1) (1975), pp. 90-109
[6] B. Greenshields, J. Bibbins, W. Channing, H. Miller A study of traffic capacity Highway Research Board Proceedings, 1935, National Research Council (USA), Highway Research Board (1935)
[7] R.A. Fisher The wave of advance of advantageous genes Ann Eugen, 7 (4) (1937), pp. 355-369
[8] A.N. Kolmogorov Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique Bull Univ Moskow, Ser-Internat, Sec A, 1 (1937), pp. 1-25
[9] N.F. Britton, et al. Reaction-Diffusion equations and their applications to biology Academic Press (1986)
[10] D.G. Aronson, H.F. Weinberger Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation Partial Differential Equations and Related Topics, Springer (1975), pp. 5-49
[11] D.A. Frank-Kamenetskii Diffusion and heat exchange in chemical kinetics Princeton University Press (2015)
[12] M.D. Bramson Maximal displacement of branching brownian motion Commun Pure Appl Math, 31 (5) (1978), pp. 531-581
[13] J. Canosa Diffusion in nonlinear multiplicative media J Math Phys, 10 (10) (1969), pp. 1862-1868
[14] J. Satsuma, M. Ablowitz, B. Fuchssteiner, M. Kruskal Topics in soliton theory and exactly solvable nonlinear equations World Scientific, Singapore City (1987)
[15] M. Nadeem, F. Li, H. Ahmad Modified laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients Computers & Mathematics with Applications, 78 (6) (2019), pp. 2052-2062
[16] A.C. Loyinmi, T.K. Akinfe Exact solutions to the family of Fisher’s reaction-diffusion equation using elzaki homotopy transformation perturbation method Engineering Reports, 2 (2) (2020), p. e12084
[17] T.K. Akinfe, A.C. Loyinmi A solitary wave solution to the generalized Burgers-Fisher’s equation using an improved differential transform method: a hybrid scheme approach Heliyon, 7 (5) (2021), p. e07001
[18] R.K. Mohanty, S. Sharma A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear burgers equations Eng Comput (Swansea) (2020)
[19] A. Kumar Verma, S. Kayenat On the stability of micken’s type NSFD schemes for generalized Burgers fisher equation Journal of Difference Equations and Applications, 25 (12) (2019), pp. 1706-1737
[20] A.-M. Wazwaz The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations Appl Math Comput, 169 (1) (2005), pp. 321-338
[21] L. Wazzan A modified tanh–coth method for solving the general Burgers–Fisher and the kuramoto–sivashinsky equations Commun Nonlinear Sci Numer Simul, 14 (6) (2009), pp. 2642-2652
[22] Z.-h. Xu, D.-q. Xian Application of exp-function method to generalized Burgers-Fisher equation Acta Mathematicae Applicatae Sinica, English Series, 26 (4) (2010), pp. 669-676
[23] J. Lu, G. Yu-Cui, X. Shu-Jiang Some new exact solutions to the Burgers-Fisher equation and generalized burgers–fisher equation Chin Phys, 16 (9) (2007), p. 2514
[24] R.T. Redi, Y. Obsie, A. Shiferaw The improved (g/g)-expansion method to the generalized Burgers-Fisher equation Mathematical Modelling and Applications, 3 (1) (2018), p. 16
[25] A.A. Hassaballa, T.M. Elzaki Applications of the improved (g’/g) expansion method for solve Burgers-Fisher equation J Comput Theor Nanosci, 14 (10) (2017), pp. 4664-4668
[26] H. Chen, H. Zhang New multiple soliton solutions to the general Burgers-Fisher equation and the kuramoto–sivashinsky equation Chaos, Solitons & Fractals, 19 (1) (2004), pp. 71-76
[27] C. Muriel, J.L. Romero New methods of reduction for ordinary differential equations IMA J Appl Math, 66 (2) (2001), pp. 111-125
[28] C. Muriel, J. Romero First integrals, integrating factors and -symmetries of second-order differential equations J Phys A: Math Theor, 42 (36) (2009), p. 365207 View PDFCrossRefView Record in ScopusGoogle Scholar
[29] J. Mendoza, C. Muriel Exact solutions and riccati-type first integrals J Nonlinear Math Phys, 24 (sup1) (2017), pp. 75-89
[30] S. Lie Klassifikation und integration von gewöhnlichen differentialgleichungen zwischen x, y, die eine gruppe von transformationen gestatten. iii Arch Mat Naturvidenskab, 8 (1883), pp. 371-458
[31] L. Duarte, F. Santos, I. Moreira Linearisation under non-point transformations Tech. Rep., SCAN/9408138 (1993)
[32] N. Euler, M. Euler Sundman symmetries of nonlinear second-order and third-order ordinary differential equations J Nonlinear Math Phys, 11 (3) (2004), pp. 399-421
[33] N. Euler, T. Wolf, P. Leach, M. Euler Linearisable third-order ordinary differential equations and generalised sundman transformations: the case Acta Applicandae Mathematica, 76 (1) (2003), pp. 89-115
[34] S. Moyo, S.V. Meleshko Application of the generalised sundman transformation to the linearisation of two second-order ordinary differential equations J Nonlinear Math Phys, 18 (sup1) (2011), pp. 213-236 View PDFCrossRefView Record in ScopusGoogle Scholar
[35] P.J. Olver Applications of lie groups to differential equations 107, Springer Science & Business Media (2000) Google Scholar
[36] E. Pucci, G. Saccomandi On the reduction methods for ordinary differential equations J Phys A Math Gen, 35 (29) (2002), p. 6145
[37] D.C. Ferraioli Nonlocal aspects of -symmetries and ODEs reduction J Phys A: Math Theor, 40 (21) (2007), p. 5479
[38] G. Bluman, S. Anco Symmetry and integration methods for differential equations 154, Springer Science & Business Media (2008)
[39] G.W. Bluman, S. Kumei Symmetries and differential equations 81, Springer Science & Business Media (2013)
[40] Cheb-Terrab E, Roche A. Abel equations: equivalence and new integrable classes. Comput Phys Commun 200; 130.
[41] E. Kamke Differentialgleichungen lȵsungsmethoden und lȵsungen. i: gewöhnliche Vieweg+Teubner Verlag (1977)
[42] H. Bateman Higher transcendental functions 1–3, McGraw-Hill Book Company (1953)
[43] F.W. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark NIST Handbook of mathematical functions hardback and CD-ROM Cambridge University Press (2010)
[44] V.F. Zaitsev, A.D. Polyanin Handbook of exact solutions for ordinary differential equations CRC press (2002)
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dc.source.spa.fl_str_mv Chaos, Solitons and Fractals
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spelling Mendoza, J.Muriel, C.2021-10-21T13:48:12Z2021-10-21T13:48:12Z202120230960-0779https://hdl.handle.net/11323/8794https://doi.org/10.1016/j.chaos.2021.111360Corporación Universidad de la CostaREDICUC - Repositorio CUChttps://repositorio.cuc.edu.co/New travelling wave solutions for a generalised Burgers-Fisher (GBF) equation are obtained. They arise from the solutions of nonlinear second-order equations that can be linearised by a generalised Sundman transformation. The reconstruction problem involves a one-parameter family of first-order equations of Chini type. Firstly we obtain a unified expression of a one-parameter family of exact solutions, few of which have been reported in the recent literature by using hitherto not interrelated procedures, such as the tanh method, the modified tanh-coth method, the Exp-function method, the first integral method, or the improved expansion method. Upon certain condition on the coefficients of the GBF equation, the procedure successes in finding all the possible travelling wave solutions, given through a single expression depending on two arbitrary parameters, and expressed in terms of the Lerch Transcendent function. Finally, the case is completely solved, classifying all the admitted travelling wave solutions into either a one-parameter family of exponential solutions, or into a two-parameter family of solutions that involve Bessel functions and modified Bessel functions. For particular subclasses of the GBF equation new families of solutions, depending on one or two arbitrary parameters and given in terms of the exponential, trigonometric, and hyperbolic functions, are also reported.Mendoza, J.Muriel, C.application/pdfengCorporación Universidad de la CostaCC0 1.0 Universalhttp://creativecommons.org/publicdomain/zero/1.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Chaos, Solitons and Fractalshttps://www.sciencedirect.com/science/article/pii/S0960077921007141#!Generalised sundman transformationλ−SymmetriesGeneralisedBurgers-Fisher equationsTravelling wave solutionsNew exact solutions for a generalised Burgers-Fisher equationPre-Publicaciónhttp://purl.org/coar/resource_type/c_816bTextinfo:eu-repo/semantics/preprinthttp://purl.org/redcol/resource_type/ARTOTRinfo:eu-repo/semantics/acceptedVersion[1] H. Bateman Some recent researches on the motion of fluids Mon Weather Rev, 43 (4) (1915), pp. 163-170[2] J.M. Burgers A mathematical model illustrating the theory of turbulence Advances in Applied Mechanics, 1, Elsevier (1948), pp. 171-199[3] J. Murray On Burgers’ model equations for turbulence J Fluid Mech, 59 (2) (1973), pp. 263-279[4] J. Yepez An efficient quantum algorithm for the one-dimensional burgers equation arXiv preprint quant-ph/0210092 (2002)[5] R.S. Hirsh Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique J Comput Phys, 19 (1) (1975), pp. 90-109[6] B. Greenshields, J. Bibbins, W. Channing, H. Miller A study of traffic capacity Highway Research Board Proceedings, 1935, National Research Council (USA), Highway Research Board (1935)[7] R.A. Fisher The wave of advance of advantageous genes Ann Eugen, 7 (4) (1937), pp. 355-369[8] A.N. Kolmogorov Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique Bull Univ Moskow, Ser-Internat, Sec A, 1 (1937), pp. 1-25[9] N.F. Britton, et al. Reaction-Diffusion equations and their applications to biology Academic Press (1986)[10] D.G. Aronson, H.F. Weinberger Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation Partial Differential Equations and Related Topics, Springer (1975), pp. 5-49[11] D.A. Frank-Kamenetskii Diffusion and heat exchange in chemical kinetics Princeton University Press (2015)[12] M.D. Bramson Maximal displacement of branching brownian motion Commun Pure Appl Math, 31 (5) (1978), pp. 531-581[13] J. Canosa Diffusion in nonlinear multiplicative media J Math Phys, 10 (10) (1969), pp. 1862-1868[14] J. Satsuma, M. Ablowitz, B. Fuchssteiner, M. Kruskal Topics in soliton theory and exactly solvable nonlinear equations World Scientific, Singapore City (1987)[15] M. Nadeem, F. Li, H. Ahmad Modified laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients Computers & Mathematics with Applications, 78 (6) (2019), pp. 2052-2062[16] A.C. Loyinmi, T.K. Akinfe Exact solutions to the family of Fisher’s reaction-diffusion equation using elzaki homotopy transformation perturbation method Engineering Reports, 2 (2) (2020), p. e12084[17] T.K. Akinfe, A.C. Loyinmi A solitary wave solution to the generalized Burgers-Fisher’s equation using an improved differential transform method: a hybrid scheme approach Heliyon, 7 (5) (2021), p. e07001[18] R.K. Mohanty, S. Sharma A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear burgers equations Eng Comput (Swansea) (2020)[19] A. Kumar Verma, S. Kayenat On the stability of micken’s type NSFD schemes for generalized Burgers fisher equation Journal of Difference Equations and Applications, 25 (12) (2019), pp. 1706-1737[20] A.-M. Wazwaz The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations Appl Math Comput, 169 (1) (2005), pp. 321-338[21] L. Wazzan A modified tanh–coth method for solving the general Burgers–Fisher and the kuramoto–sivashinsky equations Commun Nonlinear Sci Numer Simul, 14 (6) (2009), pp. 2642-2652[22] Z.-h. Xu, D.-q. Xian Application of exp-function method to generalized Burgers-Fisher equation Acta Mathematicae Applicatae Sinica, English Series, 26 (4) (2010), pp. 669-676[23] J. Lu, G. Yu-Cui, X. Shu-Jiang Some new exact solutions to the Burgers-Fisher equation and generalized burgers–fisher equation Chin Phys, 16 (9) (2007), p. 2514[24] R.T. Redi, Y. Obsie, A. Shiferaw The improved (g/g)-expansion method to the generalized Burgers-Fisher equation Mathematical Modelling and Applications, 3 (1) (2018), p. 16[25] A.A. Hassaballa, T.M. Elzaki Applications of the improved (g’/g) expansion method for solve Burgers-Fisher equation J Comput Theor Nanosci, 14 (10) (2017), pp. 4664-4668[26] H. Chen, H. Zhang New multiple soliton solutions to the general Burgers-Fisher equation and the kuramoto–sivashinsky equation Chaos, Solitons & Fractals, 19 (1) (2004), pp. 71-76[27] C. Muriel, J.L. Romero New methods of reduction for ordinary differential equations IMA J Appl Math, 66 (2) (2001), pp. 111-125[28] C. Muriel, J. Romero First integrals, integrating factors and -symmetries of second-order differential equations J Phys A: Math Theor, 42 (36) (2009), p. 365207 View PDFCrossRefView Record in ScopusGoogle Scholar[29] J. Mendoza, C. Muriel Exact solutions and riccati-type first integrals J Nonlinear Math Phys, 24 (sup1) (2017), pp. 75-89[30] S. Lie Klassifikation und integration von gewöhnlichen differentialgleichungen zwischen x, y, die eine gruppe von transformationen gestatten. iii Arch Mat Naturvidenskab, 8 (1883), pp. 371-458[31] L. Duarte, F. Santos, I. Moreira Linearisation under non-point transformations Tech. Rep., SCAN/9408138 (1993)[32] N. Euler, M. Euler Sundman symmetries of nonlinear second-order and third-order ordinary differential equations J Nonlinear Math Phys, 11 (3) (2004), pp. 399-421[33] N. Euler, T. Wolf, P. Leach, M. Euler Linearisable third-order ordinary differential equations and generalised sundman transformations: the case Acta Applicandae Mathematica, 76 (1) (2003), pp. 89-115[34] S. Moyo, S.V. Meleshko Application of the generalised sundman transformation to the linearisation of two second-order ordinary differential equations J Nonlinear Math Phys, 18 (sup1) (2011), pp. 213-236 View PDFCrossRefView Record in ScopusGoogle Scholar[35] P.J. Olver Applications of lie groups to differential equations 107, Springer Science & Business Media (2000) Google Scholar[36] E. Pucci, G. Saccomandi On the reduction methods for ordinary differential equations J Phys A Math Gen, 35 (29) (2002), p. 6145[37] D.C. Ferraioli Nonlocal aspects of -symmetries and ODEs reduction J Phys A: Math Theor, 40 (21) (2007), p. 5479[38] G. Bluman, S. Anco Symmetry and integration methods for differential equations 154, Springer Science & Business Media (2008)[39] G.W. Bluman, S. Kumei Symmetries and differential equations 81, Springer Science & Business Media (2013)[40] Cheb-Terrab E, Roche A. Abel equations: equivalence and new integrable classes. Comput Phys Commun 200; 130.[41] E. Kamke Differentialgleichungen lȵsungsmethoden und lȵsungen. i: gewöhnliche Vieweg+Teubner Verlag (1977)[42] H. Bateman Higher transcendental functions 1–3, McGraw-Hill Book Company (1953)[43] F.W. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark NIST Handbook of mathematical functions hardback and CD-ROM Cambridge University Press (2010)[44] V.F. Zaitsev, A.D. 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