• Previous Article
    Quasilinearization applied to boundary value problems at resonance for Riemann-Liouville fractional differential equations
  • DCDS-S Home
  • This Issue
  • Next Article
    Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion
October  2020, 13(10): 2703-2717. doi: 10.3934/dcdss.2020218

Symmetries and conservation laws of a time dependent nonlinear reaction-convection-diffusion equation

1. 

Department of Mathematics, College of Science, China Three Gorges University, Yichang, Hubei 443002, China

2. 

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China

3. 

African Institute for Mathematical Sciences, Muizenberg, Cape Town, South Africa

* Corresponding author: Lijun Zhang

Received  January 2019 Revised  June 2019 Published  December 2019

Fund Project: The first author is funded by the China Scholarship Fund (No.201908420198). The second author is supported by NSF grant No. 11672270. The revision was done during L Zhang's research visit to African Institute for Mathematical Sciences South Africa

In this paper, we study a nonlinear diffusion-convection-reaction equation with a variable coefficient which has applications in many fields. The Lie point symmetries of this equation are derived, according to which this equation is classified into four different kinds. Conservation laws for this equation are constructed by using the conservation theorem of Ibragimov.

Citation: Zhijie Cao, Lijun Zhang. Symmetries and conservation laws of a time dependent nonlinear reaction-convection-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2703-2717. doi: 10.3934/dcdss.2020218
References:
[1]

S. C. Anco and G. W. Bluman, Direct construction of conservation laws from field equations, Phys. Rev. Lett., 78 (1997), 2869-2873.  doi: 10.1103/PhysRevLett.78.2869.  Google Scholar

[2]

E. D. AvdoninaN. H. Ibragimov and R. Khamitova, Exact solutions of gasdynamic equaitons obtained by the method of conservation laws, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 2359-2366.  doi: 10.1016/j.cnsns.2012.12.023.  Google Scholar

[3]

G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations, Applications mathematics series, 154. Springer, 2002. Google Scholar

[4]

G. W. BlumanTe muerchaolu and S. C. Anco, New conservation laws obtained directly from symmetry action on a known conservation law, J. Math. Anal. Appl., 322 (2006), 233-250.  doi: 10.1016/j.jmaa.2005.08.092.  Google Scholar

[5]

I. L. Freire, Conservation laws for self-adjoint first order evolution equation, J. Nonlin. Math. Phys., 18 (2011), 279-290.  doi: 10.1142/S1402925111001453.  Google Scholar

[6]

I. L. Freire, New conservation laws for inviscid Burgers equation, Comp. Appl. Math., 31 (2012), 559-567.  doi: 10.1590/S1807-03022012000300007.  Google Scholar

[7]

I. L. Freire and J. C. S. Sampaio, On the nonlinear self-adjointness and local conservation laws for a class of evolution equations unifying many models, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 350-360.  doi: 10.1016/j.cnsns.2013.06.010.  Google Scholar

[8]

N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311-328.  doi: 10.1016/j.jmaa.2006.10.078.  Google Scholar

[9]

N. H. Ibragimov, Integrating factors, adjoint equations and Lagrangians, J. Math. Anal. Appl., 318 (2006), 742-757.  doi: 10.1016/j.jmaa.2005.11.012.  Google Scholar

[10]

N. H.Ibragimov, Method of conservation laws for constructing solutions to systems of PDEs, Disc. Nonlinearity compl., 1 (2012), 353-362.   Google Scholar

[11]

N. H. Ibragimov and T. Kolsrud, Lagrangian approach to evolution equations: Symmetries and conservation laws, Nonlinear Dyn., 36 (2004), 29-40.  doi: 10.1023/B:NODY.0000034644.82259.1f.  Google Scholar

[12]

A. H. Kara and F. M. Mahomed, Relationship between symmetries and conservation laws, Internat. J. Theoret. Phys., 39 (2000), 23-40.  doi: 10.1023/A:1003686831523.  Google Scholar

[13]

A. H. Kara and F. M. Mahomed, Noether-type symmetries and conservation laws via partial Lagrangians, Nonlinear Dyn., 45 (2006), 367-383.  doi: 10.1007/s11071-005-9013-9.  Google Scholar

[14]

A. Mishra and R. Kumar, Exact solutions of variable coefficient nonlinear diffusion-reaction equations with a nonliear convective term, Phys. Lett. A, 374 (2010), 2921-2924.  doi: 10.1016/j.physleta.2010.03.039.  Google Scholar

[15]

B. Muatjetjeja and C. M. Khalique, First integrals for a generalized coupled Lane-Emden system, Nonlinear Anal. Real World Appl., 12 (2011), 1202-1212.  doi: 10.1016/j.nonrwa.2010.09.013.  Google Scholar

[16]

J. D. Murray, Mathematical Biology, Springer-Verlag, New York, 1993. Google Scholar

[17]

P. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986. Google Scholar

[18]

R. Popovych, Direct methods of construction of conservation laws, Physics AUC, 16 (2006), 81-947.   Google Scholar

[19]

J. C. S. Sampaio and I. L. Freire, Nonlinear self-adjoint classification of a Burgers-KdV family of equations, Abs. Appl. Anal., 2014 (2014), 1-7.  doi: 10.1155/2014/804703.  Google Scholar

[20]

H. Steudel, Noether's theorem and the conservation laws of the Korteweg-de Vries equation, Ann. Physik, 32 (1975), 445-455.  doi: 10.1002/andp.19754870605.  Google Scholar

show all references

References:
[1]

S. C. Anco and G. W. Bluman, Direct construction of conservation laws from field equations, Phys. Rev. Lett., 78 (1997), 2869-2873.  doi: 10.1103/PhysRevLett.78.2869.  Google Scholar

[2]

E. D. AvdoninaN. H. Ibragimov and R. Khamitova, Exact solutions of gasdynamic equaitons obtained by the method of conservation laws, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 2359-2366.  doi: 10.1016/j.cnsns.2012.12.023.  Google Scholar

[3]

G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations, Applications mathematics series, 154. Springer, 2002. Google Scholar

[4]

G. W. BlumanTe muerchaolu and S. C. Anco, New conservation laws obtained directly from symmetry action on a known conservation law, J. Math. Anal. Appl., 322 (2006), 233-250.  doi: 10.1016/j.jmaa.2005.08.092.  Google Scholar

[5]

I. L. Freire, Conservation laws for self-adjoint first order evolution equation, J. Nonlin. Math. Phys., 18 (2011), 279-290.  doi: 10.1142/S1402925111001453.  Google Scholar

[6]

I. L. Freire, New conservation laws for inviscid Burgers equation, Comp. Appl. Math., 31 (2012), 559-567.  doi: 10.1590/S1807-03022012000300007.  Google Scholar

[7]

I. L. Freire and J. C. S. Sampaio, On the nonlinear self-adjointness and local conservation laws for a class of evolution equations unifying many models, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 350-360.  doi: 10.1016/j.cnsns.2013.06.010.  Google Scholar

[8]

N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311-328.  doi: 10.1016/j.jmaa.2006.10.078.  Google Scholar

[9]

N. H. Ibragimov, Integrating factors, adjoint equations and Lagrangians, J. Math. Anal. Appl., 318 (2006), 742-757.  doi: 10.1016/j.jmaa.2005.11.012.  Google Scholar

[10]

N. H.Ibragimov, Method of conservation laws for constructing solutions to systems of PDEs, Disc. Nonlinearity compl., 1 (2012), 353-362.   Google Scholar

[11]

N. H. Ibragimov and T. Kolsrud, Lagrangian approach to evolution equations: Symmetries and conservation laws, Nonlinear Dyn., 36 (2004), 29-40.  doi: 10.1023/B:NODY.0000034644.82259.1f.  Google Scholar

[12]

A. H. Kara and F. M. Mahomed, Relationship between symmetries and conservation laws, Internat. J. Theoret. Phys., 39 (2000), 23-40.  doi: 10.1023/A:1003686831523.  Google Scholar

[13]

A. H. Kara and F. M. Mahomed, Noether-type symmetries and conservation laws via partial Lagrangians, Nonlinear Dyn., 45 (2006), 367-383.  doi: 10.1007/s11071-005-9013-9.  Google Scholar

[14]

A. Mishra and R. Kumar, Exact solutions of variable coefficient nonlinear diffusion-reaction equations with a nonliear convective term, Phys. Lett. A, 374 (2010), 2921-2924.  doi: 10.1016/j.physleta.2010.03.039.  Google Scholar

[15]

B. Muatjetjeja and C. M. Khalique, First integrals for a generalized coupled Lane-Emden system, Nonlinear Anal. Real World Appl., 12 (2011), 1202-1212.  doi: 10.1016/j.nonrwa.2010.09.013.  Google Scholar

[16]

J. D. Murray, Mathematical Biology, Springer-Verlag, New York, 1993. Google Scholar

[17]

P. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986. Google Scholar

[18]

R. Popovych, Direct methods of construction of conservation laws, Physics AUC, 16 (2006), 81-947.   Google Scholar

[19]

J. C. S. Sampaio and I. L. Freire, Nonlinear self-adjoint classification of a Burgers-KdV family of equations, Abs. Appl. Anal., 2014 (2014), 1-7.  doi: 10.1155/2014/804703.  Google Scholar

[20]

H. Steudel, Noether's theorem and the conservation laws of the Korteweg-de Vries equation, Ann. Physik, 32 (1975), 445-455.  doi: 10.1002/andp.19754870605.  Google Scholar

Table 1.  The results for symmetries of equation (1.1)
Cases Sol. of determining eq. Infinitesimal generator Restriction on $ k $
$ \xi=\frac{1}{2}c_{1}x+ $ $ X= $
$ a=0 $ $ c_4\int{k(t)dt}+c_{3}, $ $ c_1(\frac12 x\partial_x+t\partial_t) $
$ b=0 $ $ \tau=c_{1}t+c_{2}, $ $ +c_2\partial_t+c_3\partial_x+c_5u\partial_u $ $ (c_5+\frac{1}{2}c_1)k+ $
$ \phi=c_{5}u+c_{4}. $ $ +c_4(\int{k(t)dt}\partial_x+\partial_u) $ $ (c_1t+c_2)k'=0 $
$ \xi=\frac{1}{2}c_{1}x+c_{3}, $ X=
$ a=0 $ $ \tau=c_{1}t+c_{2}, $ $ c_1(\frac12x\partial_x+t\partial_t-u\partial_u) $ $ (c_1t+c_2)k'- $
$ b\neq 0 $ $ \phi=-c_{1}u $ $ +c_2\partial_t+c_3\partial_x. $ $ \frac{1}{2}c_1k(t)=0 $
$ \xi=\frac{1}{2}c_{1}x+ $ $ X= $
$ a\neq 0 $ $ c_4\int{e^{at}kdt}+c_{3}, $ $ c_1(\frac12x\partial_x+t\partial_t+atu\partial_u) $
$ b=0 $ $ \tau=c_{1}t+c_{2}, $ $ +c_2\partial_t+ c_3\partial_x+c_4(e^{at}\partial_u $ $ (c_1(at+\frac12)+c_5)k $
$ \phi=(c_{1}at+c_5)u+c_4e^{at} $ $ +\int{e^{at}kdt}\partial_x)+c_5u\partial_u $ $ +(c_1t+c_2)k'=0 $
$ {\xi=c_{3}}, $ $ X= $ ${c_2k' = 0}$
$a\neq 0$ ${\tau = c_{2}}, $ ${c_2\partial_t+c_3\partial_x}.$
$b\neq0$ ${\phi = 0}$
Cases Sol. of determining eq. Infinitesimal generator Restriction on $ k $
$ \xi=\frac{1}{2}c_{1}x+ $ $ X= $
$ a=0 $ $ c_4\int{k(t)dt}+c_{3}, $ $ c_1(\frac12 x\partial_x+t\partial_t) $
$ b=0 $ $ \tau=c_{1}t+c_{2}, $ $ +c_2\partial_t+c_3\partial_x+c_5u\partial_u $ $ (c_5+\frac{1}{2}c_1)k+ $
$ \phi=c_{5}u+c_{4}. $ $ +c_4(\int{k(t)dt}\partial_x+\partial_u) $ $ (c_1t+c_2)k'=0 $
$ \xi=\frac{1}{2}c_{1}x+c_{3}, $ X=
$ a=0 $ $ \tau=c_{1}t+c_{2}, $ $ c_1(\frac12x\partial_x+t\partial_t-u\partial_u) $ $ (c_1t+c_2)k'- $
$ b\neq 0 $ $ \phi=-c_{1}u $ $ +c_2\partial_t+c_3\partial_x. $ $ \frac{1}{2}c_1k(t)=0 $
$ \xi=\frac{1}{2}c_{1}x+ $ $ X= $
$ a\neq 0 $ $ c_4\int{e^{at}kdt}+c_{3}, $ $ c_1(\frac12x\partial_x+t\partial_t+atu\partial_u) $
$ b=0 $ $ \tau=c_{1}t+c_{2}, $ $ +c_2\partial_t+ c_3\partial_x+c_4(e^{at}\partial_u $ $ (c_1(at+\frac12)+c_5)k $
$ \phi=(c_{1}at+c_5)u+c_4e^{at} $ $ +\int{e^{at}kdt}\partial_x)+c_5u\partial_u $ $ +(c_1t+c_2)k'=0 $
$ {\xi=c_{3}}, $ $ X= $ ${c_2k' = 0}$
$a\neq 0$ ${\tau = c_{2}}, $ ${c_2\partial_t+c_3\partial_x}.$
$b\neq0$ ${\phi = 0}$
[1]

María Rosa, María de los Santos Bruzón, María de la Luz Gandarias. Lie symmetries and conservation laws of a Fisher equation with nonlinear convection term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1331-1339. doi: 10.3934/dcdss.2015.8.1331

[2]

María-Santos Bruzón, Elena Recio, Tamara-María Garrido, Rafael de la Rosa. Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2691-2701. doi: 10.3934/dcdss.2020222

[3]

Chaudry Masood Khalique, Muhammad Usman, Maria Luz Gandarais. Nonlinear differential equations: Lie symmetries, conservation laws and other approaches of solving. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : i-ii. doi: 10.3934/dcdss.2020415

[4]

María Santos Bruzón, Tamara María Garrido. Symmetries and conservation laws of a KdV6 equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 631-641. doi: 10.3934/dcdss.2018038

[5]

Wen-Xiu Ma. Conservation laws by symmetries and adjoint symmetries. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 707-721. doi: 10.3934/dcdss.2018044

[6]

Giovany M. Figueiredo, Tarcyana S. Figueiredo-Sousa, Cristian Morales-Rodrigo, Antonio Suárez. Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3689-3711. doi: 10.3934/dcdsb.2018311

[7]

Elena Beretta, Cecilia Cavaterra. Identifying a space dependent coefficient in a reaction-diffusion equation. Inverse Problems & Imaging, 2011, 5 (2) : 285-296. doi: 10.3934/ipi.2011.5.285

[8]

Stephen Anco, Maria Rosa, Maria Luz Gandarias. Conservation laws and symmetries of time-dependent generalized KdV equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 607-615. doi: 10.3934/dcdss.2018035

[9]

Carlos Fresneda-Portillo. A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5097-5114. doi: 10.3934/cpaa.2020228

[10]

Alexander Bobylev, Mirela Vinerean, Åsa Windfäll. Discrete velocity models of the Boltzmann equation and conservation laws. Kinetic & Related Models, 2010, 3 (1) : 35-58. doi: 10.3934/krm.2010.3.35

[11]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493

[12]

Rafael de la Rosa, María Santos Bruzón. Differential invariants of a generalized variable-coefficient Gardner equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 747-757. doi: 10.3934/dcdss.2018047

[13]

Martin Oberlack, Andreas Rosteck. New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 451-471. doi: 10.3934/dcdss.2010.3.451

[14]

Özlem Orhan, Teoman Özer. New conservation forms and Lie algebras of Ermakov-Pinney equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 735-746. doi: 10.3934/dcdss.2018046

[15]

Iryna Pankratova, Andrey Piatnitski. Homogenization of convection-diffusion equation in infinite cylinder. Networks & Heterogeneous Media, 2011, 6 (1) : 111-126. doi: 10.3934/nhm.2011.6.111

[16]

Vitali Vougalter, Vitaly Volpert. On the solvability conditions for the diffusion equation with convection terms. Communications on Pure & Applied Analysis, 2012, 11 (1) : 365-373. doi: 10.3934/cpaa.2012.11.365

[17]

Gianluca Crippa, Laura V. Spinolo. An overview on some results concerning the transport equation and its applications to conservation laws. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1283-1293. doi: 10.3934/cpaa.2010.9.1283

[18]

Xueying Wang, Drew Posny, Jin Wang. A reaction-convection-diffusion model for cholera spatial dynamics. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2785-2809. doi: 10.3934/dcdsb.2016073

[19]

Grigori Chapiro, Lucas Furtado, Dan Marchesin, Stephen Schecter. Stability of interacting traveling waves in reaction-convection-diffusion systems. Conference Publications, 2015, 2015 (special) : 258-266. doi: 10.3934/proc.2015.0258

[20]

Jacson Simsen, Mariza Stefanello Simsen, Marcos Roberto Teixeira Primo. Reaction-Diffusion equations with spatially variable exponents and large diffusion. Communications on Pure & Applied Analysis, 2016, 15 (2) : 495-506. doi: 10.3934/cpaa.2016.15.495

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (110)
  • HTML views (312)
  • Cited by (0)

Other articles
by authors

[Back to Top]