• 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]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

[2]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[3]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[4]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[5]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[6]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[7]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[8]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[9]

Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307

[10]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[11]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164

[12]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[13]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115

[14]

D. R. Michiel Renger, Johannes Zimmer. Orthogonality of fluxes in general nonlinear reaction networks. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 205-217. doi: 10.3934/dcdss.2020346

[15]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

[16]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[17]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[18]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[19]

Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial & Management Optimization, 2021, 17 (1) : 133-149. doi: 10.3934/jimo.2019103

[20]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (125)
  • HTML views (312)
  • Cited by (0)

Other articles
by authors

[Back to Top]