Invariant Solutions of Generalized Fisher-KPP Equation


  • Mehdi Nadjafikhah Prof.


Fisher-KPP equation, ‎Lie symmetry metho, ‎Invariant solution, Conditional symmetry, ‎Approximate symmetry


‎In this paper‎, ‎we consider a hyperbolic generalized Fisher-KPP equation‎: ‎$\varepsilon^2 u_{tt}‎ + ‎g(u) u_t = ( k(u) u_x )_x‎ +‎f(u)$ where $f$‎, ‎$g $ ‎‎‎and $k$ are arbitrary smooth functions of variable $u$ and $\varepsilon$ is a speed parameter‎. ‎We find invariant solutions by Lie method‎. ‎Also‎, ‎we study standard and weak conditional and approximate symmetries‎.


Download data is not yet available.


‎\bibitem{[1]}{\sc J‎. ‎Adler}‎, ‎{\em Chemotaxis in bacteria}‎, ‎Science 153 (1966)‎, ‎708-716‎.
‎\bibitem{[2]}{\sc G.W‎. ‎Bluman} and {\sc J.D‎. ‎Cole}‎, ‎{\em Similarity methods for differential equations}‎, ‎Appl‎. ‎Math‎. ‎Sci‎, ‎No‎. ‎13‎, ‎Springer-Verlag‎, ‎New York‎, ‎1974‎.
‎\bibitem{[3]}{\sc E‎. ‎Bouin}‎, ‎{\sc V‎. ‎Calvezyz} and {\sc G‎. ‎Nadin}‎, ‎{\em Hyperbolic traveling waves driven by growth}‎, ‎2011‎, ‎‎.
‎\bibitem{[4]}{\sc V.A‎. ‎Baikov}‎, ‎{\sc R.K‎. ‎Gazizov} and {\sc N.H‎. ‎Ibragimov.}‎, ‎{\em Approximate symmetries}‎, ‎Math‎. ‎Sbornik‎ ,136(178), ‎no‎. ‎3:435-450,1988‎. ‎English trans1.‎, ‎Math.USSR Sb.‎, ‎64(1989)‎, ‎No.2‎, ‎pp.427-441‎.
‎\bibitem{[6]}{\sc G‎. ‎Cicogna}‎, ‎{\em Discussion on the different notions of symmetry of differential equations}‎, ‎Proceedings of Institute of Mathematics of NAS of Ukraine‎, ‎Vol‎. ‎50‎, ‎Part 1‎, ‎2004‎, ‎77–84‎.
‎\bibitem{[7]}{\sc S‎. ‎Dunbar} and {\sc H‎. ‎Othmer}‎, ‎{\em On a nonlinear hyperbolic equation describing transmission lines‎, ‎cell movement‎, ‎and branching random walks‎, ‎in nonlinear oscillations in biology and chemistry}‎, ‎Lecture Notes in Biomathematics‎, ‎Springer-Verlag‎, ‎1986‎.
‎\bibitem{[8]}{\sc S‎. ‎Dunbar}‎, ‎{\sc H‎. ‎Othmer} and {\sc W‎. ‎Alt}‎, ‎{\em Models of dispersal in biological systems}‎, ‎J‎. ‎Math‎. ‎Boil (1988)26:263-298‎, ‎Springer-Verlag‎, ‎1988‎.
‎\bibitem{[10]}{\sc W.I‎. ‎Fushchych}‎, ‎{\em On symmetry and particular solutions of some multidimensional physics equations}‎, ‎In Algebraic-theoretical Methods in Mathematical Physics Problems‎, ‎Kyiv‎, ‎Inst‎. ‎Math‎. ‎Acad‎. ‎Sci‎, ‎of Ukraine‎, ‎1983‎, ‎4–23‎.
‎\bibitem{[11]}{\sc W‎. ‎Fushchych}‎, ‎{\sc W‎. ‎Shtelen} and {\sc M‎. ‎Serov}‎, ‎{\em Symmetry analysis and exact solutions of nonlinear equations of mathematical physics}‎, ‎Kiev‎, ‎Naukova dumka‎, ‎1989‎, ‎Dordrecht‎, ‎Kluwer‎, ‎1993‎.
‎\bibitem{[12]}{\sc S.A‎. ‎Gourley}‎, ‎{\em Travelling front solutions of a nonlocal Fisher equation}‎, ‎J‎. ‎Math‎. ‎Biol‎. ‎41‎, ‎272–284‎, ‎Springer-verlag 2000‎.
‎\bibitem{[13]}{\sc K.P‎. ‎Hadeler}‎, ‎{\em Hyperbolic travelling fronts}‎, ‎Proc‎. ‎Edinburgh Math‎. ‎Soc‎. ‎31‎, ‎1988‎, ‎89-97‎.
‎\bibitem{[14]}{\sc K.P‎. ‎Hadeler}‎, ‎{\em Travelling fronts and free boundary value problems}‎, ‎In Numerical treatment‎
‎of free boundary value problems‎, ‎Oberwolfach Conference‎, ‎1980‎.
‎%\bibitem{[15]}{\sc K.P‎. ‎Hadeler}‎, ‎{\em Free boundary problems in biological models}‎, ‎In Free boundary problems‎: ‎Theory and applications‎, ‎Vol‎. ‎II‎, ‎Montecatini Conference‎, ‎1981‎.
‎\bibitem{[16]}{\sc N.H‎. ‎Ibragimov} and {\sc V.F‎. ‎Kovalev}‎, ‎{\em Approximate and renormgroup symmetries}‎, ‎ALGA‎
‎publications‎, ‎Karlskrona‎, ‎2009‎, ‎1-73‎.
‎\bibitem{[17]}{\sc N.H‎. ‎Ibragimov}‎, ‎{\em Perturbation methods in group analysis}‎, ‎In Differential equations and chaos‎,
‎New Dehli‎, ‎1996‎, ‎41-60‎.
‎\bibitem{[18]}{\sc M‎. ‎Nadjafikhah} and {\sc A‎. ‎Mokhtari}‎, ‎{\em Symmetry analysis of Black-Scholes equation for small values of volatility and rate of return}‎, ‎Journal of Interpolation and Approximation in Scientific Computing‎, ‎Vol‎. ‎2014‎, ‎ID jiasc-00054‎.
‎\bibitem{[21]}{\sc M‎. ‎Nadjafikhah} and {\sc A‎. ‎Mahdavi}‎, ‎{\em Approximate symmetry and approximate solution of the $\phi-$equation with a small parameter}‎, ‎Gen‎. ‎Math‎. ‎Notes‎, ‎Vol‎. ‎24‎, ‎No‎. ‎1‎, ‎Sep 2014‎, ‎40--51‎.
‎\bibitem{[19]}{\sc P.J‎. ‎Olver}‎, ‎{\em Applications of Lie groups to differential equations}‎, ‎Second Edition‎, ‎GTM‎, ‎Vol‎. ‎107‎, ‎Springer Verlage‎, ‎New York‎, ‎1993‎.
‎\bibitem{[20]}{\sc P.J‎. ‎Olver} and {\sc P‎. ‎Rosenau}‎, ‎{\em The construction of special solutions to partial differential equations}‎, ‎Phys‎. ‎Lett‎. ‎A‎, ‎1986‎, ‎V.114‎, ‎N 3,107–112; Group-invariant solutions of differential equations‎, ‎SIAM J‎. ‎Appl‎. ‎Math.‎, ‎1987‎, ‎V.47‎, ‎N 2‎, ‎263–278‎.
‎\bibitem{[23]}{\sc J‎. ‎Saragosti} et‎. ‎al.‎, ‎{\em Directional persistence of chemotactic bacteria in a traveling concentration wave}‎, ‎PNAS‎, ‎2011‎.
‎\bibitem{[24]} {\sc A‎. ‎Valenti}‎, ‎{\em Approximate symmetries for a model describing dissipative media}‎, ‎Proceedings of 10th international conference in modern group analysis 2005‎, ‎236-243‎.



How to Cite

Nadjafikhah, M. (2019). Invariant Solutions of Generalized Fisher-KPP Equation. MathLAB Journal, 2(1), 126-132. Retrieved from



Research Articles

Most read articles by the same author(s)