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**New ansätze and exact solutions for nonlinear reaction-diffusion equations arising in mathematical biology.**
*(English)*
Zbl 0954.35038

Shkil, Mykola (ed.) et al., Symmetry in nonlinear mathematical physics. Proceedings of the second international conference, Kyiv, Ukraine, July 7-13, 1997. Memorial Prof. W. Fushchych conference. Vol. 1. Kyiv: Institute of Mathematics of the National Academy of Sciences of Ukraine. 138-146 (1997).

In this paper the method of additional generating conditions, suggested by R. M. Cherniha [Dopov. Akad. Nauk Ukr., No. 4, 17-21 (1995)] is applied for the finding of new non-Lie symmetry ansätze and exact solutions of nonlinear generalized reaction-diffusion equations with convection term of the general form
\[
U_t=(\lambda+\lambda_0 U)U_{xx}+rU_x^2+pUU_x+qU^2+sU+s_0.
\]
Here \(U=U(t,x)\) is unknown function, the coefficients \(\lambda\), \(\lambda_0\), \(r\), \(p\), \(q\), \(s\) and \(s_0\) are constants or smooth functions of \(t\). The subscripts \(t\) and \(x\) denote differentiation with respect to these variables. The class of partial differential equations of the above form contains a number of well-known nonlinear second-order evolution equations arising, in particular, in mathematical biology.

The essence of the used method is to consider the initial equation together with a linear high-order homogeneous ordinary differential equation (named an additional generating condition) \(\alpha_0(t,x)U+\alpha_1(t,x)dU/dx+\cdots+\alpha_{m-1}(t,x)d^{m-1}U/dx^{m-1}+d^mU/dx^m=0,\) where \(\alpha_0(t,x)\), \(\alpha_1(t,x)\), …, \(\alpha_{m-1}(t,x)\) are arbitrary smooth functions and \(t\) is assumed as a parameter. It is equivalent to the search for solutions of the initial equation in the form \(U=\varphi_0(t)g_0(t,x)+\varphi_1(t)g_1(t,x)+\cdots+\varphi_{m-1}(t)g_{m-1}(t,x)\). Here \(g_0(t,x)\), \(g_1(t,x)\), …, \(g_{m-1}(t,x)\) are linearly independent over the field of functions of \(t\) functions, which should be determined from the condition that the substitution of the expression for \(U\) into the initial equation reduces it to a system of ordinary differential equations for the functions \(\varphi_0(t)\), \(\varphi_1(t)\), …, \(\varphi_{m-1}(t)\).

In this paper the author uses the additional generating condition with \(m=3\), \(\alpha_0=0\), \(\alpha_1=\alpha_1(t)\) and \(\alpha_2=\alpha_2(t)\). As a result new multi-parameter families of exact solutions of equations from the class under consideration (in particular, the Fisher and Murray equations) are found. Some constructed solutions can not be obtained by means of (generalized) separation of variables.

For the entire collection see [Zbl 0882.00038].

The essence of the used method is to consider the initial equation together with a linear high-order homogeneous ordinary differential equation (named an additional generating condition) \(\alpha_0(t,x)U+\alpha_1(t,x)dU/dx+\cdots+\alpha_{m-1}(t,x)d^{m-1}U/dx^{m-1}+d^mU/dx^m=0,\) where \(\alpha_0(t,x)\), \(\alpha_1(t,x)\), …, \(\alpha_{m-1}(t,x)\) are arbitrary smooth functions and \(t\) is assumed as a parameter. It is equivalent to the search for solutions of the initial equation in the form \(U=\varphi_0(t)g_0(t,x)+\varphi_1(t)g_1(t,x)+\cdots+\varphi_{m-1}(t)g_{m-1}(t,x)\). Here \(g_0(t,x)\), \(g_1(t,x)\), …, \(g_{m-1}(t,x)\) are linearly independent over the field of functions of \(t\) functions, which should be determined from the condition that the substitution of the expression for \(U\) into the initial equation reduces it to a system of ordinary differential equations for the functions \(\varphi_0(t)\), \(\varphi_1(t)\), …, \(\varphi_{m-1}(t)\).

In this paper the author uses the additional generating condition with \(m=3\), \(\alpha_0=0\), \(\alpha_1=\alpha_1(t)\) and \(\alpha_2=\alpha_2(t)\). As a result new multi-parameter families of exact solutions of equations from the class under consideration (in particular, the Fisher and Murray equations) are found. Some constructed solutions can not be obtained by means of (generalized) separation of variables.

For the entire collection see [Zbl 0882.00038].

Reviewer: R.O.Popovych (Kyïv)

### MSC:

35C05 | Solutions to PDEs in closed form |

58J70 | Invariance and symmetry properties for PDEs on manifolds |

35K57 | Reaction-diffusion equations |

92B05 | General biology and biomathematics |

35Q80 | Applications of PDE in areas other than physics (MSC2000) |

### Keywords:

new multi-parameter families of exact solutions; Fisher and Murray equations; additional generating condition; reduction; Fisher equation; Murray equation; Lie symmetry; conditional symmetry
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\textit{R. M. Cherniha}, in: Symmetry in nonlinear mathematical physics. Proceedings of the second international conference, Kyiv, Ukraine, July 7--13, 1997. Memorial Prof. W. Fushchych conference. Vol. 1. Kyiv: Institute of Mathematics of the National Academy of Sciences of Ukraine. 138--146 (1997; Zbl 0954.35038)