Reduction of a Non—Linear Parabolic Initial—Boundary Value Problem to Cauchy Problem for a System of ODEs

We consider the boundary value problem for a parabolic equation in the form $$\frac{{\partial {\text{u}}}}{{\partial t}} = \frac{1}{{p(x)}}\frac{\partial }{{\partial x}}\left( {p(x)f'(u)\frac{{\partial u}}{{\partial x}}} \right) + F(u),x \in (0,l),t0,$$ (1) $$u(0,x) = {u_0}(x),$$ (2) $$\frac{{\partial u}}{{\partial x}}{|_{x = 0}} = {f_1}\left( {{u_1}} \right),$$ (3) $$\frac{{\partial u}}{{\partial x}}{|_{x = 1}} = {f_2}\left( {{u_2}} \right),$$ (4) where u = u(t,x) is the unknown function, f 1, f 2, F, f are nonlinear functions and f′ (u) > 0, $${u_1} = {u_1}\left( t \right) \equiv u\left( {t,0} \right),{u_2} = {u_2}(t) \equiv u\left( {t,l} \right),f'\left( u \right) \equiv df(u)/du,p(x) \g…