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Uniform convergence of the implicit scheme of the finite-difference method for solving the first boundary-value problem for a nonlinear second-order parabolic equation
M. N. Yakovlev
Abstract:
Let
$u(t,x)$ be a solution of the first initial–boundary-value problem for the
quasilinear parabolic equation
\begin{equation}
\dfrac{\partial u}{\partial t}=a(t,x,u,\dfrac{\partial u}{\partial x})\dfrac{\partial^2u}{\partial x^2}+
b(t,x,u,\dfrac{\partial u}{\partial x}),\qquad 0<t\leqslant T,\quad 0<x<1
\tag{1}
\end{equation}
with the initial condition
\begin{equation}
u(0,x)=\omega(x),\quad 0<x<1
\tag{2}
\end{equation}
and the boundary conditions
\begin{equation}
u(t,0)=u(t,1)=0,\quad 0<t\leqslant T,
\tag{3}
\end{equation}
such that
$$
\biggl|\dfrac{\partial^4u}{\partial x^4}(t,x)\biggr|\leqslant C,\quad
\biggl|\dfrac{\partial^2u}{\partial t^2}(t,x)\biggr|\leqslant\dfrac{c}{t^\sigma},\quad
0\leqslant\sigma<2
$$
Assume that the functions
$a(t,x,u,p)$,
$b(t,x,u,p)$ are smooth and
in a small neighborhood of the solution under consideration. Then, the implicit
scheme of the finite-difference method converges uniformly to the solution under
consideration with the order
$h^2+\varphi(\tau)$, under the condition that
\begin{equation}
\varphi(\tau)\leqslant\beta h^\gamma,\quad \beta>0,\quad \gamma>1
\tag{4}
\end{equation}
Here
$$
\varphi(\tau)=
\begin{cases}
\tau & \text{\rm{ при }}0\leqslant\sigma<1\\
\tau\ln\dfrac{T}{\tau} & \text{\rm{ при }}\sigma=1\\
\tau^{2-\sigma} & \text{\rm{ при }}1<\sigma<2.
\end{cases}
$$
One also considers convergence conditions when the relations (4) do not hold,
convergence conditions for equations of the form
$$
\dfrac{\partial u}{\partial t}=F\biggl(t,x,u,\dfrac{\partial u}{\partial x},\dfrac{d}{dx}K(t,x,\dfrac{\partial u}{\partial x})\biggr)
$$
and weakly connected systems of such equations.
UDC:
518.517.949.8