Boundary value problems for differential equations on fractals and some difference methods for their solution (Q2713911)

The present article deals with the problems stemming from studying processes in porous media.NEWLINENEWLINENEWLINE1. Given a domain \(Q_T=\{(x,t):0<x<l,0<t<T\}\), consider the problem NEWLINE\[NEWLINE \begin{gathered} {\partial u\over\partial t}={1\over x^\alpha} {\partial\over\partial x} \left(x^\alpha k(x,t){\partial u\over\partial x}\right) -{v\over x^\alpha}{\partial\over\partial x}(x^\alpha u) -q(x,t)u+f(x,t), \\ \lim_{x\to+0}x^\alpha\left(k(x,t){\partial u\over\partial x} -vu\right)=0,\quad u(l,t)=0,\quad u(x,0)=u_0(x), \end{gathered} NEWLINE\]NEWLINE where \(k(x,t)\), \(q(x,t)\), \(f(x,t)\), and \(u_0(x)\) are known functions, \(\alpha\) and \(v\) are some constants, \(k(x,t)\geq c_1>0\), \(1<\alpha <2\), \(v>0\), \(|q(x,t)|\leq c_2\).NEWLINENEWLINENEWLINE2. Given a domain \(Q_T\), consider the problem NEWLINE\[NEWLINE \begin{gathered} D^\alpha_{0t}u={1\over x^m}{\partial\over\partial x} \left(x^mk(x,t){\partial u\over\partial x}\right)+ f(x,t), \\ \lim_{x\to+0}x^mk(x,t){\partial u\over\partial x} =0,\quad u(l,t)=0, \\ D^{\alpha-1}_{0t}\Big|_{t=0}=u_0(x), \end{gathered} NEWLINE\]NEWLINE where \(k(x,t)\), \(f(x,t)\), and \(u_0(x)\) are known functions, \(m=1\) or \(m=2\), \(k(x,t)\geq c_1>0\), \(0<\alpha<1\), \(D^\alpha_{ot}u\) is the fractional derivative of order~\(\alpha\) in the Riemann--Liouville sense.NEWLINENEWLINENEWLINEFor each problem, the authors find a priori estimates, construct a~difference solution, and show that the latter converges to the exact solution of the problem in question.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].