In this paper, we consider the existence, uniqueness and convergence of weak and strongimplicit difference solution for the first boundary problem of quasilinear parabolic system: u_t=(-1)^(M+1)A(x,t,u,…,u_x^(M-1))u_...In this paper, we consider the existence, uniqueness and convergence of weak and strongimplicit difference solution for the first boundary problem of quasilinear parabolic system: u_t=(-1)^(M+1)A(x,t,u,…,u_x^(M-1))u_x^(2M)+f(x,t,u,…,u_x^(2M-1)), (x,t)∈Q_T={0<x<l, 0<t≤T}, (1) u_xk(0,t)=u_xk(l,t)=0,(k=0,1,…,M-1), 0<t≤T, (2) u(x,0) =ψ(x), 0≤x≤l, (3)where u, ψ and f are m-dimensional vector valued functions, A is an m×m positivelydefinite matrix and u_xk denotes ?~ku/?_xk. For this problem, the estimations of the differencesolution are obtained. As h→0, △t→0, the difference solution converges weakly in W_2^(2M,1) (QT)to the unique generalized solution u(x,t)∈W_2^(2M,1)(QT) of problems (1), (2), (3). Especially,a favorable restriction condition to the step lengths △t and h for explicit and weak implicitschemes is found.展开更多
文摘In this paper, we consider the existence, uniqueness and convergence of weak and strongimplicit difference solution for the first boundary problem of quasilinear parabolic system: u_t=(-1)^(M+1)A(x,t,u,…,u_x^(M-1))u_x^(2M)+f(x,t,u,…,u_x^(2M-1)), (x,t)∈Q_T={0<x<l, 0<t≤T}, (1) u_xk(0,t)=u_xk(l,t)=0,(k=0,1,…,M-1), 0<t≤T, (2) u(x,0) =ψ(x), 0≤x≤l, (3)where u, ψ and f are m-dimensional vector valued functions, A is an m×m positivelydefinite matrix and u_xk denotes ?~ku/?_xk. For this problem, the estimations of the differencesolution are obtained. As h→0, △t→0, the difference solution converges weakly in W_2^(2M,1) (QT)to the unique generalized solution u(x,t)∈W_2^(2M,1)(QT) of problems (1), (2), (3). Especially,a favorable restriction condition to the step lengths △t and h for explicit and weak implicitschemes is found.
