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Galerkin methods for nonlinear parabolic integrodifferential equations with nonlinear boundary conditions. (English) Zbl 0703.65095
Author’s abstract: Galerkin methods are analyzed for the nonlinear parabolic integrodifferential equation \[ u_ t=\nabla \cdot \{a(u)\nabla u+\int^{t}_{0}b(t,u(x,s))\nabla u(x,s)ds\}+f(u) \] in \(\Omega\times (0,T]\), \(T>0\), \(\Omega \subset R^ d(d\leq 3)\), subject to the nonlinear boundary condition \[ a(u)\frac{\partial u}{\partial v}\int^{t}_{0}b(t,u(x,s))\frac{\partial u}{\partial v}(x,s)ds=g(u) \] on \(S_ T=\partial \Omega \times [0,T]\) and the usual initial condition. Optimal-order error estimates are derived in \(L^ 2(\Omega)\) norm for the continuous, Crank-Nicolson, and modified Crank-Nicolson Galerkin approximations. As usual, the latter will yield a system of linear algebraic equations to be solved at each level.
Reviewer: D.A.Quinney

MSC:
65R20 Numerical methods for integral equations
45G10 Other nonlinear integral equations
45K05 Integro-partial differential equations
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