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Differential quadrature for multi-dimensional problems. (English) Zbl 0557.65084
This paper uses a technique developed by R. E. Bellman and his coworkers and applies a generalisation of this technique to the solution of time dependent and time independent partial differential equations involving multiple space dimensions. This technique involves an approximation of the form \((\partial^ m/\partial x^ m)f(x_ i)\simeq \sum^{N}_{j=1}W_{ij}f(x_ j)\) where the data points \(x_ i\) are given and the weights are chosen so that the result is exact when \(f(x)=x^ k\), \(k=0,...,(N-1)\). For the time independent case the theory leads to a set of linear algebraic equations for the values of the solution at the grid points. The time dependent case leads to a similar set of linear first order differential equations in time. Examples and numerical computations are given using convection-diffusion equations.
Reviewer: B.Burrows

MSC:
65Z05 Applications to the sciences
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
92D40 Ecology
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