Differential quadrature for multi-dimensional problems.

*(English)*Zbl 0557.65084This 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 |

##### Keywords:

differential quadrature; convection-diffusion equation; steady-state dispersion of inert, neutrally buoyant pollutants; unbounded atmosphere
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\textit{F. Civan} and \textit{C. M. Sliepcevich}, J. Math. Anal. Appl. 101, 423--443 (1984; Zbl 0557.65084)

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##### References:

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