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Particular solutions of anisotropic radial basis function can be found by the same procedure as that of regular radial basis functions under Laplace operator.

It is inferred that neural networks, in particular the generalised radial basis function, can be a promising tool for predicting the chiller's performance for fault detection and other diagnosis purposes.

Then the network basis function can be formed.

Also, a Gaussian basis function can be used to sparsely represent environmental signals [37].

The projection of estimated vibration signal ( {tilde{r}}_{v0}left eta right) ) on the basis function can be expressed as Sleft({widehat{varphi}}_iright)=leftlangle {tilde{r}}_{v0}left eta right),{r}_{mathrm{vref}}left eta, {widehat{varphi}}_iright)rightrangle (22).

The basis function can be adjusted by means of shift, shear, and scaling operators, resulting in a five-dimensional parameter space for the energy density which comprises as projections the respective densities obtained from a short-time Fourier transform (time and frequency shift) and a wavelet transform (time shift and scaling).

The MSPH basis functions can be used in any meshless method to numerically solve either static or dynamic problems.

In addition, basis functions can be adaptively updated using efficient global smoothing strategies to account for multiphase flow effects.

Moreover, arbitrary high order degree of basis functions can be used and their regularity enables the use of a low number of elements.

By using the procedure described here, the coefficients of the basis functions can be obtained directly by solving an uncoupled ordinary differential equation.

So the unmatched fuzzy basis functions can be handled in stability analysis of the resulting closed-loop systems with support of these local representations.