# Difference between finite difference approximation and differential quadrature approximation

As a student of numerical analysis, I understand that a finite difference approximation (FDM) of the derivative '$$u_x$$' of a desired solution '$$u$$' at the point $$x_n$$ in the domain is just a linear combination $$\displaystyle \sum_{i=0}^Nc_i u_i,$$ where, $$u_i=u(x_i)$$.

That is, $$\displaystyle (u_x)_n=u_x(x_n)\approx c_0u_0+c_1u_1+...+c_Nu_N$$.

For instance, if $$c_i=0,$$ for $$i=0,1,2,...,n-1,n+2,...,N$$, $$c_n=\frac{-1}{h}$$, and $$c_{n+1}=\frac{1}{h},$$ with $$h=x_{n+1}-x_n$$ for all $$n$$ (equal sub interval length), then the approximation is said to be a 'forward difference approximation'. In a similar way, we can have backward differences, central differences, finite differences on non-uniform grids, non standard finite differences (NSFD), etc.

However, while analyzing the differential quadrature approximation (DQM),

$$$$\label{DQM} u_{x}(x_{i})=\left.\dfrac{du}{dx}\right\vert_{x_{i}} \approx \sum_{j=1}^{N} a_{ij} u(x_{j}),\textrm{for}~ i=1,2,\dots,N,$$$$

I feel that none of the authors introduce it as a finite difference method.

Doubt: What exactly is the conceptual difference of DQM over FDM?,

Whether it is due to the number of non-zero coefficients $$c_i$$'s to be considered? OR

Due to the difference in methods of finding the coefficients $$c_i$$? We bother about the coefficients from the beginning itself while applying FDMs, whereas we need some test functions for DQM.

The linear “integral” operator $$L$$ applied to a function $$u$$ can be written as

$$L u(x_i) \approx \sum_{j} a_{ij} u(x_j)$$

where the coefficients are determined by many ways, usually. You just treat differential operator as some kind of integral operator. The original work by Bellman seemed to use a linear system.

$$L \phi(x_i) = \sum_{j} a_{ij} \phi(x_j)$$

where $$\phi$$ are taken through polynomials of various degrees, for instance, the Lagrange polynomials on the nodes. Then you will obtain a set of coefficients for DQM.

If you take $$\phi$$ as "hat function" (its integrals) at each node, then you will get the usual finite difference method.

If you take Fourier basis for $$\phi$$, then you will get the pseudo spectral method.

That is, the difference is simply what basis functions are you applying.