OperationalMatrix

In the context of the Jacobi spectral tau method, an operational matrix is a mathematical tool used to represent the operation of differentiation (or potentially other operations) on a series of functions, such as polynomials, in terms of a matrix operation on the coefficients of those functions. Specifically, when functions are expanded in terms of a basis of polynomials (e.g., Jacobi polynomials), the derivative of such a function can be expressed as a linear combination of the same set of polynomials. The operational matrix of differentiation encodes this transformation.

Given a function $f(x)$ expanded in terms of a basis ${P_n^{(\alpha, \beta)}(x)}_{n=0}^{N}$ of Jacobi polynomials,

$$ f(x) = \sum_{n=0}^{N} a_n P_n^{(\alpha, \beta)}(x) $$

where $a_n$ are the coefficients of the expansion, the derivative of $f(x)$, denoted by $f'(x)$, can also be expanded in terms of the same polynomials:

$$ f'(x) = \sum_{n=0}^{N} b_n P_n^{(\alpha, \beta)}(x) $$

The operational matrix $D$ for differentiation is then defined such that

$$ \mathbf{b} = D \mathbf{a} $$

where $\mathbf{a} = [a_0, a_1, \ldots, a_N]^T$ and $\mathbf{b} = [b_0, b_1, \ldots, b_N]^T$ are vectors of the coefficients of $f(x)$ and its derivative $f'(x)$, respectively. The matrix $D$ thus transforms the coefficient vector of the function into the coefficient vector of its derivative.

The elements of $D$ can be determined by explicitly calculating the derivatives of the Jacobi polynomials and expressing these derivatives as linear combinations of the polynomials themselves. That is, for each polynomial $P_n^{(\alpha, \beta)}(x)$,

$$ \frac{d}{dx}P_n^{(\alpha, \beta)}(x) = \sum_{m=0}^{N} D_{mn} P_m^{(\alpha, \beta)}(x) $$

where $D_{mn}$ are the elements of the operational matrix $D$, encoding how the $n$-th polynomial's derivative is represented in terms of the polynomial basis.

This operational matrix $D$ is crucial in the spectral tau method because it enables the analytical representation of differential operators on the polynomial basis, thus facilitating the transformation of differential equations into algebraic ones. Operational matrices for higher-order derivatives or other linear operators can be constructed similarly, allowing for the efficient solution of various differential equations using polynomial approximations.