IntegralOperationalMatrix

The construction of the integral operational matrix, especially in the context of orthogonal polynomials like the shifted Jacobi polynomials, requires knowledge of several components:

1. Orthogonal Polynomial Basis:

   • The choice of the orthogonal polynomial basis (e.g., Shifted Jacobi, Shifted Legendre, etc.) is fundamental. Each polynomial basis will have its own set of properties, recursion relations, and weight functions.
   • The orthogonality property of these polynomials is crucial for the construction of the operational matrix.

2. Definition of the Operational Matrix:

   • An operational matrix is typically a square matrix, where the entry in the $i$-th row and $j$-th column represents the integral of the product of the $i$-th and $j$-th basis polynomials.
   • Mathematically, the element $P_{ij}$ of the matrix $P$ is given by:

$$P_{ij} = \int_{a}^{b} \phi_i(x) \phi_j(x) w(x)dx$$

where $\phi_i(x)$ and $\phi_j(x)$ are the basis polynomials, and $w(x)$ is the weight function associated with the chosen orthogonal polynomial.

3. Integration Limits and Weight Function:

   • The limits of integration, $a$ and $b$, and the weight function $w(x)$ depend on the choice of the orthogonal polynomial.
   • For instance, for the standard Jacobi polynomials, the weight function is $(1-x)^\alpha (1+x)^\beta$ and the domain is typically $[-1, 1]$.

4. Computational Tools:

   • To actually compute the elements of the operational matrix, one would typically use symbolic computation tools (like Mathematica, Maple, or SymPy in Python) or numerical integration methods if symbolic methods are not feasible.

5. Size of the Matrix:

   • The size of the operational matrix depends on the truncation or the number of terms considered in the polynomial expansion. If $N$ terms are considered, the matrix will be of size $N \times N$.

6. Properties of the Operational Matrix:

   • Some matrices, depending on the polynomial basis, may have specific properties that can be exploited to simplify computations (e.g., diagonal dominance, sparsity). Understanding these can help in optimizing calculations.

7. Application to Differential Equations:

   • Once the operational matrix is constructed, it can be used to transform differential equations into algebraic equations. This typically involves expressing the function in terms of the chosen polynomial basis, applying the operational matrix, and then simplifying to get a system of algebraic equations.

In the context of the paper you provided, the specifics of the shifted Jacobi polynomial and its associated weight function would be crucial in constructing the shifted Jacobi polynomial integral operational matrix.