This singular boundary integral is solved in the same paper by using a collocation method.
In case of periodic solution of period T, a discrete approximation on a mesh in [0, T] and its period are computed as solutions of the corresponding periodic boundary value problem by using a piecewise polynomial collocation.
We approximate the analytical solution y by a piecewise polynomial function p ∈ P k, h ∩ C [ 0, 1 ], p ( t ) : = p j ( t ), t ∈ [ t j, t j + 1 ], j = 0, …, I − 1, such that p satisfies ODE system (5) at the collocation points, p ′ ( t j l ) − M t j l p ( t j l ) = f ( t j l ) t j l, l = 1, …, k, j = 0, …, I − 1, (42).
But this Hilbert subspace is thus a subspace spanned by a polynomial number of vectors and is thus at most a polynomial subspace of the total Hilbert space.
In this respect, recently, a collocation algorithm based on employing Fibonacci polynomials has been analyzed for solving Volterra-Fredholm integral equations in [11].
Both were described by a polynomial equation.
Experimental points (magenta) were interpolated by a polynomial (blue line).
To this end, a collocation spectral model based on the Chebyshev polynomials for the 2D Sobolev equations is first established.
It is common to employ a collocation method or analytic method based on the use of polynomial base functions to solve Volterra integral equations.
In this work, it is shown that the polynomial finite difference method can be seen as a collocation method with finite elements of equal size with uniform distribution of collocation points within each element.
