Discrete approximations of differential equations via trigonometric interpolation
This work provides a novel numerical method for solving linear differential equations, potentially benefiting computational scientists and engineers, though it is an incremental improvement over existing spectral methods.
The paper proposes a method to approximate solutions of linear differential equations by projecting the solution space onto trigonometric polynomials via interpolation, constructing a matrix representation of the differential operator. Numerical tests demonstrate high accuracy and fast convergence for boundary and eigenvalue problems.
To approximate solutions of a linear differential equation, we project, via trigonometric interpolation, its solution space onto a finite-dimensional space of trigonometric polynomials and construct a matrix representation of the differential operator associated with the equation. We compute the ranks of the matrix representations of a certain class of linear differential operators. Our numerical tests show high accuracy and fast convergence of the method applied to several boundary and eigenvalue problems.