Executive Summary : | The Orthogonal collocation method is a weighted residual method used to discretize two point boundary value problems. Standard Lagrangian interpolating polynomials can be used as a base function, but they may not be suitable for stiff systems or equations with slip boundary conditions. Hermite interpolating splines are better suited for non-linear singularly perturbed boundary conditions with slip boundary conditions. These splines interpolate the base function and its derivatives at node points, and in space direction, quintic Hermite interpolation polynomials are introduced to approximate the trial function.
This project proposes a fully discrete meshless Hermite spline collocation, which introduces radial basis functions with Hermite interpolating polynomials. These radial basis functions reduce the domain into a meshfree domain and satisfy boundary conditions like Dirichlet's or Neumann's conditions. The hybrid technique of meshfree Hermite interpolating polynomials with radial basis functions is combined with the ADI finite difference technique to discretize the system of equations numerically, increasing convergence rates in time and space direction. The proposed project will discuss two point boundary value problems in 1D and 2D space, focusing on the stability and convergence behavior of numerical solutions obtained. The post-error estimate will be based on the singularly perturbed parameter that affects the entire solution. |