Journal of Theoretical
and Applied Mechanics

55, 2, pp. 571-582, Warsaw 2017
DOI: 10.15632/jtam-pl.55.2.571

Meshless local radial point interpolation (MLRPI) for generalized telegraph and heat diffusion equation with non-local boundary conditions

Elyas Shivanian, Arman Khodayari
In this paper, the meshless local radial point interpolation (MLRPI) method is formulated to
the generalized one-dimensional linear telegraph and heat diffusion equation with non-local
boundary conditions. The MLRPI method is categorized under meshless methods in which
any background integration cells are not required, so that all integrations are carried out
locally over small quadrature domains of regular shapes, such as lines in one dimensions,
circles or squares in two dimensions and spheres or cubes in three dimensions. A technique
based on the radial point interpolation is adopted to construct shape functions, also called
basis functions, using the radial basis functions. These shape functions have delta function
property in the frame work of interpolation, therefore they convince us to impose boundary
conditions directly. The time derivatives are approximated by the finite difference time-
-stepping method. We also apply Simpson’s integration rule to treat the non-local boundary
conditions. Convergency and stability of the MLRPI method are clarified by surveying some
numerical experiments.
Keywords: non-local boundary condition, meshless local radial point interpolation (MLRPI) method, local weak formulation, radial basis function, telegraph equation