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

Abstract

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.

A Novel Haptic Interactive Approach to Simulation of Surgery Cutting Based on Mesh and Meshless Models

“…Following [ 30 ], we utilize the L 2 Hilbert space to analyze the stability and convergence. The inner product is as follows: and its induced norm is as follows: …”

“…In 2013, the research group successfully extended this work to 3D objects [ 28 ]. Unlike the EFG method, the radial point interpolation meshless method (RPIM) does not need any integration cell and is applied to solve telegraph and heat diffusion equations and time fractional diffusion-wave equation [ 29 , 30 ]. Recently, Zou et al simulated the deformation of soft tissue based on RPIM [ 31 ].…”

Transcendental Bernstein series for solving reaction–diffusion equations with nonlocal boundary conditions through the optimization technique

“…Therefore, their analytical and numerical solutions have been an interesting area of research for many years. Recently, several numerical methods such as Crank-Nicolson Hermite cubic orthogonal spline collocation method [1], polynomial-based mean weighted residuals methods [2], finite difference schemes [3][4][5][6][7], Laplace transform method [8], finite element method [9], wavelets method [10], Ritz-Galerkin method [11], spectral meshless radial point interpolation method [12], -method [13], implicit Euler method [14] and operational approach of the Tau method [15] have been proposed to solve systems including nonlocal boundary conditions. It is well known that, Bernstein polynomials (BPs) play an important role as basis functions for various numerical techniques for solving different mathematical systems.…”

On the analysis of a kind of nonlinear Sobolev equation through locally applied pseudo‐spectral meshfree radial point interpolation

“…• Those meshless techniques benefit from collocation approach and usually applied in strong forms, like for example the Kansa's method in the frame of radial basis functions (RBFs) [11][12][13][14][15]. • Those meshless techniques benefit from weak forms and use collocation scheme at the same time [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].…”

“…Finally Equation (45) can be rewritten as If we apply the approximations (31) and (34) for the unknown function in the above equation then we will discretize spatial variable and obtain…”