Solving 2D Reaction-Diffusion Equations with Nonlocal Boundary Conditions by the RBF-MLPG Method

Shirzadi, Ahmad

doi:10.1007/s10598-014-9246-x

Cited by 4 publications

(3 citation statements)

References 21 publications

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“…, z N } ⊂ is a set of trial points. This is the common MLPG approach based on kernel approximations [12,13]. However, it is possible to construct a contour example that leads to a singular matrix [14,15].…”

Section: The Unsymmetric Mlpgmentioning

confidence: 99%

“…An unsymmetric Petrov–Galerkin method can be formulated by replacing u in by a function s of the from

s = \sum_{j = 1}^{N} c_{j} Φ (\cdot, z_{j}) \approx u,

where

X_{tr} = {z_{1}, \dots, z_{N}} \subset Ω

is a set of trial points . This is the common MLPG approach based on kernel approximations . However, it is possible to construct a contour example that leads to a singular matrix .…”

Section: The Unsymmetric Mlpgmentioning

confidence: 99%

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A greedy meshless local Petrov-Galerkin methodbased on radial basis functions

Mirzaei

2015

Numer. Methods Partial Differential Eq.

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The meshless local Petrov-Galerkin (MLPG) method with global radial basis functions (RBF) as trial approximation leads to a full final linear system and a large condition number. This makes MLPG less efficient when the number of data points is increased. We can overcome this drawback if we avoid using more points from the data site than absolutely necessary. In this article, we equip the MLPG method with the greedy sparse approximation technique of (Schaback, Numercail Algorithms 67 (2014), 531-547) and use it for numerical solution of partial differential equations. This scheme uses as few neighbor nodal values as possible and allows to control the consistency error by explicit calculation. Whatever the given RBF is, the final system is sparse and the algorithm is well-conditioned.

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Section: The Unsymmetric Mlpgmentioning

confidence: 99%

“…An unsymmetric Petrov–Galerkin method can be formulated by replacing u in by a function s of the from

s = \sum_{j = 1}^{N} c_{j} Φ (\cdot, z_{j}) \approx u,

where

X_{tr} = {z_{1}, \dots, z_{N}} \subset Ω

is a set of trial points . This is the common MLPG approach based on kernel approximations . However, it is possible to construct a contour example that leads to a singular matrix .…”

Section: The Unsymmetric Mlpgmentioning

confidence: 99%

A greedy meshless local Petrov-Galerkin methodbased on radial basis functions

Mirzaei

2015

Numer. Methods Partial Differential Eq.

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“…But in methods based on the local weak form formulation, numerical integrations are carried out over a local quadrature domains, therefore, no cells are required. As a result, they are referred to as truly meshless methods such as the meshless local Petrov-Galerkin (MLPG) method (Atluri and Zhu, 1998;Dehghan and Mirzaei, 2008;Shirzadi, 2014;Shivanian, 2015b …”

Section: Introductionmentioning

confidence: 99%

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

Shivanian

Khodayari

2017

jtam

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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.

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A Meshless Runge–Kutta Method for Some Nonlinear PDEs Arising in Physics

Mohammadi

Shirzadi

2022

Comput Math Model

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Solving 2D Reaction-Diffusion Equations with Nonlocal Boundary Conditions by the RBF-MLPG Method

Cited by 4 publications

References 21 publications

A greedy meshless local Petrov-Galerkin methodbased on radial basis functions

A greedy meshless local Petrov-Galerkin methodbased on radial basis functions

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

A Meshless Runge–Kutta Method for Some Nonlinear PDEs Arising in Physics

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