Abstract
The Fokker Planck Equation (FPE) is a partial differential equation for the prob- ability density and transition probability of a random process. Owing to its broad range of applications, the FPE has been an interesting research topic. Recently, Radial basis functions (RBFs) have emerged as a powerful numerical tool for solving partial differential equations and this paper reports an integrated RBFs (IRBFs) based numerical method for the solution of FPEs. The use of integration to construct RBF approximants helps avoid the reduction in con- vergence rate caused by differentiation (1). Numerical experiments showed that IRBF methods can yield accurate solutions on a much coarser mesh, thus reducing the computational effort required for a given degree of accuracy.
Highlights
The Fokker-Planck Equation (FPE) is used to describe a stochastic process in diverse fields, including plasma physics, biophysics, engineering, neurosciences, nonlinear hydrodynamics, and polymer physics
Numerical experiments showed that integrated RBFs (IRBFs) based methods can yield accurate solutions on a coarse mesh [13], and have the ability to reduce the computational effort required for a given degree of accuracy
We present a collocation technique incorporating the one dimensional integrated Radial basis functions (RBFs) (1D-IRBF) for a numerical solution of FPEs
Summary
The Fokker-Planck Equation (FPE) is used to describe a stochastic process in diverse fields, including plasma physics, biophysics, engineering, neurosciences, nonlinear hydrodynamics, and polymer physics. For a numerical solution of partial differential equations in engineering and sciences, RBF based collocation methods [9, 10], have increasingly been a focus of research efforts. Numerical schemes, based on the integrated RBFs (IRBFs) approach, for solving differential differential equations were reported [1, 13]. The use of integration to construct the RBF approximants is expected to overcome the problem of reduced convergence rate caused by differentiation [1]. Numerical experiments showed that IRBF based methods can yield accurate solutions on a coarse mesh [13], and have the ability to reduce the computational effort required for a given degree of accuracy. We present a collocation technique incorporating the one dimensional integrated RBFs (1D-IRBF) for a numerical solution of FPEs. The paper is organized as follows.
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