scholarly journals Numerical Approximation for Nonlinear Noisy Leaky Integrate-and-Fire Neuronal Model

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 363 ◽  
Author(s):  
Dipty Sharma ◽  
Paramjeet Singh ◽  
Ravi P. Agarwal ◽  
Mehmet Emir Koksal
Keyword(s):  
Planck Equation ◽  
Neuronal Model ◽  
Finite Difference Approximation ◽  
Computational Time ◽  
Difference Approximation ◽  
Regular Space ◽  
The Finite Element Method ◽  
Integrate And Fire ◽  
The Stability ◽  
Time Required

We consider a noisy leaky integrate-and-fire (NLIF) neuron model. The resulting nonlinear time-dependent partial differential equation (PDE) is a Fokker-Planck Equation (FPE) which describes the evolution of the probability density. The finite element method (FEM) has been proposed to solve the governing PDE. In the realistic neural network, the irregular space is always determined. Thus, FEM can be used to tackle those situations whereas other numerical schemes are restricted to the problems with only a finite regular space. The stability of the proposed scheme is also discussed. A comparison with the existing Weighted Essentially Non-Oscillatory (WENO) finite difference approximation is also provided. The numerical results reveal that FEM may be a better scheme for the solution of such types of model problems. The numerical scheme also reduces computational time in comparison with time required by other schemes.

scholarly journals The Dirihlet problem for the fractional Poisson’s equation with Caputo derivatives: A finite difference approximation and a numerical solution

2012 ◽  
Vol 16 (2) ◽  
pp. 385-394 ◽  
Author(s):  
V.D. Beibalaev ◽  
R.P. Meilanov
Keyword(s):  
Monte Carlo ◽  
Finite Difference ◽  
Fractional Derivative ◽  
Monte Carlo Methods ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Difference Problem ◽  
The Stability

A finite difference approximation for the Caputo fractional derivative of the 4-?, 1 < ? ? 2 order has been developed. A difference schemes for solving the Dirihlet?s problem of the Poisson?s equation with fractional derivatives has been applied and solved. Both the stability of difference problem in its right-side part and the convergence have been proved. A numerical example was developed by applying both the Liebman and the Monte-Carlo methods.

scholarly journals Error Analysis of An Explicit Finite Difference Approximation for the Space Fractional Wave Equations

2012 ◽  
Vol 17 (2) ◽  
pp. 245
Author(s):  
N.H. Sweilam ◽  
T.A. Assiri
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Error Analysis ◽  
Wave Equations ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Fractional Wave Equation ◽  
Fractional Wave Equations ◽  
Explicit Finite Difference ◽  
The Stability

In this paper, the space fractional wave equation (SFWE) is numerically studied, where the fractional derivative is defined in the sense of Caputo. An explicit finite difference approximation (EFDA) for SFWE is presented. The stability and the error analysis of the EFDA are discussed. To demonstrate the effectiveness of the approximated method, some test examples are presented.   

scholarly journals New Nonpolynomial Spline in Compression Method of for the Solution of 1D Wave Equation in Polar Coordinates

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Venu Gopal ◽  
R. K. Mohanty ◽  
Navnit Jha
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Mathematical Formulation ◽  
Finite Difference Approximation ◽  
Polar Coordinates ◽  
Difference Approximation ◽  
Compression Method ◽  
One Dimensional ◽  
Spline In Compression ◽  
The Stability

We propose a three-level implicit nine point compact finite difference formulation of order two in time and four in space direction, based on nonpolynomial spline in compression approximation in -direction and finite difference approximation in -direction for the numerical solution of one-dimensional wave equation in polar coordinates. We describe the mathematical formulation procedure in detail and also discussed the stability of the method. Numerical results are provided to justify the usefulness of the proposed method.

A Robust Algorithm for Nonlinear Variable-Order Fractional Control Systems with Delay

2018 ◽  
Vol 19 (3-4) ◽  
pp. 231-238 ◽  
Author(s):  
José António Tenreiro Machado ◽  
Behrouz Parsa Moghaddam
Keyword(s):  
Control Systems ◽  
High Accuracy ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Variable Order ◽  
The Novel ◽  
Fractional Differentiation ◽  
Nonlinear Feedback ◽  
The Stability ◽  
Fractional Control

AbstractIn this paper, we propose a high-accuracy linear B-spline finite-difference approximation for variable-order (VO) derivative. We consider VO fractional differentiation as a control parameter for improving the stability in systems exhibiting vibrations. The method is applied to nonlinear feedback with VO fractional derivative. The results demonstrate the efficiency and high accuracy of the novel algorithm.

scholarly journals Displacement convexity for the entropy in semi-discrete non-linear Fokker–Planck equations

2018 ◽  
Vol 30 (6) ◽  
pp. 1103-1122 ◽  
Author(s):  
JOSÉ A. CARRILLO ◽  
ANSGAR JÜNGEL ◽  
MATHEUS C. SANTOS
Keyword(s):  
A Priori ◽  
Gradient Flow ◽  
Planck Equation ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Standard Entropy ◽  
Non Linear ◽  
Finite State ◽  
Mean Function ◽  
Fokker Planck

The displacement λ-convexity of a non-standard entropy with respect to a non-local transportation metric in finite state spaces is shown using a gradient flow approach. The constant λ is computed explicitly in terms ofa prioriestimates of the solution to a finite-difference approximation of a non-linear Fokker–Planck equation. The key idea is to employ a new mean function, which defines the Onsager operator in the gradient flow formulation.

scholarly journals Solving Coupled Cluster Equations by the Newton Krylov Method

2020 ◽  
Vol 8 ◽  
Author(s):  
Chao Yang ◽  
Jiri Brabec ◽  
Libor Veis ◽  
David B. Williams-Young ◽  
Karol Kowalski
Keyword(s):  
Iterative Method ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Coupled Cluster ◽  
Regularization Technique ◽  
Krylov Method ◽  
Direct Inversion ◽  
Minimum Residual ◽  
Accelerate Convergence ◽  
The Stability

We describe using the Newton Krylov method to solve the coupled cluster equation. The method uses a Krylov iterative method to compute the Newton correction to the approximate coupled cluster amplitude. The multiplication of the Jacobian with a vector, which is required in each step of a Krylov iterative method such as the Generalized Minimum Residual (GMRES) method, is carried out through a finite difference approximation, and requires an additional residual evaluation. The overall cost of the method is determined by the sum of the inner Krylov and outer Newton iterations. We discuss the termination criterion used for the inner iteration and show how to apply pre-conditioners to accelerate convergence. We will also examine the use of regularization technique to improve the stability of convergence and compare the method with the widely used direct inversion of iterative subspace (DIIS) methods through numerical examples.

Compact Crank–Nicolson and Du Fort–Frankel Method for the Solution of the Time Fractional Diffusion Equation

2015 ◽  
Vol 12 (06) ◽  
pp. 1550041 ◽  
Author(s):  
Faoziya Al-Shibani ◽  
Ahmad Ismail
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Finite Difference Methods ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Time Fractional Diffusion Equation ◽  
The Stability ◽  
The One

In this paper, two compact implicit finite difference methods are developed and analyzed for solving the one-dimensional time fractional diffusion equation. The temporal derivative is approximated by using Grünwald–Letnikov formula. Compact finite difference approximation is used for the second-order derivative in space. The local truncation errors are discussed. The stability analysis and the convergence of the proposed methods are investigated by means of Fourier series method. A comparison between the results of these methods and the exact solution is made. Numerical tests are given to verify the feasibility and accuracy of the methods.

scholarly journals Adaptive GMRES(m) for the Electromagnetic Scattering Problem

2020 ◽  
Vol 21 (1) ◽  
pp. 191
Author(s):  
Gustavo Espínola ◽  
Juan C. Cabral ◽  
Christian Schaerer
Keyword(s):  
Finite Difference ◽  
Helmholtz Equation ◽  
Control Problem ◽  
Electromagnetic Scattering ◽  
Scattering Problem ◽  
Finite Difference Approximation ◽  
Computational Time ◽  
Difference Approximation ◽  
Standard Methods ◽  
Restarted Gmres

In this article, an adaptive version of the restarted GMRES (GMRES(m)) is introduced for the resolution of the finite difference approximation of the Helmholtz equation. It has been observed that the choice of the restart parameter m strongly affects the convergence of standard GMRES(m). To overcome this problem, the GMRES(m) is formulated as a control problem in order to adaptively combine two strategies: a) the appropriate variation of the restarted parameter m, if a stagnation in the convergence is detected; and b) the augmentation of the search subspace using vectors obtained at previous cycles. The proposal is compared with similar iterative methods of the literature based on standard GMRES(m) with fixed parameters. Numerical results for selected matrices suggest that the switching adaptive proposal method could overcome the stagnation observed in standard methods, and even improve the performance in terms of computational time and memory requirements.

Symmetric Multistep Methods Revisited: II. Numerical Experiments

1999 ◽  
Vol 173 ◽  
pp. 309-314 ◽  
Author(s):  
T. Fukushima
Keyword(s):  
Multistep Methods ◽  
Computational Time ◽  
Integration Time ◽  
Implicit Methods ◽  
Symmetric Methods ◽  
Celestial Bodies ◽  
The Stability ◽  
Predictor Corrector ◽  
Symmetric Multistep Methods ◽  
Integration Errors

AbstractBy using the stability condition and general formulas developed by Fukushima (1998 = Paper I) we discovered that, just as in the case of the explicit symmetric multistep methods (Quinlan and Tremaine, 1990), when integrating orbital motions of celestial bodies, the implicit symmetric multistep methods used in the predictor-corrector manner lead to integration errors in position which grow linearly with the integration time if the stepsizes adopted are sufficiently small and if the number of corrections is sufficiently large, say two or three. We confirmed also that the symmetric methods (explicit or implicit) would produce the stepsize-dependent instabilities/resonances, which was discovered by A. Toomre in 1991 and confirmed by G.D. Quinlan for some high order explicit methods. Although the implicit methods require twice or more computational time for the same stepsize than the explicit symmetric ones do, they seem to be preferable since they reduce these undesirable features significantly.

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