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.

scholarly journals Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 322
Author(s):  
Ricardo Almeida ◽  
Ravi P. Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Lyapunov Functions ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Hopfield Neural Network ◽  
Quadratic Lyapunov Functions ◽  
Fractional Differential ◽  
The Stability ◽  
Theoretical Results

A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results.

scholarly journals Interpolating Stabilized Element Free Galerkin Method for Neutral Delay Fractional Damped Diffusion-Wave Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mostafa Abbaszadeh ◽  
Mehdi Dehghan ◽  
Mahmoud A. Zaky ◽  
Ahmed S. Hendy
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Element Free Galerkin Method ◽  
Neutral Delay ◽  
Diffusion Wave ◽  
Element Free Galerkin ◽  
Numerical Formulation ◽  
Element Free

A numerical solution for neutral delay fractional order partial differential equations involving the Caputo fractional derivative is constructed. In line with this goal, the drift term and the time Caputo fractional derivative are discretized by a finite difference approximation. The energy method is used to investigate the rate of convergence and unconditional stability of the temporal discretization. The interpolation of moving Kriging technique is then used to approximate the space derivative, yielding a meshless numerical formulation. We conclude with some numerical experiments that validate the theoretical findings.

PERTURBATION STABLE CONDITIONAL ANALYTIC MONTE-CARLO PRICING SCHEME FOR AUTO-CALLABLE PRODUCTS

2011 ◽  
Vol 14 (02) ◽  
pp. 197-219 ◽  
Author(s):  
CHRISTIAN P. FRIES ◽  
MARK S. JOSHI
Keyword(s):  
Monte Carlo ◽  
Finite Difference ◽  
Transition Probability ◽  
Payoff Function ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Probability Of Survival ◽  
Monte Carlo Approximation ◽  
Stochastic Location ◽  
Finite Sum

In this paper, we present a generic method for the Monte-Carlo pricing of (generalized) auto-callable products (aka. trigger products), i.e., products for which the payout function features a discontinuity with a (possibly) stochastic location (the trigger) and value (the payout). The Monte-Carlo pricing of products with discontinuous payout is known to come with a high Monte-Carlo error. The numerical calculation of sensitivities (i.e., partial derivatives) of such prices by finite differences gives very noisy results, since the Monte-Carlo approximation (being a finite sum of discontinuous functions) is not smooth. Additionally, the Monte-Carlo error of the finite-difference approximation explodes as the shift size tends to zero. Our method combines a product specific modification of the underlying numerical scheme, which is to some extent similar to an importance sampling and/or partial proxy simulation scheme and a reformulation of the payoff function into an equivalent smooth payout. From the financial product we merely require that hitting of the stochastic trigger will result in an conditionally analytic value. Many complex derivatives can be written in this form. A class of products where this property is usually encountered are the so called auto-callables, where a trigger hit results in cancellation of all future payments except for one redemption payment, which can be valued analytically, conditionally on the trigger hit. From the model we require that its numerical implementation allows for a calculation of the transition probability of survival (i.e., non-trigger hit). Many models allows this, e.g., Euler schemes of Itô processes, where the trigger is a model primitive. The method presented is effective across a large range of cases where other methods fail, e.g. small finite difference shift sizes or short time to trigger reset (approaching maturity); this means that a practitioner can use this method and be confident that it will work consistently. The method itself can be viewed as a generalization of the method proposed by Glasserman and Staum (2001), both with respect to the type (and shape) of the boundaries, as well as, with respect to the class of products considered. In addition we explicitly consider the calculation of sensitivities.

A diffusion–convection problem with a fractional derivative along the trajectory of motion

2021 ◽  
Vol 36 (3) ◽  
pp. 157-163
Author(s):  
Alexander V. Lapin ◽  
Vladimir V. Shaidurov
Keyword(s):  
Finite Difference ◽  
Fractional Derivative ◽  
Characteristic Curve ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Finite Difference Schemes ◽  
Stability Estimates ◽  
Initial Boundary Value

Abstract A new mathematical model of the diffusion–convective process with ‘memory along the flow path’ is proposed. This process is described by a homogeneous one-dimensional Dirichlet initial-boundary value problem with a fractional derivative along the characteristic curve of the convection operator. A finite-difference approximation of the problem is constructed and investigated. The stability estimates for finite-difference schemes are proved. The accuracy estimates are given for the case of sufficiently smooth input data and the solution.

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 Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

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