A Newton Linearized Crank-Nicolson Method for the Nonlinear Space Fractional Sobolev Equation

In this paper, one class of finite difference scheme is proposed to solve nonlinear space fractional Sobolev equation based on the Crank-Nicolson (CN) method. Firstly, a fractional centered finite difference method in space and the CN method in time are utilized to discretize the original equation. Next, the existence, uniqueness, stability, and convergence of the numerical method are analyzed at length, and the convergence orders are proved to be O τ 2 + h 2 in the sense of l 2 -norm, H α / 2 -norm, and l ∞ -norm. Finally, the extensive numerical examples are carried out to verify our theoretical results and show the effectiveness of our algorithm in simulating spatial fractional Sobolev equation.

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Application of Fuzzy Finite Difference Scheme for the Non-homogeneous Fuzzy Heat Equation

10.21203/rs.3.rs-1003065/v1 ◽

2021 ◽

Author(s):

Samaneh Zabihi ◽

reza ezzati ◽

F Fattahzadeh ◽

J Rashidinia

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Heat Equation ◽

Difference Scheme ◽

Difference Method ◽

Taylor Expansion ◽

Truncation Error ◽

Initial Boundary ◽

Convergence Conditions ◽

Numerical Examples

Abstract A numerical framework based on fuzzy finite difference is presented for approximating fuzzy triangular solutions of fuzzy partial differential equations by considering the type of $[gH-p]-$differentiability. The fuzzy triangle functions are expanded using full fuzzy Taylor expansion to develop a new fuzzy finite difference method. By considering the type of gH-differentiability, we approximate the fuzzy derivatives with a new fuzzy finite-difference. In particular, we propose using this method to solve non-homogeneous fuzzy heat equation with triangular initial-boundary conditions. We examine the truncation error and the convergence conditions of the proposed method. Several numerical examples are presented to demonstrate the performance of the methods. The final results demonstrate the efficiency and the ability of the new fuzzy finite difference method to produce triangular fuzzy numerical results which are more consistent with existing reality.

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Finite Difference Solution to a Nonlinear Time-Fractional Logistic Model

Journal of Mathematics Research ◽

10.5539/jmr.v13n2p60 ◽

2021 ◽

Vol 13 (2) ◽

pp. 60

Author(s):

Yuanyuan Yang ◽

Gongsheng Li

Keyword(s):

Finite Difference ◽

Theoretical Analysis ◽

Difference Scheme ◽

Logistic Model ◽

Stability And Convergence ◽

Numerical Examples ◽

Implicit Finite Difference Scheme ◽

Finite Difference Solution ◽

Implicit Finite Difference ◽

Convergence Of The Scheme

We set forth a time-fractional logistic model and give an implicit finite difference scheme for solving of the model. The L^2 stability and convergence of the scheme are proved with the aids of discrete Gronwall inequality, and numerical examples are presented to support the theoretical analysis.

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A Comparative Study between Implicit and Crank-Nicolson Finite Difference Method for Option Pricing

GANIT Journal of Bangladesh Mathematical Society ◽

10.3329/ganit.v40i1.48192 ◽

2020 ◽

Vol 40 (1) ◽

pp. 13-27

Author(s):

Tanmoy Kumar Debnath ◽

ABM Shahadat Hossain

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Option Pricing ◽

Difference Method ◽

Finite Difference Methods ◽

Put Option ◽

Black Scholes ◽

Implicit Finite Difference ◽

Difference Methods ◽

Crank Nicolson

In this paper, we have applied the finite difference methods (FDMs) for the valuation of European put option (EPO). We have mainly focused the application of Implicit finite difference method (IFDM) and Crank-Nicolson finite difference method (CNFDM) for option pricing. Both these techniques are used to discretized Black-Scholes (BS) partial differential equation (PDE). We have also compared the convergence of the IFDM and CNFDM to the analytic BS price of the option. This turns out a conclusion that both these techniques are fairly fruitful and excellent for option pricing. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 13-27

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Numerical Solution of Fourth-Order Boundary Value Problems for Euler-Bernoulli Beam Equation using FDM

Journal of Physics Conference Series ◽

10.1088/1742-6596/2070/1/012052 ◽

2021 ◽

Vol 2070 (1) ◽

pp. 012052

Author(s):

M. Adak ◽

A. Mandal

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Boundary Value Problems ◽

Fourth Order ◽

Difference Method ◽

Boundary Value ◽

Stability And Convergence ◽

Beam Equation ◽

Bernoulli Beam ◽

Euler Bernoulli Beam

Abstract Euler-Bernoulli beam equation is widely used in engineering, especially civil and mechanical engineering to determine the deflection or strength of bending beam. In physical science and engineering, to predict the deflection for beam problem, bending moment, soil settlement and modeling of viscoelastic flows, fourth-order ordinary differential equation (ODE) is widely used. The analytical solution of most of the higher order ordinary differential equations with complicated boundary condition that occur in any engineering problems is not easy way. Therefore, numerical technique based on finite difference method (FDM) is comparatively easy and important for solving the boundary value problems (BVP). In this study four boundary conditions (Neumann condition) are considered for solving BVP. Absolute error calculation, numerical stability and convergence are discussed. Two examples are considered to illustrate the finite difference method for solving fourth order BVP. The numerical results are rapidly converged with exact results. The results shows that the FDM is appropriate and reliable for such type of problems. Thus present study will enhance the mathematical understanding of engineering students along with an application in different field.

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Numerical Solution of Stochastic Generalized Fractional Diffusion Equation by Finite Difference Method

Mathematical and Computational Applications ◽

10.3390/mca23040053 ◽

2018 ◽

Vol 23 (4) ◽

pp. 53 ◽

Cited By ~ 1

Author(s):

Amaneh Sepahvandzadeh ◽

Bahman Ghazanfari ◽

Nader Asadian

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Diffusion Equation ◽

Numerical Solution ◽

Difference Method ◽

Fractional Diffusion ◽

Fractional Diffusion Equation ◽

Mean Square ◽

Numerical Examples ◽

Mean Square Convergence

The present study aimed at solving the stochastic generalized fractional diffusion equation (SGFDE) by means of the random finite difference method (FDM). Moreover, the conditions of mean square convergence of the numerical solution are studied and numerical examples are presented to demonstrate the validity and accuracy of the method.

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Fully Coupled Numerical Methods for Elastohydrodynamic Lubrication Line Contact Problems by the High Order Finite Difference Methods

10.21203/rs.3.rs-914621/v1 ◽

2021 ◽

Author(s):

QUAN SHEN ◽

Bing Wu ◽

GUANGWEN XIAO

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Difference Scheme ◽

Difference Method ◽

Elastohydrodynamic Lubrication ◽

Contact Problems ◽

High Order ◽

Line Contact ◽

High Order Finite Difference ◽

Upwind Finite Difference

Abstract In this paper a high order finite difference method is constructed to solve the elastohydrodynamic lubrication line contact problems, whose cavitation condition is handled by the penalty method. The highly nonlinear equations from the discretization of the high order finite difference method are solved by the trust-region dogleg algorithm. In order to reduce the numerical dissipation and dispersion brought by the high order upwind finite difference scheme, a high order biased upwind finite difference scheme is also presented. Our method is found to achieve more accurate solutions using just a small number of nodes compared to the multilevel methods combined with the lower order finite difference method.

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