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

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.

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

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Yifan Qin ◽  
Xiaocheng Yang ◽  
Yunzhu Ren ◽  
Yinghong Xu ◽  
Wahidullah Niazi
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Scheme ◽  
Difference Method ◽  
Stability And Convergence ◽  
Sobolev Equation ◽  
Numerical Examples ◽  
Convergence Orders ◽  
Theoretical Results ◽  
Crank Nicolson

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.

scholarly journals Stable difference scheme for a nonlocal boundary value heat conduction problem

2018 ◽  
Vol 1 (1) ◽  
pp. 1-10
Author(s):  
Makhmud A. Sadybekov
Keyword(s):  
Heat Conduction ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Heat Equation ◽  
Difference Scheme ◽  
Difference Method ◽  
Heat Conduction Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Strong Regularity

AbstractIn this paper, a new finite difference method to solve nonlocal boundary value problems for the heat equation is proposed. The most important feature of these problems is the non-self-adjointness. Because of the non-self-adjointness, major difficulties occur when applying analytical and numerical solution techniques. Moreover, problems with boundary conditions that do not possess strong regularity are less studied. The scope of the present paper is to justify possibility of building a stable difference scheme with weights for mentioned type of problems above.

scholarly journals A Hybrid Finite Difference Method for Pricing Two-Asset Double Barrier Options

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Y. L. Hsiao ◽  
S. Y. Shen ◽  
Andrew M. L. Wang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Hybrid Method ◽  
Difference Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Barrier Options ◽  
Barrier Option ◽  
Double Barrier ◽  
Numerical Examples

The pricing of the two-asset double barrier option is modeled as an initial-boundary value problem of the two-dimensional Black-Scholes partial differential equation. We use the hybrid finite different method to solve the problem. The hybrid method is a combination of the Laplace transform and a finite difference method. It is more efficient than a traditional finite difference method to obtain a solution without a step-by-step process. The method is implemented on a computer. Two numerical examples are calculated to verify the performance of the hybrid method. In our numerical examples, the convergence rate of the method is approximately two. We conclude that the method is efficient for pricing two-asset barrier options.

scholarly journals Computational Analysis of the Stability of 2D Heat Equation on Elliptical Domain Using Finite Difference Method

2020 ◽  
pp. 8-19
Author(s):  
Mehwish Naz Rajput ◽  
Asif Ali Shaikh ◽  
Shakeel Ahmed Kamboh
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Heat Equation ◽  
Stability Condition ◽  
Difference Method ◽  
Stable Solution ◽  
Finite Difference Schemes ◽  
Step Size ◽  
Stability Range ◽  
The Stability

Aims: The aim and objective of the study to derive and analyze the stability of the finite difference schemes in relation to the irregularity of domain. Study Design: First of all, an elliptical domain has been constructed with the governing two dimensional (2D) heat equation that is discretized using the Finite Difference Method (FDM). Then the stability condition has been defined and the numerical solution by writing MATLAB codes has been obtained with the stable values of time domain. Place and Duration of Study: The work has been jointly conducted at the MUET, Jamshoro and QUEST, Nawabshah Pakistan from January 2019 to December 2019.  Methodology: The stability condition over an elliptical domain with the non-uniform step size depending upon the boundary tracing function is derived by using Von Neumann method. Results: From the results it was revealed that stability region for the small number of mesh points remains larger and gets smaller as the number of mesh nodes is increased. Moreover, the ranges for the time steps are defined for varied spatial step sizes that help to find the stable solution. Conclusion: The corresponding stability range for number of nodes N=10, 20, 30, 40, 50, and 60 was found respectively. Within this range the solution remains smooth as time increases. The results of this study attempt to provide the stable solution of partial differential equations on irregular domains.

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

2018 ◽  
Vol 23 (4) ◽  
pp. 53 ◽  
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.

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

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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