scholarly journals On Solution of Fredholm Integrodifferential Equations Using Composite Chebyshev Finite Difference Method

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Z. Pashazadeh Atabakan ◽  
A. Kazemi Nasab ◽  
A. Kılıçman
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Scheme ◽  
Difference Method ◽  
Integrodifferential Equations ◽  
Algebraic Equations ◽  
Block Pulse Functions ◽  
Collocation Points ◽  
Chebyshev Finite Difference Method ◽  
The Given

A new numerical method is introduced for solving linear Fredholm integrodifferential equations which is based on a hybrid of block-pulse functions and Chebyshev polynomials using the well-known Chebyshev-Gauss-Lobatto collocation points. Composite Chebyshev finite difference method is indeed an extension of the Chebyshev finite difference method and can be considered as a nonuniform finite difference scheme. The main advantage of the proposed method is reducing the given problem to a set of algebraic equations. Some examples are given to approve the efficiency and the accuracy of the proposed method.

scholarly journals Numerical Solution of Second-Order Fredholm Integrodifferential Equations with Boundary Conditions by Quadrature-Difference Method

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
Chriscella Jalius ◽  
Zanariah Abdul Majid
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Conditions ◽  
Difference Method ◽  
Numerical Experiments ◽  
Second Order ◽  
Integrodifferential Equations ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Gauss Elimination

In this research, the quadrature-difference method with Gauss Elimination (GE) method is applied for solving the second-order of linear Fredholm integrodifferential equations (LFIDEs). In order to derive an approximation equation, the combinations of Composite Simpson’s 1/3 rule and second-order finite-difference method are used to discretize the second-order of LFIDEs. This approximation equation will be used to generate a system of linear algebraic equations and will be solved by using Gauss Elimination. In addition, the formulation and the implementation of the quadrature-difference method are explained in detail. Finally, some numerical experiments were carried out to examine the accuracy of the proposed method.

FINITE DIFFERENCE METHOD FOR SOLVING THE NONLINEAR DYNAMIC EQUATION OF UNDERWATER TOWED SYSTEM

2014 ◽  
Vol 11 (04) ◽  
pp. 1350060 ◽  
Author(s):  
ZHIJIANG YUAN ◽  
LIANGAN JIN ◽  
WEI CHI ◽  
HENGDOU TIAN
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Solution ◽  
Difference Method ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Computational Time ◽  
Algebraic Equations ◽  
Nonlinear Dynamic Equation ◽  
Nonlinear Dynamic Equations

A wide body of work exists that describes numerical solution for the nonlinear system of underwater towed system. Many researchers usually divide the tow cable with less number elements for the consideration of computational time. However, this type of installation affects the accuracy of the numerical solution. In this paper, a newly finite difference method for solving the nonlinear dynamic equations of the towed system is developed. The mathematical model of tow cable and towed body are both discretized to nonlinear algebraic equations by center finite difference method. A newly discipline for formulating the nonlinear equations and Jacobian matrix of towed system are proposed. We can solve the nonlinear dynamic equation of underwater towed system quickly by using this discipline, when the size of number elements is large.

scholarly journals Chebyshev Wavelet Finite Difference Method: A New Approach for Solving Initial and Boundary Value Problems of Fractional Order

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
A. Kazemi Nasab ◽  
A. Kılıçman ◽  
Z. Pashazadeh Atabakan ◽  
S. Abbasbandy
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Finite Difference Methods ◽  
Algebraic Equations ◽  
Chebyshev Wavelets ◽  
New Approach ◽  
Nonlinear Algebraic Equations ◽  
Chebyshev Wavelet ◽  
Fractional Differential

A new method based on a hybrid of Chebyshev wavelets and finite difference methods is introduced for solving linear and nonlinear fractional differential equations. The useful properties of the Chebyshev wavelets and finite difference method are utilized to reduce the computation of the problem to a set of linear or nonlinear algebraic equations. This method can be considered as a nonuniform finite difference method. Some examples are given to verify and illustrate the efficiency and simplicity of the proposed method.

Numerical Modeling of Casting Solidification Using Generalized Finite Difference Method

2010 ◽  
Vol 638-642 ◽  
pp. 2676-2681 ◽  
Author(s):  
Bohdan Mochnacki ◽  
Ewa Majchrzak
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Energy Equation ◽  
Solidification Process ◽  
Temperature Gradients ◽  
Thermal Processes ◽  
Heating Rates ◽  
Collocation Points ◽  
The One

The system casting-mould is considered. The thermal processes proceeding in a casting sub-domain are described using the one domain approach. The model of solidification process is supplemented by the energy equation concerning the mould sub-domain, the continuity conditions given on the contact surface between casting and mould, boundary conditions on the outer surface of the system and the initial ones. To solve the problem the generalized variant of finite difference method (GFDM) is used. Temporary and local values of temperature can be found at the optional set of collocation points from the domain considered. This essential advantage of GFDM allows to locate and thicken nodes at the regions for which the temperature gradients and cooling (heating) rates are considerable. In the final part of the paper, the example of numerical simulation is shown.

Fractional Chebyshev Finite Difference Method for Solving the Fractional-Order Delay BVPs

2015 ◽  
Vol 12 (06) ◽  
pp. 1550033 ◽  
Author(s):  
M. M. Khader
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Fractional Order ◽  
Difference Method ◽  
Matrix Method ◽  
Fractional Derivatives ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Operational Matrix Method ◽  
Chebyshev Finite Difference Method

In this paper, we implement an efficient numerical technique which we call fractional Chebyshev finite difference method (FChFDM). The fractional derivatives are presented in terms of Caputo sense. The algorithm is based on a combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the fractional derivatives. This operational matrix method can be regarded as a nonuniform finite difference scheme. The error bound for the fractional derivatives is introduced. We used the introduced technique to solve numerically the fractional-order delay BVPs. The application of the proposed method to introduced problem leads to algebraic systems which can be solved by an appropriate numerical method. Several numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method.

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.

A Novel Finite Difference Method for Flow Calculation on Colocated Grids

2008 ◽  
Author(s):  
Z. Z. Xia ◽  
P. Zhang ◽  
R. Z. Wang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Solutions ◽  
Fluid Dynamic ◽  
Algebraic Equations ◽  
Duct Flow ◽  
Staggered Grids ◽  
Linear Algebraic Equations ◽  
Incomplete Lu Factorization

A new finite difference method, which removes the need for staggered grids in fluid dynamic computation, is presented. Pressure checker boarding is prevented through a dual-velocity scheme that incorporates the influence of pressure on velocity gradients. A supplementary velocity resulting from the discrete divergence of pressure gradient, together with the main velocity driven by the discretized pressure first-order gradient, is introduced for the discretization of continuity equation. The method in which linear algebraic equations are solved using incomplete LU factorization, removes the pressure-correction equation, and was applied to rectangle duct flow and natural convection in a cubic cavity. These numerical solutions are in excellent agreement with the analytical solutions and those of the algorithm on staggered grids. The new method is shown to be superior in convergence compared to the original one on staggered grids.

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