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.

scholarly journals A parallel in time/spectral collocation combined with finite difference method for the time fractional differential equations

2021 ◽  
Vol 15 ◽  
pp. 174830262110084
Author(s):  
Xianjuan Li ◽  
Yanhui Su
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Spectral Method ◽  
Fractional Differential Equations ◽  
Difference Method ◽  
Spectral Collocation ◽  
Parallel Method ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

In this article, we consider the numerical solution for the time fractional differential equations (TFDEs). We propose a parallel in time method, combined with a spectral collocation scheme and the finite difference scheme for the TFDEs. The parallel in time method follows the same sprit as the domain decomposition that consists in breaking the domain of computation into subdomains and solving iteratively the sub-problems over each subdomain in a parallel way. Concretely, the iterative scheme falls in the category of the predictor-corrector scheme, where the predictor is solved by finite difference method in a sequential way, while the corrector is solved by computing the difference between spectral collocation and finite difference method in a parallel way. The solution of the iterative method converges to the solution of the spectral method with high accuracy. Some numerical tests are performed to confirm the efficiency of the method in three areas: (i) convergence behaviors with respect to the discretization parameters are tested; (ii) the overall CPU time in parallel machine is compared with that for solving the original problem by spectral method in a single processor; (iii) for the fixed precision, while the parallel elements grow larger, the iteration number of the parallel method always keep constant, which plays the key role in the efficiency of the time parallel 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 A Comparative Study between Implicit and Crank-Nicolson Finite Difference Method for Option Pricing

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

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.

Determination of the Steady-State Response of Viscoelastically Supported Rectangular Orthotropic Mass Loaded Plates by an Energy-Based Finite Difference Method

2005 ◽  
Vol 11 (12) ◽  
pp. 1535-1552 ◽  
Author(s):  
Gökhan Altintaş ◽  
Muhiddin Bağci
Keyword(s):  
Steady State ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Plate Theory ◽  
Steady State Response ◽  
Algebraic Equations ◽  
State Response ◽  
Elastic Rectangular Plate ◽  
Spring Coefficient

A method based on a variational procedure in conjunction with a finite difference method is used to examine the free vibration characteristics and steady-state response to a sinusoidally varying force applied orthotropic elastic rectangular plate carrying masses. Using the energy-based finite difference method, the problem reduced to the solution of a system of algebraic equations. Due to the significance of the fundamental natural frequency of the plate, its variation is investigated with respect to the mechanical properties of the plate material, the translational spring coefficient of the supports, the mass distribution, the mass locations and the quantity of mass. The steady-state response of the viscoelastically supported plates was also investigated numerically for the damping coefficient of the supports and the force distribution in addition to the characteristics of the plate system. Many new results are presented and the validity of the present approach is demonstrated by comparing the results with other solutions based on the Kirchhoff-Love plate theory.

scholarly journals Application of the Finite Differences Methods to Computation of Definite Integrals

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Multistep Methods ◽  
Finite Difference Methods ◽  
Engineering Problem ◽  
Stable Method ◽  
Order Of Accuracy ◽  
Definite Integrals ◽  
The Finite Difference Method

- It is known that one of the popular methods constructed to solve the scientific and engineering problem is the finite difference method studied beginning from the known and famous scientists as the Newton, Leibniz, Euler and etc. One of the first applications of the finite difference method is defined as the conception of the derivatives for the continuous function. These methods have been successfully used in solving of the differential equations at the present time. But here have investigated the application of the finite-difference methods to compute the definite integrals. Constructed the stable methods with the high order of accuracy which are applied to computation of the definite integrals. Also has been found some connection between the constructed multistep methods, the Gauss and Chebyshev methods. The received here methods have been applied to the computation of the double definite integral. For this aim, here have been investigated the double integrals for the degenerate functions. The advantages of these methods are illustrated by using the known model integral, which has been solved by using the stable method with the degree p=6.

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.

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