scholarly journals An Energy Conserving Numerical Scheme for the Dynamics of Hyperelastic Rods

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Thorsten Fütterer ◽  
Axel Klar ◽  
Raimund Wegener
Keyword(s):  
Numerical Method ◽  
Numerical Scheme ◽  
Hyperelastic Materials ◽  
Cosserat Rods ◽  
Energy Conserving ◽  
Potential Forces

A numerical method for special Cosserat rods based on Antman's description Antman, 2005 is developed for hyperelastic materials and potential forces. This method preserves the relevant properties of the underlying PDE system, namely, the orthonormality of the directors and the conservation of the energy.

scholarly journals Numerical Solution of Stieltjes Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1571
Author(s):  
Francisco J. Fernández ◽  
F. Adrián F. Tojo
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Explicit Solution ◽  
Numerical Scheme ◽  
Linear Ordinary Differential Equation ◽  
Stieltjes Integral ◽  
Novel Method ◽  
Theoretical Results

This work is devoted to the obtaining of a new numerical scheme based on quadrature formulae for the Lebesgue–Stieltjes integral for the approximation of Stieltjes ordinary differential equations. This novel method allows us to numerically approximate models based on Stieltjes ordinary differential equations for which no explicit solution is known. We prove several theoretical results related to the consistency, convergence, and stability of the numerical method. We also obtain the explicit solution of the Stieltjes linear ordinary differential equation and use it to validate the numerical method. Finally, we present some numerical results that we have obtained for a realistic population model based on a Stieltjes differential equation and a system of Stieltjes differential equations with several derivators.

A numerical method based on Haar wavelets for the Hadamard-type fractional differential equations

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Zain ul Abdeen ◽  
Mujeeb ur Rehman
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Haar Wavelets ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Content Type ◽  
Fractional Differential

PurposeThe purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.Design/methodology/approachThe aim of this paper is to develop a numerical scheme for numerical solutions of Hadamard-type fractional differential equations. The classical Haar wavelets are modified to align them with Hadamard-type operators. Operational matrices are derived and used to convert differential equations to systems of algebraic equations.FindingsThe upper bound for error is estimated. With the help of quasilinearization, nonlinear problems are converted to sequences of linear problems and operational matrices for modified Haar wavelets are used to get their numerical solution. Several numerical examples are presented to demonstrate the applicability and validity of the proposed method.Originality/valueThe numerical method is purposed for solving Hadamard-type fractional differential equations.

An Improvement of a Nonclassical Numerical Method for the Computation of Fractional Derivatives

2009 ◽  
Vol 131 (1) ◽  
Author(s):  
Kai Diethelm
Keyword(s):  
Numerical Calculation ◽  
Numerical Method ◽  
Dynamic Systems ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Differential Operators ◽  
Standard Methods ◽  
Efficient Approach

Standard methods for the numerical calculation of fractional derivatives can be slow and memory consuming due to the nonlocality of the differential operators. Yuan and Agrawal (2002, “A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives,” ASME J. Vibr. Acoust., 124, pp. 321–324) have proposed a more efficient approach for operators whose order is between 0 and 1 that differs substantially from the traditional concepts. It seems, however, that the accuracy of the results can be poor. We modify the approach, adapting it better to the properties of the problem, and show that this leads to a significantly improved quality. Our idea also works for operators of order greater than 1.

scholarly journals A Numerical Scheme Based on the Chebyshev Functions to Find Approximate Solutions of the Coupled Nonlinear Sine-Gordon Equations with Fractional Variable Orders

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
MohammadHossein Derakhshan
Keyword(s):  
Fractional Derivative ◽  
Numerical Method ◽  
Numerical Scheme ◽  
Numerical Approximation ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variable Order ◽  
Algebraic Equations ◽  
Variable Order Fractional Derivative ◽  
Chebyshev Functions

In this article, a numerical method based on the shifted Chebyshev functions for the numerical approximation of the coupled nonlinear variable-order fractional sine-Gordon equations is shown. The variable-order fractional derivative is considered in the sense of Caputo-Prabhakar. To solve the problem, first, we obtain the operational matrix of the Caputo-Prabhakar fractional derivative of shifted Chebyshev polynomials. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional sine-Gordon equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate the accuracy and efficiency of the proposed method.

scholarly journals Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule

2021 ◽  
Vol 5 (3) ◽  
pp. 111
Author(s):  
Samaneh Soradi-Zeid ◽  
Mehdi Mesrizadeh ◽  
Carlo Cattani
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Numerical Method ◽  
Fractional Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Quadrature Rule ◽  
Differential Problem ◽  
Transform Method ◽  
Fractional Differential

This paper introduces an efficient numerical scheme for solving a significant class of fractional differential equations. The major contributions made in this paper apply a direct approach based on a combination of time discretization and the Laplace transform method to transcribe the fractional differential problem under study into a dynamic linear equations system. The resulting problem is then solved by employing the numerical method of the quadrature rule, which is also a well-developed numerical method. The present numerical scheme, which is based on the numerical inversion of Laplace transform and equal-width quadrature rule is robust and efficient. Some numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.

High Order Numerical Methods for Fractional Terminal Value Problems

2014 ◽  
Vol 14 (1) ◽  
pp. 55-70 ◽  
Author(s):  
Neville J. Ford ◽  
Maria L. Morgado ◽  
Magda Rebelo
Keyword(s):  
Numerical Methods ◽  
Boundary Value Problems ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Numerical Scheme ◽  
Initial Value ◽  
Shooting Algorithm ◽  
Effective Approximation

Abstract. In this paper we present a shooting algorithm to solve fractional terminal (or boundary) value problems. We provide a convergence analysis of the numerical method, derived based upon properties of the equation being solved and without the need to impose smoothness conditions on the solution. The work is a sequel to our recent investigation where we constructed a nonpolynomial collocation method for the approximation of the solution to fractional initial value problems. Here we show that the method can be adapted for the effective approximation of the solution of terminal value problems. Moreover, we compare the efficiency of this numerical scheme against other existing methods.

A numerical method for the fractional Schrödinger type equation of spatial dimension two

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Neville Ford ◽  
M. Manuela Rodrigues ◽  
Nelson Vieira
Keyword(s):  
Wave Function ◽  
Numerical Method ◽  
Numerical Scheme ◽  
Type Equation ◽  
Spatial Dimension ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Mechanical Probability ◽  
Laplace And Fourier Transform

AbstractThis work focuses on an investigation of the (n+1)-dimensional time-dependent fractional Schrödinger type equation. In the early part of the paper, the wave function is obtained using Laplace and Fourier transform methods and a symbolic operational form of the solutions in terms of Mittag-Leffler functions is provided. We present an expression for the wave function and for the quantum mechanical probability density. We introduce a numerical method to solve the case where the space component has dimension two. Stability conditions for the numerical scheme are obtained.

scholarly journals An Improved Finite Difference Type Numerical Method for Structural Dynamic Analysis

1994 ◽  
Vol 1 (6) ◽  
pp. 569-583
Author(s):  
Sung-Hoon Kim ◽  
Youn-sik Park
Keyword(s):  
Finite Difference ◽  
Numerical Method ◽  
Numerical Scheme ◽  
Single Step ◽  
Second Order ◽  
Partial Differential ◽  
Central Difference ◽  
Structural Dynamic ◽  
Difference Type ◽  
Natural Boundary Conditions

An improved finite difference type numerical method to solve partial differential equations for one-dimensional (1-D) structure is proposed. This numerical scheme is a kind of a single-step, second-order accurate and implicit method. The stability, consistency, and convergence are examined analytically with a second-order hyperbolic partial differential equation. Since the proposed numerical scheme automatically satisfies the natural boundary conditions and at the same time, all the partial differential terms at boundary points are directly interpretable to their physical meanings, the proposed numerical scheme has merits in computing 1-D structural dynamic motion over the existing finite difference numeric methods. Using a numerical example, the suggested method was proven to be more accurate and effective than the well-known central difference method. The only limitation of this method is that it is applicable to only 1-D structure.

scholarly journals A Positivity-Preserving Numerical Scheme for Nonlinear Option Pricing Models

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Shengwu Zhou ◽  
Wei Li ◽  
Yu Wei ◽  
Cui Wen
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Method ◽  
Numerical Scheme ◽  
Discrete Equation ◽  
Pricing Models ◽  
Positivity Preserving ◽  
Unconditionally Stable ◽  
Put Option ◽  
Black Scholes

A positivity-preserving numerical method for nonlinear Black-Scholes models is developed in this paper. The numerical method is based on a nonstandard approximation of the second partial derivative. The scheme is not only unconditionally stable and positive, but also allows us to solve the discrete equation explicitly. Monotone properties are studied in order to avoid unwanted oscillations of the numerical solution. The numerical results for European put option and European butterfly spread are compared to the standard finite difference scheme. It turns out that the proposed scheme is efficient and reliable.

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