scholarly journals New Modification on Heun’s Method Based on Contraharmonic Mean for Solving Initial Value Problems with High Efficiency

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Abushet Hayalu Workie
Keyword(s):  
Initial Value Problems ◽  
High Performance ◽  
High Efficiency ◽  
Arithmetic Mean ◽  
Initial Condition ◽  
Initial Value ◽  
Euler’S Method ◽  
Contraharmonic Mean ◽  
The Stability ◽  
Tangent Slope

In this paper, small modification on Improved Euler’s method (Heun’s method) is proposed to improve the efficiency so as to solve ordinary differential equations with initial condition by assuming the tangent slope as an average of the arithmetic mean and contra-harmonic mean. In order to validate the conclusion, the stability, consistency, and accuracy of the system were evaluated and numerical results were presented, and it was recognized that the proposed method is more stable, consistent, and accurate with high performance.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

scholarly journals Finding symmetries by incorporating initial conditions as side conditions

2008 ◽  
Vol 19 (6) ◽  
pp. 701-715 ◽  
Author(s):  
JOANNA GOARD
Keyword(s):  
Initial Value Problems ◽  
Initial Conditions ◽  
Invariant Solution ◽  
Lie Symmetry ◽  
Initial Condition ◽  
Initial Value ◽  
Side Condition ◽  
Symmetry Generator ◽  
Infinitesimal Symmetry

It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.

scholarly journals TheZ-Transform Method and Delta Type Fractional Difference Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Dorota Mozyrska ◽  
Małgorzata Wyrwas
Keyword(s):  
Matrix Function ◽  
Initial Value Problems ◽  
Stability Problem ◽  
Difference Operators ◽  
Fractional Order Systems ◽  
Initial Value ◽  
Fractional Difference ◽  
Transform Method ◽  
Delta Type ◽  
The Stability

The Caputo-, Riemann-Liouville-, and Grünwald-Letnikov-type difference initial value problems for linear fractional-order systems are discussed. We take under our consideration the possible solutions via the classicalZ-transform method. We stress the formula for the image of the discrete Mittag-Leffler matrix function in theZ-transform. We also prove forms of images in theZ-transform of the expressed fractional difference summation and operators. Additionally, the stability problem of the considered systems is studied.

scholarly journals The Order Completion Method for Systems of Nonlinear PDEs: Solutions of Initial Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Jan Harm van der Walt
Keyword(s):  
Initial Value Problems ◽  
Lower Semicontinuous ◽  
Nonlinear Pdes ◽  
Generalized Solutions ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Initial Condition ◽  
Initial Value ◽  
Extended Sense ◽  
Order Completion

We present an existence result for generalized solutions of initial value problems obtained through the order completion method. The solutions we obtain satisfy the initial condition in a suitable extended sense, and each such solution may be represented in a canonical way through its generalized partial derivatives as nearly finite normal lower semicontinuous function.

scholarly journals Explicit Solutions of Initial Value Problems for Linear Scalar Riemann-Liouville Fractional Differential Equations With a Constant Delay

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 32 ◽  
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Initial Condition ◽  
Constant Delay ◽  
Initial Value ◽  
Fractional Differential ◽  
Set Up ◽  
Ordinary Derivative ◽  
Linear Scalar

In this paper, we study Linear Riemann-Liouville fractional differential equations with a constant delay. The initial condition is set up similarly to the case of ordinary derivative. Explicit formulas for the solutions are obtained for various initial functions.

scholarly journals New Algorithm for First order Stiff Initial Value Problems

2017 ◽  
Vol 58 (1) ◽  
pp. 19-28 ◽  
Author(s):  
A. O. Adesanya ◽  
R. O. Onsachi ◽  
M. R. Odekunle
Keyword(s):  
Initial Value Problems ◽  
Numerical Experiments ◽  
System Of Nonlinear Equations ◽  
Initial Value ◽  
Unknown Parameters ◽  
First Order ◽  
Continuous Scheme ◽  
Discrete Methods ◽  
Grid Points ◽  
The Stability

AbstractIn this paper, we consider the development and implementation of algorithms for the solution of stiff first order initial value problems. Method of interpolation and collocation of basis function to give system of nonlinear equations which is solved for the unknown parameters to give a continuous scheme that is evaluated at selected grid points to give discrete methods. The stability properties of the method is verified and numerical experiments show that the new method is efficient in handling stiff problems.

A Symplectic Method for Dynamic Response of Structures

2015 ◽  
Vol 724 ◽  
pp. 7-11
Author(s):  
Wei Hua Li ◽  
Hong Ying Sun
Keyword(s):  
Dynamic Response ◽  
Initial Value Problems ◽  
Lagrange Interpolation ◽  
Interpolation Method ◽  
Symplectic Method ◽  
Initial Value ◽  
Numerical Examples ◽  
Dynamic Analyses ◽  
The Stability ◽  
Stability And Accuracy

A modified Hamilton’s principle corresponding to initial value problems of dynamics is presented. Based on this principle and combined with Lagrange interpolation method, a symplectic method is constructed. The evaluations of the stability and accuracy of the proposed method are also given in this paper. With some numerical examples introduced, the proposed method is performing well and is a powerful tool for practical dynamic analyses.

scholarly journals Derivation of Diagonally Implicit Block Backward Differentiation Formulas for Solving Stiff Initial Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Iskandar Shah Mohd Zawawi ◽  
Zarina Bibi Ibrahim ◽  
Khairil Iskandar Othman
Keyword(s):  
Initial Value Problems ◽  
Maximum Error ◽  
Computational Time ◽  
Initial Value ◽  
Stiff Initial Value Problems ◽  
Fully Implicit ◽  
Backward Differentiation Formulas ◽  
The Stability ◽  
Order Four ◽  
Differentiation Formulas

The diagonally implicit 2-point block backward differentiation formulas (DI2BBDF) of order two, order three, and order four are derived for solving stiff initial value problems (IVPs). The stability properties of the derived methods are investigated. The implementation of the method using Newton iteration is also discussed. The performance of the proposed methods in terms of maximum error and computational time is compared with the fully implicit block backward differentiation formulas (FIBBDF) and fully implicit block extended backward differentiation formulas (FIBEBDF). The numerical results show that the proposed method outperformed both existing methods.

Sign in / Sign up

Export Citation Format

Share Document