A Quasi Trefftz-Type Spectral Method for Initial Value Problems with Moving Boundaries

1997 ◽  
Vol 07 (03) ◽  
pp. 385-404 ◽  
Author(s):  
S. A. Lifits ◽  
S. Yu. Reutskiy ◽  
B. Tirozzi
Keyword(s):  
Spectral Method ◽  
Initial Value Problems ◽  
Analytic Solution ◽  
Complex Geometry ◽  
High Accuracy ◽  
Moving Boundaries ◽  
Initial Value ◽  
Stefan Problems ◽  
A New Technique ◽  
Boundary Methods

A quasi Trefftz-type spectral method is a new technique for numerical solving boundary value and initial value problems in domains with complex geometry. QTSM combines the properties of the boundary methods with the spectral approach. In the present paper QTSM is applied to the problems with moving boundaries. The problems with a known boundary motion law as well as the Stefan problems are considered. The method is tested on several one- and two-dimensional problems with exact analytic solution. A high accuracy of the calculations is achieved.

scholarly journals On Effects of a New Method for Fractional Initial Value Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Hülya Kodal Sevindir ◽  
Süleyman Çetinkaya ◽  
Ali Demir
Keyword(s):  
Iterative Method ◽  
Fractional Derivative ◽  
Infinite Series ◽  
Initial Value Problems ◽  
Analytic Solution ◽  
Initial Value ◽  
Transform Method ◽  
Series Form ◽  
Time Fractional Derivative ◽  
A New Technique

The aim of this study is to analyze nonlinear Liouville-Caputo time-fractional problems by a new technique which is a combination of the iterative and ARA transform methods and is denoted by IAM. First, the ARA transform method and its inverse are utilized to get rid of time fractional derivative. Later, the iterative method is applied to establish the solution of the problem in infinite series form. The main advantages of this method are that it converges to analytic solution of the problem rapidly and implementation of method is easy. Finally, outcomes of the illustrative examples prove the efficiency and accuracy of the method.

Quasi Trefftz Spectral Method for Stokes Problem

1997 ◽  
Vol 07 (08) ◽  
pp. 1187-1212 ◽  
Author(s):  
S. A. Lifits ◽  
S. Yu. Reutskiy ◽  
G. Pontrelli ◽  
B. Tirozzi
Keyword(s):  
Free Boundary ◽  
Spectral Method ◽  
Initial Value Problems ◽  
Moving Boundary ◽  
Stokes Problem ◽  
Boundary Value ◽  
Elliptic Operators ◽  
Initial Value ◽  
Primitive Variables

A new numerical Quasi Trefftz Spectral Method (QTSM) which was earlier suggested for solving boundary value and initial value problems with elliptic operators is applied to linear stationary hydrodynamic problems. The primitive variables [Formula: see text] are used. The method has been found to work well for different problems, including free boundary ones. The problem of the Stefan type in the domain with moving boundary is also considered. The possibilities of further developments of QTSM are discussed.

scholarly journals A Legendre Wavelet Spectral Collocation Method for Solving Oscillatory Initial Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
A. Karimi Dizicheh ◽  
F. Ismail ◽  
M. Tavassoli Kajani ◽  
Mohammad Maleki
Keyword(s):  
Differential Equation ◽  
Spectral Method ◽  
Initial Value Problems ◽  
Algebraic Equations ◽  
Initial Value ◽  
Numerical Examples ◽  
Runge Kutta Method ◽  
Collocation Points ◽  
Oscillatory Initial Value Problems ◽  
Exponentially Fitted

In this paper, we propose an iterative spectral method for solving differential equations with initial values on large intervals. In the proposed method, we first extend the Legendre wavelet suitable for large intervals, and then the Legendre-Guass collocation points of the Legendre wavelet are derived. Using this strategy, the iterative spectral method converts the differential equation to a set of algebraic equations. Solving these algebraic equations yields an approximate solution for the differential equation. The proposed method is illustrated by some numerical examples, and the result is compared with the exponentially fitted Runge-Kutta method. Our proposed method is simple and highly accurate.

scholarly journals Lucas polynomials semi-analytic solution for fractional multi-term initial value problems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Mahmoud M. Mokhtar ◽  
Amany S. Mohamed
Keyword(s):  
Initial Value Problems ◽  
Analytic Solution ◽  
Collocation Methods ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Differentiation ◽  
Initial Value ◽  
Lucas Polynomials ◽  
Error Analyses ◽  
Generalized Lucas Polynomials

AbstractHerein, we use the generalized Lucas polynomials to find an approximate numerical solution for fractional initial value problems (FIVPs). The method depends on the operational matrices for fractional differentiation and integration of generalized Lucas polynomials in the Caputo sense. We obtain these solutions using tau and collocation methods. We apply these methods by transforming the FIVP into systems of algebraic equations. The convergence and error analyses are discussed in detail. The applicability and efficiency of the method are tested and verified through numerical examples.

scholarly journals A new stepsize change technique for Adams methods

2016 ◽  
Vol 1 (2) ◽  
pp. 547-558 ◽  
Author(s):  
M. Calvo ◽  
J.I. Montijano ◽  
L. Rández
Keyword(s):  
Numerical Solution ◽  
Initial Value Problems ◽  
Error Term ◽  
Computational Cost ◽  
New Technique ◽  
Initial Value ◽  
Change Technique ◽  
Adams Methods ◽  
Interpolation Technique ◽  
A New Technique

AbstractIn this paper a new technique for stepsize changing in the numerical solution of Initial Value Problems for ODEs by means of Adams type methods is considered. The computational cost of the new technique is equivalent to those of the well known interpolation technique (IT). It is seen that the new technique has better stability properties than the IT and moreover, its leading error term is smaller. These facts imply that the new technique can outperform the IT.

scholarly journals New Analytic Solution to the Lane-Emden Equation of Index 2

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
S. S. Motsa ◽  
S. Shateyi
Keyword(s):  
Analytical Methods ◽  
Initial Value Problems ◽  
Analytic Solution ◽  
Mathematical Physics ◽  
Numerical Results ◽  
Initial Value ◽  
Emden Equation ◽  
New Methods ◽  
Index 2 ◽  
Good Agreement

We present two new analytic methods that are used for solving initial value problems that model polytropic and stellar structures in astrophysics and mathematical physics. The applicability, effectiveness, and reliability of the methods are assessed on the Lane-Emden equation which is described by a second-order nonlinear differential equation. The results obtained in this work are also compared with numerical results of Horedt (1986) which are widely used as a benchmark for testing new methods of solution. Good agreement is observed between the present results and the numerical results. Comparison is also made between the proposed new methods and existing analytical methods and it is found that the new methods are more efficient and have several advantages over some of the existing analytical methods.

scholarly journals Chebyshev neural network model with linear and nonlinear active functions

2016 ◽  
Vol 5 (3) ◽  
pp. 182
Author(s):  
Sarkhosh Seddighi Chaharborj ◽  
Yaghoub Mahmoudi
Keyword(s):  
Neural Network ◽  
Neural Network Model ◽  
Initial Value Problems ◽  
High Accuracy ◽  
Numerical Results ◽  
Initial Value ◽  
Linear Ordinary Differential Equations ◽  
Chebyshev Neural Network ◽  
Active Function ◽  
Linear And Nonlinear

In this paper the second order non-linear ordinary differential equations of Lane-Emden type as singular initial value problems using Chebyshev Neural Network (ChNN) with linear and nonlinear active functions has been studied. Active functions as, \(\texttt{F(z)=z}, \texttt{sinh(x)}, \texttt{tanh(z)}\) are considered to find the numerical results with high accuracy. Numerical results from Chebyshev Neural Network shows that linear active function has more accuracy and is more convenient compare to other functions.

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