scholarly journals Two numerical methods for Abel's integral equation with comparison

2012 ◽  
Vol Volume 15, 2012 ◽  
Author(s):  
Abdallah Ali Badr
Keyword(s):  
Integral Equation ◽  
Numerical Methods ◽  
Fractional Order ◽  
Quadrature Formula ◽  
Approximate Formula ◽  
Simple Pole ◽  
Numerical Examples ◽  
International Audience ◽  
Modified Formula

International audience Analogy between Abel's integral equation and the integral of fractional order of a given function, j^α f(t), is discussed. Two different numerical methods are presented and an approximate formula for j^α f(t) is obtained. The first approach considers the case when the function, f(t), is smooth and a quadrature formula is obtained. A modified formula is deduced in case the function has one or more simple pole. In the second approach, a procedure is presented to weaken the singularities. Both two approaches could be used to solve numerically Abel's integral equation. Some numerical examples are given to illustrate our results.

scholarly journals Fractional Order Operational Matrix Method for Solving Two-Dimensional Nonlinear Fractional Volterra Integro-Differential Equations

2021 ◽  
Vol 45 (4) ◽  
pp. 571-585
Author(s):  
AMIRAHMAD KHAJEHNASIRI ◽  
◽  
M. AFSHAR KERMANI ◽  
REZZA EZZATI ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Volterra Integral Equation ◽  
Matrix Method ◽  
Operational Matrix ◽  
Basis Functions ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Operational Matrix Method

This article presents a numerical method for solving nonlinear two-dimensional fractional Volterra integral equation. We derive the Hat basis functions operational matrix of the fractional order integration and use it to solve the two-dimensional fractional Volterra integro-differential equations. The method is described and illustrated with numerical examples. Also, we give the error analysis.

THE RESOLUTION CORRECTION IN THE SCINTILLATION SPECTROMETRY OF CONTINUOUS X RAYS

1958 ◽  
Vol 36 (12) ◽  
pp. 1624-1633 ◽  
Author(s):  
W. R. Dixon ◽  
J. H. Aitken
Keyword(s):  
Integral Equation ◽  
Numerical Analysis ◽  
Numerical Methods ◽  
Analytical Procedure ◽  
Pulse Height ◽  
Matrix Inversion ◽  
Height Distribution ◽  
X Rays ◽  
Smooth Solutions ◽  
Pulse Height Distribution

The problem of making resolution corrections in the scintillation spectrometry of continuous X rays is discussed. Analytical solutions are given to the integral equation which describes the effect of the statistical spread in pulse height. The practical necessity of making some kind of numerical analysis is pointed out. Difficulties with numerical methods arise from the fact that the observed pulse-height distribution cannot be defined precisely. As a result it is possible in practice only to find smooth "solutions". Additional difficulties arise if the numerical method is based on an invalid analytical procedure. For example matrix inversion is of doubtful value in making the resolution correction because there does not appear to be an inverse kernel for the integral equation in question.

scholarly journals Hybrid Bernstein Block-Pulse Functions Method for Second Kind Integral Equations with Convergence Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu ◽  
Fereshteh Babaei
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Convergence Analysis ◽  
Bernstein Polynomials ◽  
Linear Equations ◽  
The Other ◽  
New Combination ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Block Pulse Functions

We introduce a new combination of Bernstein polynomials (BPs) and Block-Pulse functions (BPFs) on the interval [0, 1]. These functions are suitable for finding an approximate solution of the second kind integral equation. We call this method Hybrid Bernstein Block-Pulse Functions Method (HBBPFM). This method is very simple such that an integral equation is reduced to a system of linear equations. On the other hand, convergence analysis for this method is discussed. The method is computationally very simple and attractive so that numerical examples illustrate the efficiency and accuracy of this method.

On Robust Control of Fractional Order Plants: Invariant Phase Margin

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Mohammad Hossein Basiri ◽  
Mohammad Saleh Tavazoei
Keyword(s):  
Robust Control ◽  
Fractional Order ◽  
Phase Margin ◽  
Crossover Frequency ◽  
Large Uncertainty ◽  
Robust Controller ◽  
Numerical Examples

Recently, a robust controller has been proposed to be used in control of plants with large uncertainty in location of one of their poles. By using this controller, not only the phase margin and gain crossover frequency are adjustable for the nominal case but also the phase margin remains constant, notwithstanding the variations in location of the uncertain pole of the plant. In this paper, the tuning rule of the aforementioned controller is extended such that it can be applied in control of plants modeled by fractional order models. Numerical examples are provided to show the effectiveness of the tuned controller.

Legendre Spectral Collocation Technique for Advection Dispersion Equations Included Riesz Fractional

2021 ◽  
Vol 6 (1) ◽  
pp. 9
Author(s):  
Mohamed M. Al-Shomrani ◽  
Mohamed A. Abdelkawy
Keyword(s):  
Numerical Methods ◽  
Numerical Algorithm ◽  
Spectral Collocation ◽  
Theoretical Formulation ◽  
Numerical Examples ◽  
Dispersion Equations ◽  
Advection Dispersion ◽  
Collocation Technique ◽  
Collocation Approach

The advection–dispersion equations have gotten a lot of theoretical attention. The difficulty in dealing with these problems stems from the fact that there is no perfect answer and that tackling them using local numerical methods is tough. The Riesz fractional advection–dispersion equations are quantitatively studied in this research. The numerical methodology is based on the collocation approach and a simple numerical algorithm. To show the technique’s performance and competency, a comprehensive theoretical formulation is provided, along with numerical examples.

scholarly journals Solving Abel’s Type Integral Equation with Mikusinski’s Operator of Fractional Order

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Ming Li ◽  
Wei Zhao
Keyword(s):  
Integral Equation ◽  
Fractional Order ◽  
Operational Calculus ◽  
Point Of View ◽  
Alternative Method ◽  
Type Integral

This paper gives a novel explanation of the integral equation of Abel’s type from the point of view of Mikusinski’s operational calculus. The concept of the inverse of Mikusinski’s operator of fractional order is introduced for constructing a representation of the solution to the integral equation of Abel’s type. The proof of the existence of the inverse of the fractional Mikusinski operator is presented, providing an alternative method of treating the integral equation of Abel’s type.

scholarly journals Numerical methods for a Volterra integral equation with non-smooth solutions

2006 ◽  
Vol 189 (1-2) ◽  
pp. 412-423 ◽  
Author(s):  
Teresa Diogo ◽  
Neville J. Ford ◽  
Pedro Lima ◽  
Svilen Valtchev
Keyword(s):  
Integral Equation ◽  
Numerical Methods ◽  
Volterra Integral Equation ◽  
Smooth Solutions

scholarly journals On the Interior Stress Problem for Elastic Bodies

2000 ◽  
Vol 67 (4) ◽  
pp. 658-662 ◽  
Author(s):  
J. Helsing
Keyword(s):  
Integral Equation ◽  
Computational Domain ◽  
Original Formulation ◽  
Unknown Quantity ◽  
Numerical Examples ◽  
Stress Problem ◽  
Elastic Bodies ◽  
Interior Stress

The classic Sherman-Lauricella integral equation and an integral equation due to Muskhelishvili for the interior stress problem are modified. The modified formulations differ from the classic ones in several respects: Both modifications are based on uniqueness conditions with clear physical interpretations and, more importantly, they do not require the arbitrary placement of a point inside the computational domain. Furthermore, in the modified Muskhelishvili equation the unknown quantity, which is solved for, is simply related to the stress. In Muskhelishvili’s original formulation the unknown quantity is related to the displacement. Numerical examples demonstrate the greater stability of the modified schemes. [S0021-8936(00)01304-0]

scholarly journals Controllability of Flow-Conservation Transportation Networks with Fractional-Order Dynamics

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Wei Chen ◽  
Bo Zhou
Keyword(s):  
Theoretical Analysis ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Location Problem ◽  
Transportation Networks ◽  
Qr Decomposition ◽  
Rank Condition ◽  
Numerical Examples ◽  
Flow Conservation

In this paper, we adapt the fractional derivative approach to formulate the flow-conservation transportation networks, which consider the propagation dynamics and the users’ behaviors in terms of route choices. We then investigate the controllability of the fractional-order transportation networks by employing the Popov-Belevitch-Hautus rank condition and the QR decomposition algorithm. Furthermore, we provide the exact solutions for the full controllability pricing controller location problem, which includes where to locate the controllers and how many controllers are required at the location positions. Finally, we illustrate two numerical examples to validate the theoretical analysis.

scholarly journals Optimal quadrature formula in the sense of Sard in K2(P3) space

2014 ◽  
Vol 95 (109) ◽  
pp. 29-47 ◽  
Author(s):  
Abdullo Hayotov ◽  
Gradimir Milovanovic ◽  
Kholmat Shadimetov
Keyword(s):  
Hilbert Space ◽  
Sobolev Space ◽  
Quadrature Formula ◽  
Asymptotic Optimality ◽  
Trigonometric Functions ◽  
Numerical Examples ◽  
Optimal Quadrature Formula ◽  
Optimal Quadrature ◽  
Optimal Coefficients ◽  
Optimal Formula

We construct an optimal quadrature formula in the sense of Sard in the Hilbert space K2(P3). Using Sobolev?s method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order of the convergence of the optimal formula and prove an asymptotic optimality of such a formula in the Sobolev space L (3)2 (0, 1). The obtained optimal quadrature formula is exact for the trigonometric functions sin x, cos x and for constants. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.

Sign in / Sign up

Export Citation Format

Share Document