scholarly journals New Solutions for System of Fractional Integro-Differential Equations and Abel’s Integral Equations by Chebyshev Spectral Method

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Hassan A. Zedan ◽  
Seham Sh. Tantawy ◽  
Yara M. Sayed
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Integral Equations ◽  
Spectral Method ◽  
Operational Matrix ◽  
Numerical Results ◽  
Efficient Technique ◽  
Test Problems ◽  
Chebyshev Spectral Method

Chebyshev spectral method based on operational matrix is applied to both systems of fractional integro-differential equations and Abel’s integral equations. Some test problems, for which the exact solution is known, are considered. Numerical results with comparisons are made to confirm the reliability of the method. Chebyshev spectral method may be considered as alternative and efficient technique for finding the approximation of system of fractional integro-differential equations and Abel’s integral equations.

scholarly journals On theRF-pair operations for the exact solution of some classes of nonlinear Volterra integral equations

2006 ◽  
Vol 2006 ◽  
pp. 1-11 ◽  
Author(s):  
P. Darania ◽  
M. Hadizadeh
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Original Equation ◽  
Nonlinear Integral Equations ◽  
Integrable Equations

We study the exact solution of some classes of nonlinear integral equations by series of some invertible transformations andRF-pair operations. We show that this method applies to several classes of nonlinear Volterra integral equations as well and give some useful invertible transformations for converting these equations into differential equations of Emden-Fowler type. As a consequence, we analyze the effect of the proposed operations on the exact solution of the transformed equation in order to find the exact solution of the original equation. Some applications of the method are also given. This approach is effective to find a great number of new integrable equations, which thus far, could not be integrated using the classical methods.

scholarly journals A New Numerical Technique for Solving Volterra Integral Equations Using Chebyshev Spectral Method

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Ahmed A. Khidir
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Integral Equations ◽  
Numerical Technique ◽  
Volterra Integral Equations ◽  
New Method ◽  
Algebraic Equations ◽  
Chebyshev Spectral Collocation ◽  
Chebyshev Spectral Method ◽  
Chebyshev Spectral Collocation Method

In this work, we propose a new method for solving Volterra integral equations. The technique is based on the Chebyshev spectral collocation method. The application of the proposed method leads Volterra integral equation to a system of algebraic equations that are easy to solve. Some examples are presented and compared with some methods in the literature to illustrate the ability of this technique. The results demonstrate that the new method is more efficient, convergent, and accurate to the exact solution.

Numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types)

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Jiao Wang
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Numerical Algorithm ◽  
Numerical Solutions ◽  
Integral Transformation ◽  
Operational Matrix ◽  
Fredholm Integral Equations ◽  
Two Dimensional ◽  
Content Type ◽  
Mixed Types

Purpose This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types). Design/methodology/approach The main idea of the presented algorithm is to combine Bernoulli polynomials approximation with Caputo fractional derivative and numerical integral transformation to reduce the studied two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations to easily solved algebraic equations. Findings Without considering the integral operational matrix, this algorithm will adopt straightforward discrete data integral transformation, which can do good work to less computation and high precision. Besides, combining the convenient fractional differential operator of Bernoulli basis polynomials with the least-squares method, numerical solutions of the studied equations can be obtained quickly. Illustrative examples are given to show that the proposed technique has better precision than other numerical methods. Originality/value The proposed algorithm is efficient for the considered two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations. As its convenience, the computation of numerical solutions is time-saving and more accurate.

scholarly journals INTEGRAL EQUATIONS AND ACTUARIAL RISK MANAGEMENT: SOME MODELS AND NUMERICS

2003 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
A. Makroglou
Keyword(s):  
Risk Management ◽  
Differential Equations ◽  
Integral Equations ◽  
Computational Methods ◽  
Finite Time ◽  
Numerical Results ◽  
Insurance Company ◽  
Type Approximation ◽  
Actuarial Risk ◽  
Ruin Problem

The problem of the estimation of the probability R(z, t) (here t is time, z is initial reserve) of the finite time non‐ruin problem for a risk business such as an insurance company is considered, with respect to presenting models that have been used in the literature in the form of integral / integro-differential equations, reviewing some analytical and computational methods used for their solution, presenting numerical results obtained with one method (a global Lagrange type approximation in the z—space).

scholarly journals Haar Wavelet Method for the System of Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Hassan A. Zedan ◽  
Eman Alaidarous
Keyword(s):  
Exact Solution ◽  
Numerical Solution ◽  
Integral Equations ◽  
Haar Wavelet ◽  
Fredholm Integral Equations ◽  
Test Problems ◽  
Wavelet Method ◽  
System Of Integral Equations ◽  
Haar Wavelet Method ◽  
Comparison Of The Results

We employed the Haar wavelet method to find numerical solution of the system of Fredholm integral equations (SFIEs) and the system of Volterra integral equations (SVIEs). Five test problems, for which the exact solution is known, are considered. Comparison of the results is obtained by the Haar wavelet method with the exact solution.

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