Exact Solution of a Family of Integral Equations of Anisotropic Scattering

1970 ◽  
Vol 11 (5) ◽  
pp. 1575-1578 ◽  
Author(s):  
H. Kagiwada ◽  
R. Kalaba
Keyword(s):  
Exact Solution ◽  
Integral Equations ◽  
Anisotropic Scattering

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.

Radiative Heat Transfer in Multidimensional Emitting, Absorbing, and Anisotropic Scattering Media—Mathematical Formulation and Numerical Method

1989 ◽  
Vol 111 (1) ◽  
pp. 141-147 ◽  
Author(s):  
Zhiqiang Tan
Keyword(s):  
Heat Transfer ◽  
Integral Equations ◽  
Radiative Heat Transfer ◽  
Radiative Heat ◽  
Numerical Technique ◽  
Mathematical Formulation ◽  
Internal Heat ◽  
Anisotropic Scattering ◽  
Scattering Media ◽  
Zonal Method

Thermal radiative transmission in multidimensional emitting, absorbing, and anisotropic scattering media is studied in this paper. In the first part, starting from basic formulae of radiative heat transfer, a set of integral equations for the problem is derived. Then the product-integration method is applied to discretize the integral equations. This method, while analogous to Hottel’s zonal method or Razzaque’s finite element method, requires evaluation of only three or two-dimensional integrals for three-dimensional systems. Finally the formulation and the numerical technique are applied to the problems of thermal radiation in emitting, absorbing, and linearly anisotropic scattering planar and square media with gray surfaces and with or without internal heat generations. Computed results are discussed and compared with available data.

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.

Bilateral set-valued stochastic integral equations

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3253-3274
Author(s):  
Marek Malinowski ◽  
Donal O'Regan
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Integral Equations ◽  
Existence And Uniqueness ◽  
Stochastic Integral ◽  
Approximate Solutions ◽  
Existence And Uniqueness Theorem ◽  
Initial State ◽  
Stochastic Integral Equations ◽  
Stability Of The Solution

We investigate bilateral set-valued stochastic integral equations and these equations combine widening and narrrowing set-valued stochastic integral equations studied in literature. An existence and uniqueness theorem is established using approximate solutions. In addition stability of the solution with respect to small changes of the initial state and coefficients is established, also we provide a result on boundedness of the solution, and an estimate on a distance between the exact solution and the approximate solution is given. Finally some implications for deterministic set-valued integral equations are presented.

On Solution of Singular Integral Equations by Operator Method

2019 ◽  
Vol 14 ◽  
pp. 41-48
Author(s):  
Najem A. Mohammad ◽  
Mohammad Shami Hasso
Keyword(s):  
Exact Solution ◽  
Integral Equations ◽  
Singular Integral ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Operator Method ◽  
Singular Integral Equations ◽  
Transform Method ◽  
Suggested Technique ◽  
Adomian Decomposition

In this paper, we study the exact solution of singular integral equations using two methods, including Adomian Decomposition Method and Elzaki Transform Method. We propose an analytical method for solving singular integral equations and system of singular integral equations, and have some goals in our paper related to suggested technique for solving singular integral equations. The primary goal is for giving analytical solutions of such equations with simple steps, another goal is to compare the suggested method with other methods used in this study.

Slow viscous flow due to motion of an annular disk; pressure-driven extrusion through an annular hole in a wall

1991 ◽  
Vol 231 ◽  
pp. 51-71 ◽  
Author(s):  
A. M. J. Davis
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Viscous Flow ◽  
Integral Equations ◽  
Bessel Functions ◽  
Annular Disk ◽  
Slow Viscous Flow ◽  
Triple Integral Equations ◽  
Flow Through ◽  
Pressure Driven

The description of the slow viscous flow due to the axisymmetric or asymmetric translation of an annular disk involves the solution of respectively one or two sets of triple integral equations involving Bessel functions. An efficient method is presented for transforming each set into a Fredholm integral equation of the second kind. Simple, regular kernels are obtained and the required physical constants are readily available. The method is also applied to the pressure-driven extrusion flow through an annular hole in a wall. The velocity profiles in the holes are found to be flatter than expected with correspondingly sharper variation near a rim. For the sideways motion of a disk, an exact solution is given with bounded velocities and both components of the rim pressure singularity minimized. The additional drag experienced by this disk when the fluid is bounded by walls parallel to the motion is then determined by solving a pair of integral equations, according to methods given in an earlier paper.

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