scholarly journals Electromagnetic Diffraction by a Plane Slit Aperture

1968 ◽  
Vol 21 (1) ◽  
pp. 35 ◽  
Author(s):  
HS Tan
Keyword(s):  
Exact Solution ◽  
Asymptotic Solution ◽  
Integral Equations ◽  
Excellent Agreement ◽  
Approximate Solutions ◽  
Plane Slit ◽  
Electromagnetic Diffraction

An asymptotic solution of the integral equations for diffraction by an E-polarized slit aperture is given. The results for the diffracted field are in excellent agreement with the exact solution and compare favourably with the results given by other approximate solutions.

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.

scholarly journals Convergence Analysis for the Homotopy Perturbation Method for a Linear System of Mixed Volterra-Fredholm Integral Equations

2020 ◽  
Vol 17 (3(Suppl.)) ◽  
pp. 1010
Author(s):  
Pakhshan M. Hasan ◽  
Nejmaddin Abdulla Sulaiman
Keyword(s):  
Exact Solution ◽  
Linear System ◽  
Perturbation Method ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Error Estimation ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Homotopy Perturbation

           In this paper, the homotopy perturbation method (HPM) is presented for treating a linear system of second-kind mixed Volterra-Fredholm integral equations. The method is based on constructing the series whose summation is the solution of the considered system. Convergence of constructed series is discussed and its proof is given; also, the error estimation is obtained. Algorithm is suggested and applied on several examples and the results are computed by using MATLAB (R2015a). To show the accuracy of the results and the effectiveness of the method, the approximate solutions of some examples are compared with the exact solution by computing the absolute errors.

scholarly journals Asymptotic solution to convolution integral equations on large and small intervals

2021 ◽  
Vol 477 (2248) ◽  
Author(s):  
Dmitry Ponomarev
Keyword(s):  
Asymptotic Solution ◽  
Integral Equations ◽  
Kernel Function ◽  
Special Functions ◽  
Homogeneous Equation ◽  
Approximate Solutions ◽  
Convolution Integral ◽  
Interval Method ◽  
Spheroidal Harmonics ◽  
Convolution Integral Equations

We consider convolution integral equations on a finite interval with a real-valued kernel of even parity, a problem equivalent to finding a Wiener–Hopf factorization of a notoriously difficult class of 2 × 2 matrices. The kernel function is assumed to be sufficiently smooth and decaying for large values of the argument. Without loss of generality, we focus on a homogeneous equation and we propose methods to construct explicit asymptotic solutions when the interval size is large and small. The large interval method is based on a reduction of the original equation to an integro-differential equation on a half-line that can be asymptotically solved in a closed form. This provides an alternative to other asymptotic techniques that rely on fast (typically exponential) decay of the kernel function at infinity, which is not assumed here. We also consider the problem on a small interval and show that finding its asymptotic solution can be reduced to solving an ODE. In particular, approximate solutions could be constructed in terms of readily available special functions (prolate spheroidal harmonics). Numerical illustrations of the obtained results are provided and further extensions of both methods are discussed.

scholarly journals Exact solution and high frequency asymptotic methods in the wedge diffraction problem

2016 ◽  
Vol 14 (38) ◽  
pp. 9-28
Author(s):  
Hernan G. Triana ◽  
Andrés Navarro Cadavid
Keyword(s):  
Numerical Simulation ◽  
Exact Solution ◽  
Asymptotic Solution ◽  
High Frequency ◽  
Asymptotic Methods ◽  
Diffraction Problem ◽  
Approximate Solutions ◽  
Point Of View ◽  
Simulation Tool ◽  
Computational Point

AbstractThe Sommerfeld exact solution for canonical 2D wedge diffraction problem with perfectly conducting surfaces is presented. From the integral formulation of the problem, the Malyuzhinets solution is obtained and this result is extended to obtain the general impedance solution of canonical 2D wedge problem. Keller’s asymptotic solution is developed and the general formulation of exact solution it’s used to obtain general asymptotic methods for approximate solutions useful from the computational point of view. A simulation tool is used to compare numerical calculations of exact and asymptotic solutions. The numerical simulation of exact solution is compared to numerical simulation of an asymptoticmethod, and a satisfactory agreement found.  Accuracy dependence with frequency is verified.

A stochastic operational matrix method for numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations

2020 ◽  
Vol 28 (3) ◽  
pp. 209-216
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Block Pulse Function

AbstractIn this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.

Approximate solutions of nonlinear two‐dimensional Volterra integral equations

2021 ◽  
Author(s):  
Sumbal Ahsan ◽  
Rashid Nawaz ◽  
Muhammad Akbar ◽  
Kottakkaran Sooppy Nisar ◽  
Dumitru Baleanu
Keyword(s):  
Integral Equations ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Two Dimensional

On the asymptotic solution of the Orr–Sommerfeld equation by the method of multiple-scales

1968 ◽  
Vol 34 (1) ◽  
pp. 145-158 ◽  
Author(s):  
K. Kuen Tam
Keyword(s):  
Exact Solution ◽  
Velocity Profile ◽  
Asymptotic Solution ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Order Equation ◽  
Zeroth Order ◽  
Independent Variables ◽  
Sommerfeld Equation ◽  
Special Case

The method of multiple-scales is used to obtain the asymptotic solution of the Orr–Sommerfeld equation. For the special case of a linear velocity profile, the solution so obtained agrees well with an approximation of the exact solution which is known. For the general case, transformations on both the dependent and independent variables are introduced to obtain a zeroth-order equation which differs from the inner equation studied so far. On the ground of the favourable comparison for the special case, the asymptotic solution constructed is expected to be uniformly valid.

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