Approximate solution of dual integral equations using Chebyshev polynomials

2015 ◽  
Vol 94 (3) ◽  
pp. 493-502 ◽  
Author(s):  
S. Ahdiaghdam ◽  
S. Shahmorad ◽  
K. Ivaz
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Chebyshev Polynomials ◽  
Dual Integral Equations

scholarly journals The Approximate Solution of Dual Integral Equations by Variational Methods

1958 ◽  
Vol 11 (2) ◽  
pp. 115-126 ◽  
Author(s):  
B. Noble
Keyword(s):  
Approximate Solution ◽  
Variational Methods ◽  
Integral Equations ◽  
Separation Of Variables ◽  
Circular Disc ◽  
Cylindrical Coordinates ◽  
Dual Integral Equations ◽  
Particular Solution ◽  
Image Position

The classic application of dual integral equations occurs in connexion with the potential of a circular disc (e.g. Titchmarsh (9), p. 334). Suppose that the disc lies in z = 0, 0≤ρ≤1, where we use cylindrical coordinates (p, z). Then it is required to find a solution ofsuch that on z = 0Separation of variables in conjunction with the conditions that ø is finite on the axis and ø tends to zero as z tends to plus infinity yields the particular solution.

scholarly journals On the Fredholm integral equation associated with pairs of dual integral equations

1969 ◽  
Vol 16 (3) ◽  
pp. 185-194 ◽  
Author(s):  
V. Hutson
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Bessel Function ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Unknown Function ◽  
Sufficient Conditions ◽  
Neumann Series ◽  
Dual Integral Equations ◽  
Image Position

Consider the Fredholm equation of the second kindwhereand Jv is the Bessel function of the first kind. Here ka(t) and h(x) are given, the unknown function is f(x), and the solution is required for large values of the real parameter a. Under reasonable conditions the solution of (1.1) is given by its Neumann series (a set of sufficient conditions on ka(t) for the convergence of this series is given in Section 4, Lemma 2). However, in many applications the convergence of the series becomes too slow as a→∞ for any useful results to be obtained from it, and it may even happen that f(x)→∞ as a→∞. It is the aim of the present investigation to consider this case, and to show how under fairly general conditions on ka(t) an approximate solution may be obtained for large a, the approximation being valid in the norm of L2(0, 1). The exact conditions on ka(t) and the main result are given in Section 4. Roughly, it is required that 1 -ka(at) should behave like tp(p>0) as t→0. For example, ka(at) might be exp ⌈-(t/ap)⌉.

Approximate Solution of a Singular Integral Cauchy-Kernel Equation of the First Kind

2010 ◽  
Vol 10 (4) ◽  
pp. 359-367 ◽  
Author(s):  
M. Kashfi ◽  
S. Shahmorad
Keyword(s):  
Approximate Solution ◽  
Numerical Solution ◽  
Integral Equations ◽  
Singular Integral ◽  
Chebyshev Polynomials ◽  
Estimation Error ◽  
Singular Integral Equations ◽  
Cauchy Kernel ◽  
Cauchy Type ◽  
Finite Segment

AbstractIn this paper we present a method for the numerical solution of Cauchy type singular integral equations of the first kind on a finite segment which is unbounded at the end points of the segment. Chebyshev polynomials of the first and second kinds are used to derive an approximate solution. Moreover, an estimation error is computed for the approximate solution.

Orthogonal Approximate Solution of Cauchy-type Singular Integral Equations

2003 ◽  
Vol 3 (2) ◽  
pp. 330-356
Author(s):  
R. Smarzewski ◽  
M. A. Sheshko
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Singular Integral ◽  
Chebyshev Polynomials ◽  
Singular Integral Equations ◽  
Approximate Solutions ◽  
Cauchy Type

AbstractChebyshev polynomials of the first and second kind are used to derive approximate solutions of the Cauchy-type singular integral equations.

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