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Singularities of Solutions of Singular Integral Equations
Ukrainian Mathematical Journal, 2002This paper deals with a singular integral equation \[ Sq+Tq=f,\tag{1} \] where \(q(x)\) is an unknown function, \[ Sq(x):=aq(x)+\frac{1}{\pi }\text{v.p.} \int_{-1}^{1} \frac{q(\tau)}{\tau -x} d\tau,\;Tq(x):=\int_{-1}^{1}K(x,\tau)q(\tau) d\tau. \] It is assumed that the functions \(f\) and \(K\) smoothly depend on additional parameters.
Kapustyan, V. E., Il'man, V. M.
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2003
In this chapter we discuss some recent results for Fredholm and Volterra integral equations, which deal with the existence of positive (and possibly multiple) solutions of certain classes of these equations. In Section 3.2 we provide some existence results for the nonsingular Fredholm integral equations.
Ravi P. Agarwal, Donal O’Regan
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In this chapter we discuss some recent results for Fredholm and Volterra integral equations, which deal with the existence of positive (and possibly multiple) solutions of certain classes of these equations. In Section 3.2 we provide some existence results for the nonsingular Fredholm integral equations.
Ravi P. Agarwal, Donal O’Regan
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2002
The celebrated Cauchy singular integral operator on a Jordan curve, or more precisely, its 1-periodic counterpart is perhaps the most important brick in the theory of periodic integral and pseudodifferential operators. In this chapter, we first treat the Cauchy singular operators in the Holder spaces C α (Γ) and after that we extend the results to L 2 ...
Jukka Saranen, Gennadi Vainikko
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The celebrated Cauchy singular integral operator on a Jordan curve, or more precisely, its 1-periodic counterpart is perhaps the most important brick in the theory of periodic integral and pseudodifferential operators. In this chapter, we first treat the Cauchy singular operators in the Holder spaces C α (Γ) and after that we extend the results to L 2 ...
Jukka Saranen, Gennadi Vainikko
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1995
Let the function f be defined on I=[a,b] and, possibly, be singular at an interior point c∈(a,b). Recall that the improper integral was defined by $$\int\limits_{a}^{b} {f\left( x \right)} dx: = \mathop{{\lim }}\limits_{{\mathop{{{{\varepsilon }_{1}} \to 0}}\limits_{{{{\varepsilon }_{1}} > 0}} }} \int\limits_{a}^{{c - {{\varepsilon }_{1}}}} {f\left(
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Let the function f be defined on I=[a,b] and, possibly, be singular at an interior point c∈(a,b). Recall that the improper integral was defined by $$\int\limits_{a}^{b} {f\left( x \right)} dx: = \mathop{{\lim }}\limits_{{\mathop{{{{\varepsilon }_{1}} \to 0}}\limits_{{{{\varepsilon }_{1}} > 0}} }} \int\limits_{a}^{{c - {{\varepsilon }_{1}}}} {f\left(
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Solving a Singular Integral Equation
Computational Mathematics and Modeling, 2014zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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2012
The theory introduced in previous chapters, especially the Fredholm Theory, was presented under the restrictive assumptions that the kernel was continuous on its domain of definition and that the interval of integration was finite. There is no guarantee that those results or similar ones will hold if the kernel has an infinite discontinuity or if the ...
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The theory introduced in previous chapters, especially the Fredholm Theory, was presented under the restrictive assumptions that the kernel was continuous on its domain of definition and that the interval of integration was finite. There is no guarantee that those results or similar ones will hold if the kernel has an infinite discontinuity or if the ...
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1989
In this chapter we will consider one-dimensional singular integral equations involving Cauchy principal values that arise from boundary value problems for holomorphic functions. The investigations of these integral equations with Cauchy kernels by Gakhov, Muskhelishvili, Vekua, and others have had a great impact on the further development of the ...
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In this chapter we will consider one-dimensional singular integral equations involving Cauchy principal values that arise from boundary value problems for holomorphic functions. The investigations of these integral equations with Cauchy kernels by Gakhov, Muskhelishvili, Vekua, and others have had a great impact on the further development of the ...
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Abel’s Integral Equation and Singular Integral Equations
2011Abel’s integral equation occurs in many branches of scientific fields [1], such as microscopy, seismology, radio astronomy, electron emission, atomic scattering, radar ranging, plasma diagnostics, X-ray radiography, and optical fiber evaluation. Abel’s integral equation is the earliest example of an integral equation [2].
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Product Integration for Singular Integrals and Singular Integral Equations
1979Integral equations with weakly singular kernels often have solutions which have derivative singularities at the end points of the range of integration. The error analysis of a product integration method for such integral equations depends on the error analysis of the product integration method applied to integrals of the form \(\int\limits_0^1 {g\left(
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