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On a Class of Fractional Type Integral Equations in Variable Exponent Spaces

Rafeiro, Humberto, Samko, Stefan (2007)

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: 45A05, 45B05, 45E05,45P05, 46E30We obtain a criterion of Fredholmness and formula for the Fredholm index of a certain class of one-dimensional integral operators M with a weak singularity in the kernel, from the variable exponent Lebesgue space L^p(·) ([a, b], ?) to the Sobolev type space L^α,p(·) ([a, b], ?) of fractional smoothness. We also give formulas of closed form solutions ϕ ∈ L^p(·) of the 1st kind integral equation M0ϕ = f, known as the generalized...

On a generalized Wiener-Hopf integral equation

Malcolm T. McGregor (1997)

Archivum Mathematicum

Let α be such that 0 < α < 1 2 . In this note we use the Mittag-Leffler partial fractions expansion for F α ( θ ) = Γ 1 - α - θ π Γ ( α ) / Γ α - θ π Γ ( 1 - α ) to obtain a solution of a Wiener-Hopf integral equation.

On the efficient use of the Galerkin-method to solve Fredholm integral equations

Wolfgang Hackbusch, Stefan A. Sauter (1993)

Applications of Mathematics

In the present paper we describe, how to use the Galerkin-method efficiently in solving boundary integral equations. In the first part we show how the elements of the system matrix can be computed in a reasonable time by using suitable coordinate transformations. These techniques can be applied to a wide class of integral equations (including hypersingular kernels) on piecewise smooth surfaces in 3-D, approximated by spline functions of arbitrary degree. In the second part we show, how to use the...

On the Fourier cosine—Kontorovich-Lebedev generalized convolution transforms

Nguyen Thanh Hong, Trinh Tuan, Nguyen Xuan Thao (2013)

Applications of Mathematics

We deal with several classes of integral transformations of the form f ( x ) → D ∫ ℝ + 2 1 u ( e - u cosh ( x + v ) + e - u cosh ( x - v ) ) h ( u ) f ( v ) d u d v , where D is an operator. In case D is the identity operator, we obtain several operator properties on L p ( ℝ + ) with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on L 2 ( ℝ + ) and define the inversion formula. Further, for an other class of differential operators of finite...

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