The generalized singular tricomi equation as a convolution equation

Petrov, V. E.

doi:10.1134/s1064562406060299

Cited by 14 publications

(9 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The problem consists of choosing the path of integration, particular solution of Whittaker equation and determination of subintegral multiplier A(p) such that If we consider this equality with fixed , then after changing to variable 1/η the integral equation for function = reduces to the equa tion of convolution on an interval [13,14]. Using results from [13,14], this equation can be solved explicitly. After simple transformations and comparing the integral representation for the function f(μ) with the integral representation [11] for Whittaker function W, we find (10) It is worth reminding that η > 1, ν > 0 here.…”

Section: Problems Of Diffraction By Convexmentioning

confidence: 99%

Calculation of diffraction by strongly elongated bodies of revolution

Andronov

2012

Acoust. Phys.

View full text Add to dashboard Cite

An approach to construct approximations for the diffracted field on the surface of strongly elon gated bodies is suggested. The approach is based on high frequency asymptotic decomposition which is con structed under the supposition that transverse curvature of the surface is asymptotically large. Leading order terms of asymptotic decompositions for the fields diffracted by spheroid, one sheeted and two sheeted hyperboloids and on narrow cone are derived.

show abstract

Section: Problems Of Diffraction By Convexmentioning

confidence: 99%

Calculation of diffraction by strongly elongated bodies of revolution

Andronov

2012

Acoust. Phys.

View full text Add to dashboard Cite

show abstract

“…Now we invert the integral transform [20] with the help of (6) and get the finite difference equation…”

Section: Satisfying the Boundary Conditionsmentioning

confidence: 99%

Diffraction by an Impedance Strip at Almost Grazing Incidence

Andronov

Petrov²

2016

IEEE Trans. Antennas Propagat.

View full text Add to dashboard Cite

The high frequency asymptotic approach to diffraction by strongly elongated bodies is generalized to the case of the impedance boundary conditions. However, this is done only in the limiting case of infinitely flat elliptic cylinder. The representations for the field in the boundary-layer near the surface of the strip and for the radar cross-section in the case of small scattering angles are derived and tested.

show abstract

“…The present investigation is inspired by papers of V.E. Petrov [Pe06a,Pe06b,SP20], where the author applied the finite interval Fourier transformation…”

Section: Introductionmentioning

confidence: 99%

“…Equations (2)-( 6) have ample of applications in Mechanics and Mathematical physics and were investigated by many authors (see surveys in [Pe06a,SP20], [Du79, § 20], [Ka75] and the recent papers [AA21,AP16]). Equations (3) and (4) were solved by V. E. Petrov in [Pe06a] (also see [Pe06b,SP20]) by using the Fourier transformation, which was defined as the equivalent transformation to the classical Fourier transformation on the real axes under the diffeomorphism of J to R and its inverse:…”

Section: Introductionmentioning

confidence: 99%

“…The Fourier transformation (1) applied to equations (2)-(5) after some modifications, transforms them into multiplication of the Fourier transformed unknown function by the "symbol", the Fourier transformed Kernel of the equation. Equations ( 2)-(4) and some boundary value problem for equation (5) on the part of the unit sphere, were solved in [Pe06a,Pe06b] in the general Banach spaceless setting, while in [SP20] equation (3) was investigated in the Bessel potential space setting H s (J ), −1 s 1, which was defined as the image of the corresponding Bessel potential space on the axes H s (J ) := t * H s (R) under the isomorphism t * ϕ(x) := ϕ(t(x)) (cf. ( 8)).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Convolution equations on the Lie group (-1,1)

Duduchava¹

2022

Preprint

View full text Add to dashboard Cite

The interval J = (−1, 1) turns into a Lie group under the group operation x • y := (x + y)(1 + xy) −1 ,x, y ∈ J . This enables definition of the invariant measure dJ (x) := (1 − x 2 ) −1 dx and the Fourier transform F J on the interval J and, as a consequence, we can consider Fourier convolution operators W 0 J ,a := F J −1 aF J on J . This class of convolutions includes celebrated Prandtl, Tricomi and Lavrentjev-Bitsadze equations and, also, differential equations of arbitrary order with the natural weighted derivative D J u(x) = −(1 − x 2 )u ′ (x), t ∈ J . Equations are solved in the scale of Bessel potential H s p (J , dJ (x)), 1 p ∞, and Hölder-Zygmound Z ν (J ), 0 < µ, ν < ∞ spaces, adapted to the group J . Boundedness of convolution operators (the problem of multipliers) is discussed. The symbol a(ξ), ξ ∈ R, of a convolution equation W 0 J ,a u = f defines solvability: the equation is uniquely solvable if and only if the symbol a is elliptic. The solution is written explicitly with the help of the inverse symbol.We touch shortly the multidimensional analogue-the Lie group J n .

show abstract

The generalized singular tricomi equation as a convolution equation

Cited by 14 publications

References 1 publication

Calculation of diffraction by strongly elongated bodies of revolution

Calculation of diffraction by strongly elongated bodies of revolution

Diffraction by an Impedance Strip at Almost Grazing Incidence

Convolution equations on the Lie group (-1,1)

Contact Info

Product

Resources

About