Abstract
This paper deals with the extension of the H-R transform (the fractional integral) into two dimensional integral. Some of the properties of this integral are developed and discussed. Also, some of its applications to partial differential equations of hyperbolic type have been introduced.
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References
M. A. Al-Bassam, "Some properties of Holmgren-Riesz transform," Ann. della Scuola Norm. Sup. di Pisa, Series III, Vol. XV, I–II (1961), 1–24.
_____, "Some properties of Holmgren-Riesz transform in two dimensions," Ann. della Scuola Norm. Sup. di Pisa, III, Vol. XVI, I (1962), 75–90.
_____, "Some existence theorems on differential equations of generalized order," Journal für die reine und angewandte Mathematik, Vol. 218 (1965), 70–78.
The integral on the right hand side may be written in the form \(- \int_v^u {(u - z)^{\alpha + n - 1} \left[ {D_v^m \int_v^z {(v - t)^{\beta + m - 1} F(z,t)dt} } \right]dz} \).
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© 1975 Springer-Verlag
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Al-Bassam, M.A. (1975). H-R transform in two dimensions and some of its applications to partial differential equations. In: Ross, B. (eds) Fractional Calculus and Its Applications. Lecture Notes in Mathematics, vol 457. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0067099
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DOI: https://doi.org/10.1007/BFb0067099
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