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Analysis on α-time scales and its applications to Cauchy-Euler equation |
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PP: 1051-1074 |
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doi:10.18576/amis/180512
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Author(s) |
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Burcu Silindir,
Sec ̧il Gergu ̈n,
Ahmet Yantir,
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Abstract |
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| This article is devoted to present the α-power function, calculus on α-time scale, the α-logarithm and their applications on α-difference equations. We introduce the α-power function as an absolutely convergent infinite product. We state that the α-power function verifies the fundamentals of α-time scale and adheres to both the additivity and the power rule for α-derivative. Next, we propose an α-analogue of Cauchy-Euler equation whose coefficient functions are α-polynomials and then construct its solution in terms of α-power function. As illustration, we present examples of the second order α-Cauchy-Euler equation. Consequently, we construct α-analogue of logarithm function which is determined in terms of α-integral. Finally, we propose a second order BVP for α-Cauchy-Euler equation with two point unmixed boundary conditions and compute its solution by the use of Green’s function.
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