Joshua Beaudin - Academia.edu (original) (raw)
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Papers by Joshua Beaudin
Fractal and Fractional, 2021
The goal of this paper is to study the uniqueness of solutions of several nonlinear Liouville–Cap... more The goal of this paper is to study the uniqueness of solutions of several nonlinear Liouville–Caputo integro-differential equations with variable coefficients and initial conditions, as well as an associated coupled system in Banach spaces. The results derived are new and based on Banach’s contractive principle, the multivariate Mittag–Leffler function and Babenko’s approach. We also provide a few examples to demonstrate the use of our main theorems by convolutions and the gamma function.
Mathematics
The goal of this paper is to construct an integral representation for the generalized Riesz deriv... more The goal of this paper is to construct an integral representation for the generalized Riesz derivative R Z D x 2 s u ( x ) for k < s < k + 1 with k = 0 , 1 , ⋯ , which is proved to be a one-to-one and linearly continuous mapping from the normed space W k + 1 ( R ) to the Banach space C ( R ) . In addition, we show that R Z D x 2 s u ( x ) is continuous at the end points and well defined for s = 1 2 + k . Furthermore, we extend the generalized Riesz derivative R Z D x 2 s u ( x ) to the space C k ( R n ) , where k is an n-tuple of nonnegative integers, based on the normalization of distribution and surface integrals over the unit sphere. Finally, several examples are presented to demonstrate computations for obtaining the generalized Riesz derivatives.
Mathematics
The goal of this paper is to construct an integral representation for the generalized Riesz deriv... more The goal of this paper is to construct an integral representation for the generalized Riesz derivative R Z D x 2 s u ( x ) for k < s < k + 1 with k = 0 , 1 , ⋯ , which is proved to be a one-to-one and linearly continuous mapping from the normed space W k + 1 ( R ) to the Banach space C ( R ) . In addition, we show that R Z D x 2 s u ( x ) is continuous at the end points and well defined for s = 1 2 + k . Furthermore, we extend the generalized Riesz derivative R Z D x 2 s u ( x ) to the space C k ( R n ) , where k is an n-tuple of nonnegative integers, based on the normalization of distribution and surface integrals over the unit sphere. Finally, several examples are presented to demonstrate computations for obtaining the generalized Riesz derivatives.
Fractal and Fractional, 2021
The goal of this paper is to study the uniqueness of solutions of several nonlinear Liouville–Cap... more The goal of this paper is to study the uniqueness of solutions of several nonlinear Liouville–Caputo integro-differential equations with variable coefficients and initial conditions, as well as an associated coupled system in Banach spaces. The results derived are new and based on Banach’s contractive principle, the multivariate Mittag–Leffler function and Babenko’s approach. We also provide a few examples to demonstrate the use of our main theorems by convolutions and the gamma function.
Mathematics
The goal of this paper is to construct an integral representation for the generalized Riesz deriv... more The goal of this paper is to construct an integral representation for the generalized Riesz derivative R Z D x 2 s u ( x ) for k < s < k + 1 with k = 0 , 1 , ⋯ , which is proved to be a one-to-one and linearly continuous mapping from the normed space W k + 1 ( R ) to the Banach space C ( R ) . In addition, we show that R Z D x 2 s u ( x ) is continuous at the end points and well defined for s = 1 2 + k . Furthermore, we extend the generalized Riesz derivative R Z D x 2 s u ( x ) to the space C k ( R n ) , where k is an n-tuple of nonnegative integers, based on the normalization of distribution and surface integrals over the unit sphere. Finally, several examples are presented to demonstrate computations for obtaining the generalized Riesz derivatives.
Mathematics
The goal of this paper is to construct an integral representation for the generalized Riesz deriv... more The goal of this paper is to construct an integral representation for the generalized Riesz derivative R Z D x 2 s u ( x ) for k < s < k + 1 with k = 0 , 1 , ⋯ , which is proved to be a one-to-one and linearly continuous mapping from the normed space W k + 1 ( R ) to the Banach space C ( R ) . In addition, we show that R Z D x 2 s u ( x ) is continuous at the end points and well defined for s = 1 2 + k . Furthermore, we extend the generalized Riesz derivative R Z D x 2 s u ( x ) to the space C k ( R n ) , where k is an n-tuple of nonnegative integers, based on the normalization of distribution and surface integrals over the unit sphere. Finally, several examples are presented to demonstrate computations for obtaining the generalized Riesz derivatives.