Generalized Heat Equation Under Conform Derivative
DOI:
https://doi.org/10.5644/SJM.15.02.10Keywords:
Conform derivative, Colombeau algebra, generalized conformable semigroup, generalized solutionAbstract
In the present work, we establish the existence and uniqueness result of the linear heat equation with Conform derivative in Colombeau generalized algebra. We using for the first time the notion of a generalized conformable semigroup and the purpose of introducing Conform derivative is regularizing it in Colombeau.
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References
T.Abdeljawad, On Conformable Fractional Calculus}. Journal of Computational and Applied Mathematics, Vol. 279, 57--66, (2015).
A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative. Open Mathematics, Open Math. 13 889--898, (2015).
A. V. Balakrishnan, Fractional Powers Of Closed Operators And The Semigroups Generated By Them, Pacific Journal of Mathematics ,10, pp. 419-439, 1960.
H.A. Biagioni, A nonlinear theory of generalized functions, Lecture Notes in Mathematics, vol. 1421, Springer, Berlin, 1990.
J. F. Colombeau, New generalized functions and Multiplication of distributions, North-Holland, Amsterdam, (1984).
J. F. Colombeau and A. Y. L. Roux, Multiplications of distributions in elasticity and hydrodynamics, J. Math. Phys., 29 (1988), 315--319.
J. F. Colombeau, Elementary Introduction to New Generalized Functions, North-Holland Math. Studies Vol. 113, North-Holland, Amsterdam (1985).
I. M. Gel'fand and G. E. Shilov, Generalized functions, Academic press, New York, Vol. I, (1964).
M. Horani, R. Khalil, T. Abdeljawad, Conformable Fractional Semigroups of Operators. arXiv:1502.06014v1 [math.FA] 21 Nov (2014).
G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results, Computers and Mathematics with Applications, vol. 51, no. 9--10, pp. 1367--1376, (2006).
R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new Definition of Fractional Derivative}, J. Comput. Appl. Math. 264. pp. 65--0, (2014).
A. A. Kilbas, H. M. Srivastava, JJ. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006)
S.Mirjana, Nonlinear Schrödinger equation with singular potential and initial data, Nonlinear Analysis 64 (2006) 1460--1474.
A. Pazy (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York Inc.
D. Rajter-Ciric, Fractional derivatives of Colombeau Generalized stochastic processes defined on $R^+$}, Appl. Anal. Discrete Math. 5 (2011), 283--297.
M. Stojanović, Extension of Colombeau algebra to derivatives of arbitrary order$ mathcal{D}^{alpha}, :alphainR_{+}cup{0}$. Application to ODEs and PDEs with entire and fractional derivatives, Nonlinear Analysis 5 (2009), 5458--5475.
