2011
DOI: 10.2478/s13540-012-0010-7
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Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation

Abstract: In this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-fractional diffusion equation. In its turn, the maximum principle is used to show the uniqueness of solution to the initial-boundary-value problems for the time-fractional diffusion equation. The generalized solution in the sense of Vladimiro… Show more

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Cited by 183 publications
(107 citation statements)
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“…The proof of the fact that the series (37) is indeed a solution of the initialboundary-value problem (22)-(23) for the one-dimensional time-fractional diffusion equation (21) closely follows the lines of the proof presented in [13] for the case of the one-dimensional fractional diffusion equation with the Caputo fractional equation and is omitted here. Instead, let us consider an example of the initial-boundary-value problem (22)-(23) for the one-dimensional time-fractional diffusion equation (21) …”
mentioning
confidence: 87%
See 1 more Smart Citation
“…The proof of the fact that the series (37) is indeed a solution of the initialboundary-value problem (22)-(23) for the one-dimensional time-fractional diffusion equation (21) closely follows the lines of the proof presented in [13] for the case of the one-dimensional fractional diffusion equation with the Caputo fractional equation and is omitted here. Instead, let us consider an example of the initial-boundary-value problem (22)-(23) for the one-dimensional time-fractional diffusion equation (21) …”
mentioning
confidence: 87%
“…The equation (13) and the two last formulas produce now the estimate (12). Indeed, we get the following chain of equalities and inequalities…”
Section: Maximum Principles For Equations With the Riemann-liouvillementioning
confidence: 99%
“…There are various studies in literature supporting this conclusion [1][2][3][4][5][6][7][8]. By making use of Mittag-Leffler function, characteristic equations of fractional ODEs are solved and solutions of them are constructed efficiently.…”
Section: Introductionmentioning
confidence: 99%
“…Ertürk [5] used the fractional differential transform method to compute the eigenvalues of Sturm--Liouville problems of fractional order. Luchko [6] used the Fourier series to solve this problem. Neamatz et al [7] and Shi et al [8] used the method of Haar wavelet operational matrix.…”
Section: Introductionmentioning
confidence: 99%