Abstract
We investigate the oscillation of class of time fractional partial dierential equationof the formfor (x; t) 2 R+ = G; R+ = [0;1); where is a bounded domain in RN with a piecewisesmooth boundary @ ; 2 (0; 1) is a constant, D +;t is the Riemann-Liouville fractional derivativeof order of u with respect to t and is the Laplacian operator in the Euclidean N- space RNsubject to the Neumann boundary conditionWe will obtain sucient conditions for the oscillation of class of fractional partial dierentialequations by utilizing generalized Riccatti transformation technique and the integral averagingmethod. We illustrate the main results through examples.
Highlights
Fractional differential equations, that is differential equations involving fractional order derivatives seems to be a natural description of observed evolution phenomena of several real world problems
We investigate the oscillation of class of time fractional partial differential equation of the form
The study of oscillation and other asymptotic properties of solutions of fractional order differential equations has attracted a good bit of attention in the past few years [4,5,6, 8]
Summary
Fractional differential equations, that is differential equations involving fractional order derivatives seems to be a natural description of observed evolution phenomena of several real world problems. Suppose that V (t) is non oscillatory solution of (8) Without loss of generality we may assume that V is an eventually positive solution of (8). We establish sufficient conditions for the oscillation of all solutions of (E),(B2) For this we need the following: The smallest eigen value 0 of the Dirichlet problem (x) (x) = 0 in (x) = 0 on , is positive and the corresponding eigen function (x) is positive in. Theorem 4.2 Let the conditions of Theorem 3.4 hold; every solution V of (30) is oscillatory or satisfies t lim t s V (s)ds = 0. Theorem 4.3 Let the conditions of Theorem 3.5 hold; every solution V of (30) is oscillatory or satisfies t. The proofs of Corollaries 4.1 and 4.2 and Theorems 4.2 and 4.3 are similar to that of in Section 3 and the details are omitted
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