Abstract
Averaging method of the fractional general partial differential equations and a special case of these equations are studied, without any restrictions on the characteristic forms of the partial differential operators. We use the parabolic transform, existence and stability results can be obtained.
Highlights
IntroductionConsider the following fractional partial differential equations:. |q| = q1 + ... + qn, x = (x1, ..., xn) ∈ n, n is the n−dimensional Euclidean space, 0 < t < T
Consider the following fractional partial differential equations: ∂αu(x, t)∂tα = εL(x, t, D)u(x, t), (1) u(x, 0) = φ(x), (2) where L(x, t, D) = aq(x, t)Dq, |q|≤m where
We study a special case for problem (1), (2) when α = 1:
Summary
Consider the following fractional partial differential equations:. |q| = q1 + ... + qn, x = (x1, ..., xn) ∈ n, n is the n−dimensional Euclidean space, 0 < t < T. Consider the following fractional partial differential equations:. Let Cb( n) be the set of all bounded continuous functions on n. Consider the following Cauchy problem [8]:. Where C( n) is the set of all continuous functions on n, N is a sufficiently large positive integer. For sufficiently large N, we find γ ∈ (0, 1) and a constant M > 0 such that: max |Dqu(x, t)|. N where c1 ≥ 0, c2 ≥ 0, tj, t ∈ [0, T ], j = 1, ..., r and W (y, t1, ..., tr) is a continuous bounded function on n × [0, T ]γ. Compare [1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 13]
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