Sparse representations of approximation to identity via time–space fractional heat equations

Abstract

In this paper, we investigate sparse representations of approximation to identity via time–space fractional heat equations: { ∂ t β u ( t , x ) = − ν ( − Δ ) α / 2 u ( t , x ) , ( t , x ) ∈ R + 1 + n ; u ( 0 , x ) = f ( x ) ,   x ∈ R n . Due to the time-fractional derivative, the semigroup property is invalid for the solutions u ( ⋅ , ⋅ ) to the above problem. This deficiency makes it difficult to verify the boundary vanishing condition of u ( ⋅ , ⋅ ) , which is essential for getting the sparse representations. We develop a new method to avoid using the semigroup property. The analogous results are obtained for stochastic time–space fractional heat equations. As an application, we apply the adaptive Fourier decomposition to establish sparse representations of the solutions to the concerned equations.