Abstract
We give representations for solutions of time-fractional differential equations that involve operators on Lebesgue spaces of sequences defined by discrete convolutions involving kernels through the discrete Fourier transform. We consider finite difference operators of first and second orders, which are generators of uniformly continuous semigroups and cosine functions. We present the linear and algebraic structures (in particular, factorization properties) and their norms and spectra in the Lebesgue space of summable sequences. We identify fractional powers of these generators and apply to them the subordination principle. We also give some applications and consequences of our results.
Highlights
In this work, we study the following semidiscrete Cauchy problem: ⎧⎨∂tu(n, t) = Bu(n, t) + g(n, t), n ∈ Z, t > 0,⎩u(n, 0) = φ(n), n ∈ Z, (1.1)where B is the convolution operator in the discrete variable, that is, Bu(n, t) = b(n – j)u(j, t) (1.2)j∈Z with b belonging to the Banach algebra 1(Z)
Our key observation concerning this issue is that the discrete fractional Laplacian can be obtained from (1.2) by allowing the fractional powers of b to be an element of the Banach algebra 1(Z). This original approach, which we provide in this paper, allows us to obtain new insights by introducing a completely new method to analyze both qualitative behavior and fundamental solutions of (1.1) in a unified way
7 Applications to special functions we present some new formulae obtained as applications of the results proved in this paper
Summary
We study the following semidiscrete Cauchy problem:. where B is the convolution operator in the discrete variable, that is, Bu(n, t) = b(n – j)u(j, t). As the definition shows, we may follow a general methodology to treat fractional powers of elements in 1(Z), analogously to the case of operators in Banach spaces; see [47, p. Proof Since the algebra 1(Z) is semisimple (see Theorem 2.1), the formulae in (i) and (ii) are direct consequences of the scalar identities, which in case 0 < α < 1 can be found in [36, Sect. In [34, Theorem 3.3] and [26, Theorem 3.1] the time/space fractional evolution equations (1.4) and (1.5) of orders 0 < β ≤ 1 and 1 < β ≤ 2, respectively, are solved, where B = –(– d)α, and d is the discrete Laplacian operator Both proofs are based on the explicit expressions of vector-valued Mittag-Leffler functions Eβ,1(–tβ Kdα), Eβ,2(–tβ Kdα), and Eβ,β(–tβ Kdα).
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