Non-local initial problem for second order time-fractional and space-singular equation

On a Non–Local Problem for a Multi–Term Fractional Diffusion-Wave Equation

“…Investigation of different initial-value problems for the equation (1.2) was performed in [Luc11, JLTB12, LLY15, Liu17, KMR18]. The recently published paper on these topics is [AKMR17,KMR18]. In [KMR18] the authors dealt with the non-local initial boundary problem for multi-term time-fractional PDE with Bessel operator.…”

“…The recently published paper on these topics is [AKMR17,KMR18]. In [KMR18] the authors dealt with the non-local initial boundary problem for multi-term time-fractional PDE with Bessel operator. They used Fourier-Bessel series expansion in order to find the explicit solution for the considered problem, yielding also its existence.…”

“…u(t, π) + π(R) u(t, π) = f (t, π).Taking into account (5.3), we rewrite the matrix equation (5.10) componentwise as an infinite system of equations of the form (5.11)∂ α +0,t u(t, π) l,k − u(t, π) l,k + π 2 l u(t, π) l,k = f (t, π) l,k ,for all π ∈ G, and any l, k ∈ N. Now let us decouple the system given by the matrix equation (5.10). For this, we fix an arbitrary representation π, and a general entry (l, k) and we treat each equation given by (5.11) individually.According to[LG99,KMR18], the solutions of the equations (5.11) satisfying initial conditions (5.12)u(0, π) l,k + n i=1 µ i u(T i , π) l,k = 0, [α]∂ t u(0, π) l,k = 0, can be represented in the form (5.13) u(t, π) l,k = F (t, π) l,k − n i=1 µ i F (T i , π) l,k 1 + n i=1 µ i θ(T i , π) l θ(t, π) l ,for all π ∈ G and any l, k ∈ N, whereF (t, π) l,k = t 0 s α−1 E (α−α 1 ,...,α−αm,α),α a 1 s α−α 1 , ..., a m s α−αm , −π 2 l s α f (t − s, π) l,k ds, θ(t, π) l = E (α−α 1 ,...,α−αm,α),1 a 1 t α−α 1 , ..., a m t α−αm , −π 2 l t α . Then there exists a solution of Problem 5.1, and it can be written as (5.14) u(t, x) = G Tr[ K(t, π)π(x)]dµ(π),…”

A boundary-value problem for a mixed type equation involving hyper-Bessel fractional differential operator and Hilfer's bi-ordinal fractional derivative

“…Proof: Applying the Laplace transform ( 9) by means of ( 8) and considering initial conditions (10), (11) yield…”

“…Initial-value problems (IVPs) and boundary-value problems (BVPs) involving the Riemann-Liouville and Caputo derivatives atract most interest (see, for instance, [6], [7], [8]). Especially, studying IVPs and BVPs for the sub-diffusion, fractional wave equations are well-studied (see [9], [10], [11]). BVPs for mixed type equations are also an interesting target for many authors (see [12]- [16]).…”