Analysis and Numerical Simulation of Time-Fractional Derivative Contact Problem with Friction in Thermo-Viscoelasticity

Definition A.1 (Riemann–Liouville Fractional Integral).

Let X be a Banach space and let ( 0 , T ) be a finite time interval. The Riemann–Liouville fractional integral of order α > 0 for a given function f ∈ L 1 ⁢ ( 0 , T ; X ) is defined by

where Γ ⁢ ( ⋅ ) stands for the Gamma function defined by

Definition A.2 (Caputo Derivative of Order 0 < α ≤ 1 ).

Let X be a Banach space, let 0 < α ≤ 1 and let ( 0 , T ) be a finite time interval. For a given function f ∈ A ⁢ C ⁢ ( 0 , T ; W ) , the Caputo fractional derivative of f is defined by

The notation A ⁢ C ⁢ ( 0 , T ; X ) refers to the space of all absolutely continuous functions from ( 0 , T ) into X.

Proposition A.1.

Let X be a Banach space and α , β > 0 . Then the following statements hold:

1. For y ∈ L 1 ⁢ ( 0 , T ; X ) , we have

   I t α 0 ⁢ I t β 0 ⁢ y ⁢ ( t ) = 0 I t α + β ⁢ y ⁢ ( t )   for a.e.  t ∈ ( 0 , T ) .

2. For y ∈ A ⁢ C ⁢ ( 0 , T ; X ) and α ∈ ( 0 , α ] , we have

   I t α 0 ⁢ D t α 0 C ⁢ y ⁢ ( t ) = y ⁢ ( t ) - y ⁢ ( 0 )   for a.e.  ⁢ t ∈ ( 0 , T ) ,

3. For y ∈ L 1 ⁢ ( 0 , T ; X ) , we have D t α 0 C ⁢ I t α 0 ⁢ y ⁢ ( t ) = y ⁢ ( t ) for a.e. t ∈ ( 0 , T ) .

Theorem A.1.

Let T > 0 , γ ∈ ( 0 , 1 ) and p ∈ [ 1 , ∞ ) . Let M, B, Y be Banach spaces such that M ↪ B compactly and B ↪ Y continuously. Suppose W ⊂ L loc 1 ⁢ ( ( 0 , T ) ; M ) satisfies the following:

1. There exist r 1 ∈ [ 1 , ∞ ) and C 1 > 0 such that for all u ∈ W ,

   (A.1) sup t ∈ ( 0 , T ) ⁡ J γ ⁢ ( ∥ u ∥ M r 1 ) = sup t ∈ ( 0 , T ) ⁡ 1 Γ ⁢ ( γ ) ⁢ ∫ 0 t ( t - s ) γ - 1 ⁢ ∥ u ∥ M r 1 ⁢ ( s ) ⁢ 𝑑 s ≤ C 1 ,

   where

   J α ⁢ u ⁢ ( t ) = 1 Γ ⁢ ( α ) ⁢ ∫ 0 t ( t - s ) α - 1 ⁢ u ⁢ ( s ) ⁢ 𝑑 s .

2. There exists p 1 ∈ ( p , ∞ ] such that W is bounded in L p 1 ⁢ ( ( 0 , T ) ; B ) .

3. There exist r 2 ∈ [ 1 , ∞ ) and C 2 > 0 such that for all u ∈ W , there is an assignment of initial value u 0 so that the weak Caputo derivative satisfies

   (A.2) ∥ D c γ ⁢ u ∥ L r 2 ⁢ ( ( 0 , T ) ; Y ) ≤ C 2 .

   Then W is relatively compact in L p ⁢ ( ( 0 , T ) ; B ) .

Proposition A.2.

Let - ∞ ≤ α ≤ ∞ . Assume f : [ 0 , ∞ ) × ( α , β ) → R is continuous and locally Lipschitz in the second variable. In other words, for all A > 0 and K ⊂ ( α , β ) compact, there exists L A , K > 0 such that

Let 0 < γ < 1 and v 0 ∈ ( α , β ) . Then the IVP

has a unique continuous solution v ⁢ ( ⋅ ) on [ 0 , T b ) , where

is the largest time of existence satisfies T b ∈ ( 0 , ∞ ] . If T b < ∞ , then lim inf t → T b - ⁡ v ⁢ ( t ) = α or lim sup t → T b - ⁡ v ⁢ ( t ) = β .