Well-posedness and numerical approximation of tempered fractional terminal value problems

Lemma 2.1

([7]) If the functionf(t, u) is continuous, then the initial value problem

is equivalent to the nonlinear Volterra integral equation of the second kind

wheren − 1 < α < nandα ≥ 0.

In the particular case where 0 < α < 1, we have

Theorem 2.1

([7]) Letn − 1 < α < n, n ∈ ℕ⁺, λ ≥ 0 anda ∈ ℝ such thata > 0. Furthermore, let the functionf(t, y) be a continuous function such that f(t, y) ∈ L([0, a]), for anyy ∈ B, where B is an open set in ℝ, andf(t, y) also satisfies a Lipschitz condition with respect to the second variable, then the fractional differential equation(2.1)has a unique solutionu(t) ∈ Cⁿ([0, a]).

Next, we extend these results to terminal boundary problems. In what follows, the Caputo tempered fractional derivative of order α will be simply denoted by 𝔻^{α, λ}(y(t)).

The Caputo tempered fractional derivative of order α, with α ∈ (0,1) satisfies

where 𝕀^α,λ is the Riemann-Liouville tempered fractional integral given by

and 𝕀^α denotes the Riemann-Liouville fractional integral.

If y satisfies the fractional differential equation (1.1), then applying the Riemann-Liouville fractional integral at the both sides of equation and taking property (2.3) into account, we conclude that the solution y satisfies the following integral equation

Therefore, if y is a solution of the fractional boundary value problem (FBVP) (1.1)-(1.2) then y is a solution of the integral equation

Next, we establish sufficient conditions for the existence and uniqueness of solutions of the FBVP (1.1)-(1.2). The proof will be based on the Banach’s fixed point theorem. We just establish the existence and uniqueness of the solution on the interval [0, a], since the existence and uniqueness for t > a is inherited from the corresponding initial value problem theory (see [7] for details).

Define the set Ωγ={y∈C([0,a]):∥y−yae−λt∥[0,a]≤γ}, where the norm ║ ⋅ ║_{[0, a]} is defined by ∥⋅∥[0,a]=maxt∈[0,a]|g(t)|, for all g ∈ C([0, a]) and

The set Ω_γ is a closed subset of the Banach space of all continuous functions on [0, a], equipped with the norm ║ ║_{[0, a]}, and since the function y(t) = y_ae^−λt ∈ Ω_γ, it is nonempty. On Ω_γ, let us define the operator

Using this operator, the integral equation can be rewritten as y = 𝓐y, and if the operator 𝓐 has a unique fixed point on Ω_γ then the FBVP (1.1)-(1.2) has a unique continuous solution. Using the Banach’s fixed point theorem, under some assumptions on f, we prove the existence and uniqueness result on the next theorem.

Theorem 2.2

LetD=[0,a]×[yae−λa−γ,yae−λa+γ],withγgiven by(2.8), and assume that the functionf : D → ℝ is continuous for all t ∈ [0, a]. We further assume that the functionffulfills a Lipschitz condition with respect to the second variable, meaning that there existsL > 0 such that it holds

If the Lipschitz constant L is such thatL<Γ(α+1)2aαeλa,then 𝓐 maps Ω_γinto itself and it is a contraction:

Hence equation (2.7) has a unique solutiony^* ∈ Ω_γ, which is the unique fixed point of 𝓐.

Proof

Let y ∈ Ω_γ. First, we show that 𝓐y ∈ Ω_γ.

From the definition of 𝓐 we have

Then

which implies that (𝓐y) ∈ Ω_γ.

Now we prove that 𝓐 is a contraction on Ω_γ, with γ defined by (2.8).

For y, z ∈ Ω_γ, t ∈ [0, a], we have

Then, the operator 𝓐 is a contraction on Ω_γ. Finally, by the Banach fixed point principle the proof of the theorem is complete.□

If the assumptions of Theorem 2.2 are satisfied, then the FBVP (1.1)-(1.2) has an unique continuous solution, y(t), on the interval [0, a] and, in particular, a unique value for y(0) exists. Therefore, there is an exact correspondence between tempered fractional boundary value problems and tempered fractional initial value problems.

Theorem 3.1

Letyand z be the unique solutions of problems(1.1)-(1.2)and(3.2)-(3.3), respectively. Then

whereβ=1−2LeλaaαΓ(α+1).

Proof

Taking (3.1) and (3.10) into account, for any t ∈ [0, a], we have

Hence

According to the upper bound on the Lipschitz condition L, established in Theorem 2.2, we have

and therefore, we conclude that

and the theorem is proved.□

Theorem 3.2

Letyand z be the unique solutions of problems(1.1)-(1.2)and(3.4)-(3.5), respectively, where in the later we assume thatδis such that 0 < α − δ < 1. Then

Proof

Taking (3.1) and (3.11) into account, for any t ∈ [0, a], we have

Since e^−λt ≤ 1, for all t ≥ 0 and λ > 0, and e^λs ≤ e^λa, for all 0 ≤ s, then

Let us look at the first integral on the right-hand side of (3.15) first:

Considering the function G(x)=(a−s)xΓ(x+1), using the mean value theorem, we easily conclude that (a−s)α−1Γ(α)−(a−s)α−δ−1Γ(α−δ)≤Cδ, where C=maxx∈[α−δ−1,α−δ]|G′(x)|, and therefore

Proceeding similarly with the second integral on the right-hand side of (3.15), we could conclude that

and therefore

or

where β is given by (3.14). This completes the proof of the theorem.□

Theorem 3.3

Letyand z be the unique solutions of problems(1.1)-(1.2)and(3.6)-(3.7), respectively. Then

Proof

Taking (3.1) and (3.12) into account, for any t ∈ [0, a], we have

By using the mean value theorem with the function g(x) = e^−xt, we conclude that

where C1=maxx∈[λ−δ,λ]g′(x).

Concerning the first integral on the right-hand side of (3.16):

where by the mean value theorem, C2=maxx∈[λ−δ,λ]h′(x), where h(x) = e^xs, s ∈ [0, a].

Proceeding analogously with the second integral in (3.16), we conclude that

Hence

or

with β defined in (3.14). Thus, the theorem is proved.□

Theorem 3.4

Letyand z be the unique solutions of problems(1.1)-(1.2)and(3.8)-(3.9), respectively. Then

Proof

Taking (3.1) and (3.13) into account, for any t ∈ [0, a], we have

where I1=∫0a(a−s)α−1f(s,y(s))−f~(s,z(s)) ds and I2=∫0t(t−s)α−1f(s,y(s))−f~(s,z(s)) ds. Since

and, analogously

then

and the result of the theorem follows.

Theorem 4.1

Letα ∈ (0,1) and assume that f : [0, b] × [c, d] → ℝ is continuous and satisfies a Lipschitz condition with respect to the second variable.

Ify₁andy₂are are two solutions of the tempered differential equations

subject to the initial conditionsy_j(0) = y_j0, j = 1, 2, respectively, wherey₁₀ ≠ y₂₀.

Then for all t ∈ [0, b] we havey₁(t) ≠ y₂(t).

From Theorem 4.1 we can conclude that a solution of a tempered fractional differential equation of order α ∈ (0, 1) is uniquely defined by a condition that can be specified at any point t ∈ [0, b] ([0, b] is the interval where the solution of the fractional problem exist).

On the other hand, Theorem 4.1 will be crucial to properly define the ideas of the numerical methods that we present next.

From Theorem 4.1 it follows that for the solution of (1.1) that passes through the point (a, exp(−λa)y_a), we are able to find at most one point (0, y₀) that also lies on the same solution trajectory. In order to obtain an approximation of y(0) we propose a shooting algorithm based on the bisection method. Let y₁ and y₂ be the solutions of (4.6) with initial values y₀₁ and y₀₂, such that y₁(a) < exp(−λa)y_a < y₂(a), the bisection method provide successive approximations for y(0) until the distance between the two last approximations does not exceed a given tolerance ε.

In our numerical experiments, for illustration purposes, we have used the bisection method to find the initial value, but obviously, other methods can be used as easily as this, as the secant method, for instance.

To evaluate the value of y(a) we need a numerical method to solve the initial value problems

This will be straightforward if we take relationship (1.3) into account. In fact, defining the functions u and g as in (4.3), we can use any available solver for (non-tempered) Caputo-type initial value problems to determine the solution u of

and then the solution of (4.7)-(4.8) will be given by y(t) = e^−λtu(t).