A fractional derivative with two singular kernels and application to a heat conduction problem

In this article, we suggest a new notion of fractional derivative involving two singular kernels. Some properties related to this new operator are established and some examples are provided. We also present some applications to fractional differential equations and propose a numerical algorithm based on a Picard iteration for approximating the solutions. Finally, an application to a heat conduction problem is given.

In [1], Almeida introduced the notion of ψ-Caputo fractional derivative as a generalization of the Caputo derivative. Namely, given ψ ∈ C n ([a, b], R) with ψ > 0, and f ∈ C n ([a, b], R), the left-sided fractional derivative order α ∈ (n -1, n) of f with respect to ψ is defined by The right-sided fractional derivative of order α of f with respect to ψ is defined by In the particular case ψ(t) = t, C D α,ψ a reduces to the left-sided Caputo fractional derivative, and C D α,ψ b reduces to the right-sided Caputo fractional derivative. For other examples of ψ, one obtains other known fractional operators, as for example the fractional derivative of Caputo-Hadamard (see [7]) and the fractional derivative of Caputo-Erdélyi-Kober (see [9]). In all the above notions, the fractional derivatives involve only one singular kernel.
In this paper, a new concept of fractional derivative with two singular kernels k 1 (t, s) = 1 Γ (θ+1) ϕ (s)(ϕ(t)ϕ(s)) θ and k 2 (s, τ ) = 1 Γ (μ+1) ψ (τ )(ψ(s)ψ(τ )) μ , where -1 < θ , μ < 0, is proposed. We establish some properties related to this introduced operator and present some applications to fractional differential equations. Namely, we investigate the existence and uniqueness of solutions of a nonlinear fractional boundary value problem of a higher order, and provide a numerical technique based on a Picard iteration for approximating solutions. An application to a heat conduction problem is also provided.
In Sect. 2, the fractional derivative operator with two singular kernels is introduced and some properties are established. The special case ϕ = ψ is discussed in Sect. 3. In Sect. 4, we study a nonlinear fractional boundary value problem of a higher order. Namely, using Banach fixed point theorem, we establish the existence and uniqueness of solutions, and provide a numerical algorithm based on Picard iterations for approximating the solution. In Sect. 5, an application to a heat conduction problem is given.

Fractional derivative with two singular kernels
First, we fix some notations. We denote by N the set of positive integers. Let n ∈ N and a, b ∈ R with a < b. Let The right-sided (ϕ, ψ)-fractional derivative of f with parameters (α, β) is defined by Similarly, from (2), for all a ≤ t < b, one has In C([a, b], R) we consider the norm where ψ ∈ Φ (n) .

The case 2n -2 < α + β < 2n -1
In this case, using (13), for a < t ≤ b, one has Integrating by parts, one obtains where Now, we discuss two cases.
• n = 1. In this case, one has Hence, by (15), one deduces that • n ≥ 2. In this case, one has Hence, by (15), one deduces that Similarly, using (14), for a ≤ t < b and n ≥ 2, one obtains and for n = 1, Hence, we have the following results.

The case α + β = 2n -1
In this case, using (13), for a < t ≤ b, one has Similarly, using (14), for a ≤ t < b, one obtains Hence, we obtain the following.
Theorem 3.7 Let ρ > 0 and 0 < α, β < 1 with 0 < α + β < 1. Let where ϕ ∈ Φ (1) . Then Proof By Theorem 3.3, one has for all a < t ≤ b. On the other hand, an elementary calculation gives us for all a < t ≤ b. Hence, combining the above equalities, we obtain the desired result.

Proposition 4.1 Problem
Proof Let y be the function given by (18). One observes easily that Next, using the semigroup property, we have Since σ (a) = 0, one deduces that On the other hand, one can check easily that for all k = 0, 1, . . . , n -1. Therefore, the function y given by (18) solves (17). Now, suppose that y ∈ C n ([a, b], R) is a solution of (18). By (1), one has Then we have On the other hand, one has (see [1]) Using the initial conditions, one obtains Further, combining (19) with (20), one deduces that Consider now the nonlinear problem where f : [a, b] × R → R is a continuous function. We suppose that Using (22), we have Hence, On the other hand, by (23), one has 0 < L < 1. Therefore, A is a contraction.

Conclusion
The goal of this article was to propose a new notion of fractional derivative involving two singular kernels. Some properties of this introduced operator were proved and some examples were provided. We also presented some applications to fractional differential equations. Namely, an existence and uniqueness result was established for a nonlinear fractional boundary value problem with a higher order, and a numerical algorithm based on Picard iteration was provided for approximating the unique solution. Moreover, an application to a heat conduction problem was presented. It will be interesting to develop new numerical methods for solving fractional differential equations (or partial differential equations that are fractional in time) involving this new concept, in particular in the case ϕ = ψ.