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A fractional derivative with two singular kernels and application to a heat conduction problem
Advances in Difference Equations volume 2020, Article number: 252 (2020)
Abstract
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.
1 Introduction
In many applications in applied sciences, the use of fractional derivatives with singular kernels allows us to obtain more realistic models than those derived using the standard derivative (see e.g. [2–7, 10, 11, 13, 14]). The literature contains various notions of fractional derivatives with singular kernels. The best known are the Riemann–Liouville fractional derivative and the Caputo fractional derivative (see e.g. [12, 22]). For other definitions, see, for example [1, 8, 15–21] and the references therein.
In [1], Almeida introduced the notion of ψ-Caputo fractional derivative as a generalization of the Caputo derivative. Namely, given \(\psi \in C^{n}([a,b],\mathbb{R})\) with \(\psi '>0\), and \(f\in C^{n}([a,b],\mathbb{R})\), the left-sided fractional derivative order \(\alpha \in (n-1,n)\) of f with respect to ψ is defined by
where
The right-sided fractional derivative of order α of f with respect to ψ is defined by
where
In the particular case \(\psi (t)=t\), \({}^{C}\!D_{a}^{\alpha ,\psi }\) reduces to the left-sided Caputo fractional derivative, and \({}^{C}\!D_{b}^{\alpha ,\psi }\) 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 \(k_{1}(t,s)=\frac{1}{\varGamma (\theta +1)}\varphi '(s)(\varphi (t)- \varphi (s))^{\theta }\) and \(k_{2}(s,\tau )=\frac{1}{\varGamma (\mu +1)}\psi '(\tau )(\psi (s)-\psi ( \tau ))^{\mu }\), where \(-1<\theta \), \(\mu <0\), 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 \(\varphi =\psi \) 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.
2 Fractional derivative with two singular kernels
First, we fix some notations. We denote by \(\mathbb{N}\) the set of positive integers. Let \(n\in \mathbb{N}\) and \(a,b\in \mathbb{R}\) with \(a< b\). Let
For \(\varphi \in \varPhi ^{(n)}\), let
Definition 2.1
Let \(\alpha ,\beta \in (n-1,n)\), \(\varphi \in \varPhi ^{(1)}\), \(\psi \in \varPhi ^{(n)}\) and \(f\in C^{n}([a,b],\mathbb{R})\). The left-sided \((\varphi ,\psi )\)-fractional derivative of f with parameters \((\alpha ,\beta )\) is defined by
The right-sided \((\varphi ,\psi )\)-fractional derivative of f with parameters \((\alpha ,\beta )\) is defined by
Remark 2.1
From (1), for all \(a< t\leq b\), one has
Similarly, from (2), for all \(a\leq t< b\), one has
In \(C([a,b],\mathbb{R})\) we consider the norm
We endow \(C^{n}([a,b],\mathbb{R})\) with the norm
where \(\psi \in \varPhi ^{(n)}\).
Theorem 2.1
Let\(\alpha ,\beta \in (n-1,n)\), \(\varphi \in \varPhi ^{(1)}\), \(\psi \in \varPhi ^{(n)}\)and\(f\in C^{n}([a,b],\mathbb{R})\). Then
and
Proof
Let \(a< t\leq b\). Then
which proves (3). Using similar estimates, one obtains (4). □
Corollary 2.1
Let\(\alpha ,\beta \in (n-1,n)\), \(\varphi \in \varPhi ^{(1)}\), \(\psi \in \varPhi ^{(n)}\)and\(f\in C^{n}([a,b],\mathbb{R})\). Then
and
Proof
Taking the limit as \(t\to a^{+}\) in (3), (5) follows. Similarly, taking the limit as \(t\to b^{-}\) in (4), (6) follows. □
Taking
one deduces from (5) that \(D_{a}^{(\alpha ,\beta ),(\varphi ,\psi )}f\in C([a,b],\mathbb{R})\). Similarly, taking
one deduces from (6) that \(D_{b}^{(\alpha ,\beta ),(\varphi ,\psi )}f\in C([a,b],\mathbb{R})\). Therefore, by Theorem 3.1, one obtains the following.
Corollary 2.2
Let\(\alpha ,\beta \in (n-1,n)\), \(\varphi \in \varPhi ^{(1)}\)and\(\psi \in \varPhi ^{(n)}\). Then, for any\(g\in C^{n}([a,b],\mathbb{R})\), we have
where
Lemma 2.1
Let\(\varphi \in \varPhi ^{(1)}\)and\(f\in C^{1}([a,b],\mathbb{R})\). Then
and
Proof
Let \(\theta >0\). One has
Integrating by parts, one obtains
Passing to the limit as \(\theta \to 0^{+}\) in the above equality, (7) follows. Similarly, one has
Integrating by parts, one obtains
Passing to the limit as \(\theta \to 0^{+}\) in the above equality, (8) follows. □
Theorem 2.2
Let\(n-1<\beta <n\), \(\varphi \in \varPhi ^{(1)}\), \(\psi \in \varPhi ^{(n)}\)and\(f\in C^{n}([a,b],\mathbb{R})\).
- (I)
If\({}^{C}D_{a}^{\beta ,\psi }f\in C^{1}([a,b],\mathbb{R})\), then
$$ \lim_{\alpha \to n^{-}} \bigl(D_{a}^{(\alpha ,\beta ),(\varphi , \psi )}f \bigr) (t) = \bigl({}^{C}D_{a}^{\beta ,\psi }f \bigr) (t),\quad a< t\leq b. $$ - (II)
If\({}^{C}D_{b}^{\beta ,\psi }f\in C^{1}([a,b],\mathbb{R})\), then
$$ \lim_{\alpha \to n^{-}} \bigl(D_{b}^{(\alpha ,\beta ),(\varphi , \psi )}f \bigr) (t) = \bigl({}^{C}D_{b}^{\beta ,\psi }f \bigr) (t),\quad a \leq t< b. $$
Proof
Using (1) and (7), (I) follows. Similarly, using (2) and (8), (II) follows. □
Theorem 2.3
Let\(\alpha ,\beta \in (n-1,n)\), \(\varphi \in \varPhi ^{(1)}\), \(\psi \in \varPhi ^{(n+1)}\)and\(f\in C^{n+1}([a,b],\mathbb{R})\). For all\(a< t\leq b\),
For all\(a\leq t< b\),
Proof
Equation (9) follows from (1) and [1, Theorem 1]. (10) follows from (2) and [1, Theorem 1]. □
Corollary 2.3
Let\(\varphi \in \varPhi ^{(1)}\), \(\psi \in \varPhi ^{(n+1)}\)and\(f\in C^{n+1}([a,b],\mathbb{R})\). Then
and
Proof
Let \(a< t\leq b\). From (9), for \(n-1<\alpha <n\), one has
Hence, taking the limit as \(\alpha \to n^{-}\), and using (7), (11) follows. Similarly, for \(a\leq t< b\), using (10) and (8), (12) follows. □
3 The case \(\varphi =\psi \)
Let \(\alpha ,\beta \in (n-1,n)\), \(\psi =\varphi \in \varPhi ^{(n)}\) and \(f\in C^{n}([a,b],\mathbb{R})\). In this case, by (1), for all \(a< t\leq b\), one obtains
Using the semigroup property (see [1]), we have
Similarly, by (2), one obtains
3.1 The case \(2n-1< \alpha +\beta <2n\)
In this case, using (13), one has
i.e.
Similarly, using (14), one obtains
Hence, the following result holds.
Theorem 3.1
Let\(\alpha ,\beta \in (n-1,n)\), \(\varphi \in \varPhi ^{(n)}\)and\(f\in C^{n}([a,b],\mathbb{R})\). Suppose that\(2n-1< \alpha +\beta <2n\). Then
and
3.2 The case \(2n-2< \alpha +\beta <2n-1\)
In this case, using (13), for \(a< t\leq b\), 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
• \(n\geq 2\). In this case, one has
Hence, by (15), one deduces that
Similarly, using (14), for \(a\leq t< b\) and \(n\geq 2\), one obtains
and for \(n=1\),
Hence, we have the following results.
Theorem 3.2
Let\(\alpha ,\beta \in (n-1,n)\), \(n\geq 2\), \(\varphi \in \varPhi ^{(n)}\)and\(f\in C^{n}([a,b],\mathbb{R})\). Suppose that\(2n-2< \alpha +\beta <2n-1\). Then
and
Theorem 3.3
Let\(0<\alpha , \beta <1\), \(\varphi \in \varPhi ^{(1)}\)and\(f\in C^{1}([a,b],\mathbb{R})\). Suppose that\(0< \alpha +\beta <1\). Then
and
3.3 The case \(\alpha +\beta =2n-1\)
In this case, using (13), for \(a< t\leq b\), one has
Similarly, using (14), for \(a\leq t< b\), one obtains
Hence, we obtain the following.
Theorem 3.4
Let\(\alpha ,\beta \in (n-1,n)\), \(\varphi \in \varPhi ^{(n)}\)and\(f\in C^{n}([a,b],\mathbb{R})\). Suppose that\(\alpha +\beta =2n-1\). Then
and
Example 3.1
Let \(0<\alpha , \beta <1\). Consider the function
where \(\varphi \in \varPhi ^{(1)}\). By (13), one has
that is,
Using the change of variable \(z=\frac{\varphi (s)-\varphi (0)}{\varphi (t)-\varphi (0)}\), one obtains
where B is the beta function. Observe that
which confirms (11). Figures 1–3 show some graphs of \((D_{0}^{(\alpha ,\beta ),(\varphi ,\varphi )}f )(t)\) for different functions φ and different values of \((\alpha ,\beta )\).
Following a similar argument to above, one obtains a theorem.
Theorem 3.5
Let\(\alpha ,\beta \in (0,1)\)and\(\theta >0\). Let
where\(\varphi \in \varPhi ^{(1)}\). Then
The Mittag-Leffler function \(E_{\theta }\), \(\theta >0\), is defined by
Theorem 3.6
Let\(\rho >0\)and\(0<\alpha , \beta <1\)with\(1< \alpha +\beta <2\). Let
where\(\varphi \in \varPhi ^{(1)}\). Then
Proof
By Theorem 3.1, one has
Next, using [1, Lemma 2], the desired result follows. □
Theorem 3.7
Let\(\rho >0\)and\(0<\alpha , \beta <1\)with\(0< \alpha +\beta <1\). Let
where\(\varphi \in \varPhi ^{(1)}\). Then
Proof
By Theorem 3.3, one has
for all \(a< t\leq b\). On the other hand, an elementary calculation gives us
for all \(a< t\leq b\). Hence, combining the above equalities, we obtain the desired result. □
Remark 3.1
By Theorems 3.6 and 3.7, one observes that, if \(0<\rho <1\), then
4 Applications to fractional differential equations
Let \(\alpha ,\beta \in (n-1,n)\), \(\varphi \in \varPhi ^{(1)}\) and \(\psi \in \varPhi ^{(n)}\). We first consider the problem
where \(\sigma \in C^{1}([a,b],\mathbb{R})\) and \(\sigma (a)=0\).
Proposition 4.1
Problem (17) has a unique solution\(y\in C^{n}([a,b],\mathbb{R})\), which is given by
Proof
Let y be the function given by (18). One observes easily that
Hence, using (1), one has
Using the property (see [1]) \({}^{C}\!D_{a}^{\beta ,\psi }I_{a}^{\beta ,\psi }f=f\), one obtains
Next, using the semigroup property, we have
Since \(\sigma (a)=0\), one deduces that
On the other hand, one can check easily that
for all \(k=0,1,\ldots ,n-1\). Therefore, the function y given by (18) solves (17).
Now, suppose that \(y\in C^{n}([a,b],\mathbb{R})\) is a solution of (18). By (1), one has
which yields
i.e.
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 \(f: [a,b]\times \mathbb{R}\to \mathbb{R}\) is a continuous function. We suppose that
for all \(t\in [a,b]\) and \(\lambda ,\eta \in \mathbb{R}\), where
Theorem 4.1
Problem (21) admits one and only one solution\(y^{*}\in C^{n}([a,b],\mathbb{R})\). Moreover, for any\(y_{0}\in C([a,b],\mathbb{R})\), the Picard sequence\(\{y_{n}\}\subset C([a,b],\mathbb{R})\)defined by
converges uniformly to\(y^{*}\).
Proof
Let A be the self-mapping defined in \(C([a,b],\mathbb{R})\) by
i.e.
By Proposition 4.1, \(y\in C^{n}([a,b],\mathbb{R})\) is a solution of (21) if and only if \(y\in C([a,b],\mathbb{R})\) is a fixed point of A. We shall show that A is a contraction in \((C([a,b],\mathbb{R}),\|\cdot \|_{\infty })\), and then, by the fixed point theorem of Banach, we obtain the desired result. For any \(y,z\in C([a,b],\mathbb{R})\), one has
Using (22), we have
where
Hence,
for all \(y,z\in C([a,b],\mathbb{R})\). On the other hand, by (23), one has \(0< L<1\). Therefore, A is a contraction. □
Example 4.1
Consider the fractional boundary value problem
where \(0<\alpha , \beta <1\), \(\varphi (t)=t\), \(\psi (t)=\ln (t+1)\) and \(\varGamma (\alpha +1)\varGamma (\beta +1)\rho >(\ln 2)^{\beta }\). Problem (24) is a particular case of problem (21) with \((a,b)=(0,1)\), \(n=1\), \(\mu _{0}=0\) and
For all \(t\in [0,1]\) and \(\lambda ,\eta \in \mathbb{R}\), one has
where \(C_{f}=\rho ^{-1}\). On the other hand, one has
Hence, by Theorem 4.1, problem (24) admits a unique solution \(y^{*}\in C^{1}([0,1],\mathbb{R})\). Moreover, for any \(y_{0}\in C([0,1],\mathbb{R})\), the Picard sequence
for all \(\leq t\leq 1\), converges uniformly to \(y^{*}\).
5 Fractional model of a heat conduction problem
The standard Fourier law of thermal conduction in one dimension is given by
where ρ is the material’s thermal conductivity, z is the density of the heat flux and y is the temperature. Replacing \(\frac{d}{dx}\) by \(D_{0}^{(\alpha ,\beta ),(\varphi ,\psi )}\), where \(\alpha ,\beta \in (0,1)\), we obtain the fractional version of (25)
If \(z(0)=0\) and \(y(0)=y_{0}\), by Proposition 4.1, the unique solution of (26) is given by
i.e.
Example 5.1
Consider (26) with \(\varphi =\psi \) and \(z(x)=\varphi (x)-\varphi (0)\). In this case, by (27), one has
which yields
Observe that in the case \(\varphi (x)=x\) one has
which is the unique solution of (25) with \(z(x)=x\) and \(y(0)=y_{0}\). Figures 4–6 show some graphs of y for different functions φ and different values of \((\alpha ,\beta )\).
6 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 \(\varphi \neq \psi \).
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Acknowledgements
M. Jleli is supported by Researchers Supporting Project RSP-2019/57, King Saud University, Riyadh, Saudi Arabia.
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Researchers Supporting Project RSP-2019/57, King Saud University, Riyadh, Saudi Arabia.
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Baleanu, D., Jleli, M., Kumar, S. et al. A fractional derivative with two singular kernels and application to a heat conduction problem. Adv Differ Equ 2020, 252 (2020). https://doi.org/10.1186/s13662-020-02684-z
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DOI: https://doi.org/10.1186/s13662-020-02684-z