Abstract
By mixing the idea of 2-arrays, continued fractions, and Caputo-Fabrizio fractional derivative, we introduce a new operator entitled the infinite coefficient-symmetric Caputo-Fabrizio fractional derivative. We investigate the approximate solutions for two infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential problems. Finally, we analyze two examples to confirm our main results.
Similar content being viewed by others
1 Introduction
Fractional calculus has many real-world applications in various fields of science and engineering [1–10]. During the recent years, the researchers started to think how to enlarge the range of fractional calculus by constructing operators with different nonlocal kernels. For example, a new nonlocal derivative without singular kernel was introduced in [11]. After that, this new fractional operator was utilized to get more information from solving different fractional differential equations corresponding to complex phenomena (the reader can see, for example, [11–20], and the references therein). Let use consider \(b>0\) and \(x\in H^{1}(0,b)\) together with \(\alpha\in(0,1)\). For a function x, Caputo and Fabrizio defined its fractional derivative (CF) of order α as \({}^{\mathrm{CF}}C^{\alpha}x(p)=\frac{(2-\alpha)M(\alpha)}{2(1-\alpha)}\int _{0}^{p}\exp (\frac{-\alpha}{1-\alpha}(p-w))x^{\prime}(w)\,dw\), where \(t\geq0\), and \(M(\alpha)\) is such that \(M(0)=M(1)=1\) [11]. The corresponding fractional integral of order α for the function x is \({}^{\mathrm{CF}}I^{\alpha} x(p)=\frac{2(1-\alpha)}{(2-\alpha)M(\alpha)}x(p) +\frac{2\alpha}{(2-\alpha)M(\alpha)}\int_{0}^{p} x(w) \,dw\) whenever \(0<\alpha<1\) [21]. Also, the values of the function M were found as \(M(\alpha)=\frac{2}{2-\alpha}\) for all \(0\leq\alpha\leq1\) [21]. Taking into account the results mentioned, for a given function x, its fractional CF of order α becomes \({}^{\mathrm{CF}}C^{\alpha}x(p)=\frac{1}{1-\alpha}\int_{0}^{p}\exp(-\frac {\alpha}{1-\alpha}(p-w))x^{\prime}(w)\,dw\) for \(t\geq0\) and \(0<\alpha<1\) [21]. In this way a new type of fractional calculus was established. The aim of the manuscript is to propose a new operator named the infinite coefficient-symmetric Caputo-Fabrizio fractional derivative and to study some its properties.
2 Basic tools and new fractional operators
We further introduce some basic notation.
Lemma 2.1
[21]
Let us consider the equation \({}^{\mathrm{CF}}C^{\alpha}x(t)=y(t)\) such that \(x(0)=c\) and \(0<\alpha<1\). The solutions of this equation has the form \(x(p)=c+a_{\alpha}(y(p)-y(0))+b_{\alpha}\int_{0}^{p} y(z)\,dz\), where \(a_{\alpha}=\frac{2(1-\alpha)}{(2-\alpha)M(\alpha)}=1-\alpha\) and \(b_{\alpha}=\frac{2\alpha}{(2-\alpha)M(\alpha)}=\alpha\).
Let \(\varepsilon> 0\). We consider a metric space \((Z,d_{1})\), a selfmap G on Z, and a mapping \(\alpha : Z\times Z \to[ 0 , \infty) \). As a result, we say that G is α-admissible whenever \(\alpha(t,s) \geq1\) implies \(\alpha (Gt , Gs )\geq1\) [22]. An element \(z_{0}\in Z \) is called an ε-fixed point of G if \(d(G z_{0},z_{0}) \leq\varepsilon \). We say that G possess the approximate fixed point property if G possesses an ε-fixed point for all \(\varepsilon> 0\) [22]. Denote by \(\mathcal{R}\) the set of all continuous mappings \(j : [0,\infty)^{5} \to[0,\infty)\) satisfying \(j(1,1,1,2,0)= j(1,1,1,0,2):=l \in(0,1)\), \(j(\mu t_{1},\mu t_{2},\mu t_{3},\mu t_{4},\mu t_{5}) \leq\mu j(t_{1}, t_{2},t_{3},t_{4},t_{5})\) for all \((t_{1},t_{2},t_{3},t_{4},t_{5}) \in[0,\infty)^{5} \) and \(\mu\geq0 \) and also \(j( t_{1},t_{2},t_{3},0,t_{4}) \leq j( s_{1},s_{2},s_{3},0,s_{4})\) and \(j(t_{1},t_{2},t_{3},t_{4},0)\leq j(s_{1},s_{2},s_{3},s_{4},0)\) whenever \(t_{1},\dots,t_{4},s_{1},\dots,s_{4} \in[0,\infty)\) with \(t_{k}< s_{k} \) for \(k=1,2,3,4\) [22]. Next, we recall that G is called a generalized α-contractive mapping if there exists \(j \in\mathcal{R}\) such that \(\alpha(t,s)d_{1}(Gt,Gs)\leq j(d_{1}(t_{1},s_{1}),d_{1}(t_{1},Gt_{1}),d_{1}(s_{1},Gs_{1}),d_{1}(t_{1},Gs_{1}), d_{1}(s_{1},Gt_{1}))\) for all \(t_{1},s_{1} \in Z\) [22]. We need the following key result.
Theorem 2.2
[22]
Suppose that there exists \(t_{0}\in Z\) such that \(\alpha(t_{0},Gt_{0}) \geq1\). Then G possesses an approximate fixed point, where \((Z,d)\) is a metric space, \(\alpha: Z\times Z \to [0,\infty)\) denotes a mapping, and G represents a generalized α-contractive and α-admissible selfmap on Z.
Let \(\{L_{i,2^{i}}\}_{i\geq1}\) be a sequence of operators on a set. For reduction and approximation in large and infinite potential-driven flow networks, there is a method of using 2-arrays and continued fractions (see [23] and [24]). In fact, it is sufficient to arrange the operators \(\{L_{i,2^{i}}\}_{i\geq1}\) symmetrically on a 2-array, and by using a continued fraction we make a new operator \(L_{N}\) from the operators \(L_{i,2^{i}}\), where N is a natural number (see [23] and [24]). First, we arrange the operators \(L_{i,2^{i}}\) on a 2-array (tree) as in Figure 1 (see [23]).
Now, using a finite continued fraction, consider the new operator \(L_{N}\) defined by
Here, we replace symmetrically the operators \(L_{ij}\) with \({}^{\mathrm{CF}}C^{\alpha}\) for j odd (the upper branch) and \({}^{\mathrm{CF}}C^{\beta}\) for j even (the lower branch) as in Figure 2.
Put
and
Continuing this process, we can define the new operator \({}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\beta)}_{N}\). Now, we define the infinite symmetric CF fractional derivative by \({}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\beta)}_{\infty}=\lim_{N \to \infty} {}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\beta)}_{N}\). A simple calculation shows that \({}^{\mathrm{CF}}\mathbb{C}^{(\alpha,\beta)}_{\infty} =({}^{\mathrm{CF}}C^{\alpha} {}^{\mathrm{CF}}C^{\beta})^{\frac{1}{2}}\). Similarly, we can define the infinite symmetric CF fractional integral \({}^{\mathrm{CF}}\mathbb{I}^{(\alpha,\beta)}_{\infty}\) by
Let \(\mu\geq0\), \(\mu\neq2\). Putting \(\mu^{i-1} {}^{\mathrm{CF}}C^{\alpha}\) on the upper branch and \(\mu^{i-1} {}^{\mathrm{CF}}C^{\beta}\) on the lower branch in the ith stage as in Figure 3, we can make the infinite coefficient-symmetric CF fractional derivative as a generalization for last case.
In fact, we define
and so
This implies that
3 Results
To show our results, we recall below two lemmas [15] under the assumption that \(x,y\in H^{1}(0,1)\).
Lemma 3.1
[15]
If there exists a real number \(K_{1}\) such that \(\vert x(p)-y(p) \vert \leq K_{1}\) for all \(p\in[0,1]\), then \(\vert {}^{\mathrm{CF}}C^{\alpha}x(p)-{{}^{\mathrm{CF}}}C^{\alpha}y(p) \vert \leq \frac{2-\alpha}{(1-\alpha)^{2}}K_{1}\) for all \(p \in[0,1]\).
Lemma 3.2
[15]
Assume that \(x(0)=y(0)\) and there exists a real number \(K_{1}\) such that \(\vert x(p)-y(p) \vert \leq K_{1}\) for \(p\in[0,1]\). Then \(\vert {}^{\mathrm{CF}}C^{\alpha}x(p)-{{}^{\mathrm{CF}}}C^{\alpha}y(p) \vert \leq\frac{1}{(1-\alpha)^{2}}K_{1}\) for all \(p\in[0,1]\).
Let \(x,y\in C_{\mathbb{R}}[0,1]\).
Lemma 3.3
[15]
If there is \(K_{1}\geq0\) such that \(\vert x(p)-y(p) \vert \leq K_{1}\) for all \(p\in[0,1]\), then \(\vert {}^{\mathrm{CF}}I^{\alpha}x(p)-{}^{\mathrm{CF}}I^{\alpha}y(p) \vert \leq K_{1}\) for \(p\in[0,1]\).
Now we are ready to show our main results. Using Lemmas 3.1 and 3.2, we obtain the next key results.
Lemma 3.4
Let \(x,y\in H^{1}\). If there exists a real number \(K_{1}\) such that \(\vert x(p)-y(p) \vert \leq K_{1}\) for all \(p\in[0,1]\), then \(\vert {}^{\mathrm{CF}}\mathbb{C}_{\infty}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{C}_{\infty}^{(\alpha,\alpha)}y(p) \vert \leq\frac{2-\alpha}{(1-\alpha)^{2}}K_{1}\) for all \(p\in[0,1]\).
Lemma 3.5
Let \(x,y\in H^{1}\) with \(x(0)=y(0)\) and \(K_{1}\in\mathbb{R}\). If \(\vert x(p)-y(p) \vert \leq K_{1}\) for \(p\in[0,1]\), then \(\vert {}^{\mathrm{CF}}\mathbb{C}_{\infty}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{C}_{\infty}^{(\alpha,\alpha)}y(p) \vert \leq\frac{1}{(1-\alpha)^{2}}K_{1}\) for all \(p\in[0,1]\).
Using Lemmas 3.4 and 3.5 and (*), we get the following results.
Lemma 3.6
Let \(x,y\in H^{1}\). If there exists a real number \(K_{1}\) such that \(\vert x(p)-y(p) \vert \leq K_{1}\) for all \(p\in[0,1]\), then \(\vert {}^{\mathrm{CF}}\mathbb{C}_{(\mu,\infty)}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{C}_{(\mu,\infty)}^{(\alpha,\alpha)}y(p) \vert \leq\frac{(2-\alpha)}{(2-\mu)(1-\alpha)^{2}} K_{1}\) for all \(p\in [0,1]\).
Lemma 3.7
Let \(x,y\in H^{1}\) with \(x(0)=y(0)\) and \(K_{1}\in\mathbb{R}\). If \(\vert x(p)-y(p) \vert \leq K_{1}\) for all \(p\in[0,1]\), then \(\vert {}^{\mathrm{CF}}\mathbb{C}_{(\mu,\infty)}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{C}_{(\mu,\infty)}^{(\alpha,\alpha)}y(p) \vert \leq\frac{1}{(2-\mu)(1-\alpha)^{2}} K_{1}\) for all \(p\in[0,1]\).
Lemma 3.8
Let \(x,y\in C_{\mathbb{R}}[0,1]\). Let \(K_{1}\) be a real number such that \(\vert x(p)-y(p) \vert \leq K_{1}\) for all \(p\in[0,1]\), then \(\vert {}^{\mathrm{CF}}\mathbb{I}_{\infty}^{(\alpha,\alpha )}x(p)-{{}^{\mathrm{CF}}}\mathbb{I}_{\infty}^{(\alpha,\alpha)}y(p) \vert \leq K_{1}\) for all \(p\in[0,1]\).
Using Lemma 2.1, we can prove the next key result.
Lemma 3.9
Let \(\alpha\in(0,1)\) and \(c\in\mathbb{R}\). The unique solution of the problem
with boundary condition \(x(0)=c\) is given by \(x(p)=c+a_{\alpha}(y(p)-y(0))+b_{\alpha}\int_{0}^{t} y (s)\,ds\).
Also, using Lemma 2.1 and (*), we can prove the next key result.
Lemma 3.10
Let \(\alpha\in(0,1)\) and \(c\in\mathbb{R}\). The unique solution of the problem
with boundary condition \(x(0)=c\) is given by
Let \(I=[0,1]\), and let \(\gamma,\lambda:[0,1] \times[0,1]\to [0,\infty)\) be two continuous maps such that \(\sup_{p\in I} \vert \int_{0}^{p} \lambda(p,r) \,dr \vert <\infty\) and \(\sup_{p\in I} \vert \int_{0}^{p} \gamma(p,r) \,dr \vert <\infty\). We introduce the following maps ϕ and φ defined by \((\phi u)(p)= \int_{0}^{p} \gamma(p,r)u(r)\,dr \) and \((\varphi u)(p)= \int_{0}^{p} \lambda(p,r)u(r)\,dr \), respectively. Let us consider \(\gamma_{0}=\sup \vert \int_{0}^{p} \gamma(p,r) \,dr \vert \) and \(\lambda_{0}=\sup \vert \int_{0}^{p} \lambda(p,r) \,dr \vert \), respectively. Let \(\eta(p)\in L^{\infty}(I)\) with \(\eta^{\ast}=\sup_{p\in I} \vert \eta(p) \vert \). We further are going to investigate the infinite CF fractional integro-differential problem, namely
with \(u_{1}^{\prime}(0)=0\). Here \(\alpha,\beta,\gamma,\theta ,\delta\in (0,1)\), and \(\mu\geq0\).
Theorem 3.11
Let \(f^{\prime}:[0,1]\times\mathbb{R}^{5}\rightarrow\mathbb{R}\) be a continuous function satisfying
for all \(r\in I\) and \(x_{1},y_{1},w_{1},x_{1}^{\prime},y_{1}^{\prime },w_{1}^{\prime},u_{1}, u_{2},v_{1},v_{2} \in\mathbb{R}\). If \(\Delta= [\eta^{*}(2+\gamma _{0} + \lambda_{0} + \frac{1}{(1-{\delta})^{2}})+{ \mu}( \frac{1}{(1-{\gamma})^{2} }+\frac{1}{(1-{\beta})^{2}})]<1\), then problem (1) possesses an approximate solution.
Proof
Let \(H^{1}\) be equipped with \(d(u_{1}^{\prime},v_{1}^{\prime })= \Vert u_{1}^{\prime}-v_{1}^{\prime} \Vert \), where \(\Vert u_{1}^{\prime} \Vert =\sup_{t\in I} \vert u_{1}^{\prime}(t) \vert \). Now, consider the selfmap \(F:H^{1}\to H^{1}\) defined by
for all \(r\in I\) and \(u_{1}^{\prime},v_{1}^{\prime}\in H^{1}\), where \(a_{\alpha}\) and \(b_{\alpha}\) have the meaning given in Lemma 3.9. Now, utilizing Lemmas 3.5 and 3.8, we get
for all \(r\in I\) and \(u_{1}^{\prime},v_{1}^{\prime}\in H^{1}\). Hence,
for all \(u_{1}^{\prime},v_{1}^{\prime}\in H^{1}\). Consider the mappings \(j:[0,\infty)^{5} \to[0,\infty) \) and \(\alpha :H^{1}\times H^{1}\to[0,\infty)\) defined by \(j(t_{1},t_{2},t_{3},t_{4},t_{5})= \Delta t_{1}\) and \(\alpha(t,s) =1\) for all \(t,s\in H^{1}\). We can check that \(j \in\mathcal{R}\) and F is a generalized α-contraction. From Theorem 2.2 we conclude that F possesses an approximate fixed point, which is an approximate solution for problem (1). □
Let c be a real number, and k, s, and q bounded functions on \(I=[0,1]\) with \(M_{1}=\sup_{p\in I} \vert k(p) \vert <\infty\), \(M_{2}=\sup_{p\in I} \vert s(p) \vert <\infty\), and \(M_{3}=\sup_{t\in I} \vert q(p) \vert <\infty\). We investigate the infinite coefficient-symmetric CF fractional integro-differential problem
with \(x(0)=c\), where \(\lambda,\rho\geq0\) and \(\alpha, \gamma ,\delta,\theta\in(0,1)\).
Theorem 3.12
Let \(\xi_{1},\xi_{2},\xi_{3}\geq0\), and let \(f:[0,1]\times\mathbb {R}^{3}\rightarrow\mathbb{R}\) be a bounded integrable function satisfying \(\vert f(p,x_{1},y_{1},w_{1})-f(p,x_{1}^{\prime },y_{1}^{\prime},w_{1}^{\prime}) \vert \leq \xi_{1} \vert x_{1} -x_{1}^{\prime} \vert +\xi_{2} \vert y_{1} -y_{1}^{\prime} \vert +\xi_{3} \vert w_{1}-w_{1}^{\prime} \vert \) for all \(p \in I\) and \(x_{1},y_{1},w_{1},v_{1},x_{1}^{\prime },y_{1}^{\prime},w_{1}^{\prime}\in\mathbb{R}\). If \(\Delta= \vert 2-\mu \vert [\lambda\frac{M_{1}}{(1-\delta )^{2}}+\rho M_{2}+ \xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} \frac {M_{3}}{(1-\gamma)^{2} \vert 2-m \vert }]<1\), then problem (2) admits an approximate solution.
Proof
Let \(H^{1}\) be equipped with \(d(x ,y )= \Vert x -y \Vert \), where \(\Vert x \Vert =\sup_{t\in I} \vert x(t) \vert \). Now, consider the selfmap \(\mathcal{F}:H^{1}\to H^{1}\) defined by
for all \(p\in I\) and \(x,y\in H^{1}\), where \(a_{\alpha}\) and \(b_{\alpha}\) are given in Lemma 3.10. As a result, utilizing Lemmas 3.5, 3.7, and 3.8, we get
for all \(p\in I\) and \(x,y\in H^{1}\). As a result, we get
for all \(p \in I\) and \(x,y \in H^{1}\). Now we consider the mappings \(j:[0,\infty)^{5} \to[0,\infty)\) and \(\alpha:H^{1}\times H^{1}\to[0,\infty)\) defined by \(\alpha(t,s) = 1\) and \(j(t_{1}, t_{2},t_{3},t_{4},t_{5})=\frac{\Delta}{3}(t_{1} +2t_{2})\). We can check that \(j \in\mathcal{R}\) and \(\mathcal{F}\) is a generalized α-contraction. With the help of Theorem 2.2, we conclude that \(\mathcal{F}\) possesses an approximate fixed point, which represents an approximate solution for the investigated problem (2). □
The next step is to study two applications to describe the reported results.
Example 1
Let us define \(\eta\in L^{\infty}([0,1])\) and \(\gamma ,\lambda:[0,1]\times[0,1]\to[0,\infty)\) by \(\eta(p)=\frac{\pi}{e^{(p+12)}}\), \(\gamma(p,s)=e^{ p-s}\) and \(\lambda(p,s)= \ln(5^{\sin(\pi p -s)})\). Then, we have \(\eta^{*}=\frac{\pi}{e^{12}}\), \(\gamma_{0}\leq e\), and \(\lambda_{0}\leq\ln5\). Let us consider \(\alpha=\frac{1}{5}\), \(\mu= \frac{1}{20}\), \(\beta =\frac{1}{4}\), \(\gamma=\frac{1}{2}\), \(\theta=\frac{3}{4}\), and \(\delta=\frac{3}{5}\). Consider the problem
with \(u_{1}^{\prime}(0)=0\). Considering \(f(p,x,y,w,u_{1},u_{2})= e^{-\pi(p+12)}(p+x+y+w+u_{1}+u_{2})\), we note that \(\Delta= [\eta^{*}(2+\gamma_{0} + \lambda_{0} + \frac{1}{(1-{\delta })^{2}})+{ \mu}( \frac{1}{(1-{\gamma})^{2}(1-{\beta})^{2}} )]<0/4447<1\). Now, by Theorem 3.11 problem (3) admits an approximate solution.
Example 2
Consider the function \(\lambda:[0,1] \times[0,1]\to[0,\infty)\) by \(\lambda(p,s )=\frac{e^{2p-s}}{e}\). Thus, \({\lambda_{0}\leq e}\). Let us consider \(\mu=3\), \(m=\frac{1}{2}\), \(\alpha=\frac{1}{4}\), \(\delta=\frac{1}{4}\), \(\theta=\frac{1}{2}\), \(\gamma=\frac{1}{2}\), \(\lambda= \frac{1}{200}\), \(\rho=\frac{1}{122}\), \(\xi_{1}=\frac{1}{320}\), \(\xi_{2}=\frac{1}{40}\), and \(\xi_{3}=\frac {1}{119}\). Let \(k(t)=\frac{2-p}{p+1}\), \(s(p)=\sin p\) and \(q(p)=\tan^{-1}(p)\). Then, \(M_{1}=\sup_{p\in[0,1]} \vert k(p) \vert =2\), \(M_{2}=\sup_{t\in[0,1]} \vert s(p) \vert =1\), and \(M_{3}=\sup_{t\in[0,1]} \vert q(p) \vert =\frac{\pi}{2} \). As a next step, we consider the problem
with \(x(0)=0\). Considering \(f(p,x_{1},y_{1},w_{1})= \frac{2}{56}p+\xi _{1}x_{1}+\xi_{2}y_{1}+\xi_{3}w_{1}\) for all \(p\in I\) and \(x_{1},y_{1},w_{1},v\in\mathbb{R}\), we note that
Now, by Theorem 3.12, problem (4) admits an approximate solution.
4 Conclusion
Fractional derivatives with nonsingular kernels started to be utilized from both theoretical and applied viewpoints. Particularly, the fractional Caputo-Fabrizio derivative was applied to models possessing memory effect of exponential type. Therefore, new generalizations of this operator should be investigated and applied to the dynamics of real-world problems. In this manuscript, we suggested a new operator called the infinite coefficient-symmetric CF fractional derivative. Besides, its properties were investigated, and two examples clearly show the advantages of the newly introduced concept.
References
Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999)
Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993)
Magin, RL: Fractional Calculus in Bioengineering. Begell House Publishers (2006)
Kilbas, A, Srivastava, HM, Trujillo, JJ: Theory and Application of Fractional Differential Equations. Mathematics Studies, vol. 204. North-Holland, Amsterdam (2006)
Hilfer, R: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Lorenzo, CF, Hartley, TT: Variable order and distributed order fractional operators. Nonlinear Dyn. 29, 57-98 (2002)
Agrawal, OP: Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368-374 (2002)
Doha, EH, Bhrawy, AH, Baleanu, D, Hafez, RM: A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations. Appl. Numer. Math. 77, 43-54 (2014)
Baleanu, D: About fractional quantization and fractional variational principles. Commun. Nonlinear Sci. Numer. Simul. 14(6), 2520-2523 (2009)
Agila, A, Baleanu, D, Eid, R, Irfanoglu, B: Applications of the extended fractional Euler-Lagrange equations model to freely oscillating dynamical systems. Rom. J. Phys. 61(3-4), 350-359 (2016)
Caputo, M, Fabrizio, M: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73-85 (2015)
Atangana, A: On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation. Appl. Math. Comput. 273(6), 948-956 (2016)
Atangana, A, Nieto, JJ: Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel. Adv. Mech. Eng. 7, 1-7 (2015)
Aydogan, SM, Baleanu, D, Mousalou, A, Rezapour, Sh: On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations. Adv. Differ. Equ. 2017, 221 (2017)
Baleanu, D, Mousalou, A, Rezapour, Sh: A new method for investigating some fractional integro-differential equations involving the Caputo-Fabrizio derivative. Adv. Differ. Equ. 2017, 51 (2017)
Gomez-Aguilar, JF, Yepez-Martinez, H, Calderon-Ramon, C, Cruz-Orduna, I, Escobar-Jimenez, RF, Olivares-Peregrino, VH: Modeling of a mass-spring-damper system by fractional derivatives with and without a singular kernel. Entropy 17(9), 6289-6303 (2015)
Doungmo, G, Emile, F, Pene, MK, Mwambakana, JN: Duplication in a model of rock fracture with fractional derivative without singular kernel. Open Math. 13, 839-846 (2015)
Atangana, A, Baleanu, D: Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer. J. Eng. Mech. 143(5), Article ID D4016005 (2017)
Kolade, QM, Atangana, A: Numerical approximation of nonlinear fractional parabolic differential equations with Caputo-Fabrizio derivative in Riemann-Liouville sense. Chaos Solitons Fractals 9, 171-179 (2017)
Baleanu, D, Agheli, B, Mohamed, QM: Fractional advection differential equation within Caputo and Caputo-Fabrizio derivatives. Adv. Mech. Eng. 8(12), Article ID 1687814016683305 (2016)
Losada, J, Nieto, JJ: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 87-92 (2015)
Miandaragh, MA, Postolache, M, Rezapour, Sh: Some approximate fixed point results for generalized α-contractive mappings. Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys. 75(2), 3-10 (2013)
Mayes, J: Reduction and approximation in large and infinite potential-driven flow networks. Thesis (Ph.D.), University of Notre Dame, Ann Arbor, MI (2012)
Mayes, J, Sen, M: Approximation of potential-driven flow dynamics in large-scale self-similar tree networks. Proc. R. Soc. Lond. Ser. A 467(2134), 2810-2824 (2011)
Acknowledgements
The second and third authors were supported by Azarbaijan Shahid Madani University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Baleanu, D., Mousalou, A. & Rezapour, S. On the existence of solutions for some infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential equations. Bound Value Probl 2017, 145 (2017). https://doi.org/10.1186/s13661-017-0867-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-017-0867-9