Solving Abel integral equations of first kind via fractional calculus

Definition 2.1

Let $\alpha >0$ with $n-1<\alpha ⩽n\text{,}n\in \mathbb{N}$ , and $a<x<b$ . The left and right Riemann–Liouville fractional integrals of order α of a given function f are defined by $${}_a{J}_x^{\alpha }f(x)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_a^x{(x-t)}^{\alpha -1}f(t)\mathit{dt}$$ and $${}_x{J}_b^{\alpha }f(x)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_x^b{(t-x)}^{\alpha -1}f(t)\mathit{dt}\text{,}$$ respectively, where Γ is Euler’s gamma function, that is, $$\mathrm{\Gamma }(x)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}\mathit{dt}\text{.}$$

Definition 2.2

The left and right Riemann–Liouville fractional derivatives of order $\alpha >0$ , $n-1<\alpha ⩽n\text{,}n\in \mathbb{N}$ , are defined by $${}_a{D}_x^{\alpha }f(x)=\frac{1}{\mathrm{\Gamma }(n-\alpha )}\frac{d^n}{{\mathit{dx}}^n}{\int }_a^x{(x-t)}^{n-\alpha -1}f(t)\mathit{dt}$$ and $${}_x{D}_b^{\alpha }f(x)=\frac{{(-1)}^n}{\mathrm{\Gamma }(n-\alpha )}\frac{d^n}{{\mathit{dx}}^n}{\int }_x^b{(t-x)}^{n-\alpha -1}f(t)\mathit{dt}\text{,}$$ respectively.

Definition 2.3

The left and right Caputo fractional derivatives of order $\alpha >0$ , $n-1<\alpha ⩽n\text{,}n\in \mathbb{N}$ , are defined by $${{}_a{}^{C}D}_x^{\alpha }f(x)=\frac{1}{\mathrm{\Gamma }(n-\alpha )}{\int }_a^x{(x-t)}^{n-\alpha -1}{f}^{(n)}(t)\mathit{dt}$$ and $${{}_x{}^{C}D}_b^{\alpha }f(x)=\frac{{(-1)}^n}{\mathrm{\Gamma }(n-\alpha )}{\int }_x^b{(t-x)}^{n-\alpha -1}{f}^{(n)}(t)\mathit{dt}\text{,}$$ respectively.

Definition 2.4

Let $\alpha >0$ . The Grunwald–Letnikov fractional derivatives are defined by $$D^{\alpha }f(x)={\displaystyle \underset{h\to 0}{\lim }}{h}^{-\alpha }{\displaystyle \sum _{r=0}^{\infty }}{(-1)}^r\left(\begin{array}{c}\alpha \hfill \\ r\hfill \end{array}\right)f(x-\mathit{rh})$$ and $$D^{-\alpha }f(x)={\displaystyle \underset{h\to 0}{\lim }}{h}^{\alpha }{\displaystyle \sum _{r=0}^{\infty }}\left[\begin{array}{c}\hfill \alpha \hfill \\ \hfill r\hfill \end{array}\right]f(x-\mathit{rh})\text{,}$$ where $$\left[\begin{array}{c}\hfill \alpha \hfill \\ \hfill r\hfill \end{array}\right]=\frac{\alpha (\alpha +1)(\alpha +2)\cdots (\alpha +r-1)}{r!}\text{.}$$

Remark 2.5

The Caputo derivatives (Definition 2.3) have some advantages over the Riemann–Liouville derivatives (Definition 2.2). The most well known is related with the Laplace transform method for solving fractional differential equations. The Laplace transform of a Riemann–Liouville derivative leads to boundary conditions containing the limit values of the Riemann–Liouville fractional derivative at the lower terminal $x=a$ . In spite of the fact that such problems can be solved analytically, there is no physical interpretation for such a type of boundary conditions. In contrast, the Laplace transform of a Caputo derivative imposes boundary conditions involving integer-order derivatives at $x=a$ , which usually are acceptable physical conditions. Another advantage is that the Caputo derivative of a constant function is zero, whereas for the Riemann–Liouville it is not. For details see Sousa (2012).

Remark 2.6

The Grunwald–Letnikov definition gives a generalization of the ordinary discretization formulas for derivatives with integer order. The series in Definition 2.4 converge absolutely and uniformly for each $\alpha >0$ and for every bounded function f. The discrete approximations derived from the Grunwald–Letnikov fractional derivatives, e.g., (1.2), present some limitations. First, they frequently originate unstable numerical methods and henceforth usually a shifted Grunwald–Letnikov formula is used instead. Another disadvantage is that the order of accuracy of such approximations is never higher than one. For details see Baker (1977).

Theorem 3.1

[See Odibat, 2006, 2009] Let $b>0\text{,}\alpha >0$ , and suppose that the interval $[0\text{,}b]$ is subdivided into k subintervals $[x_j\text{,}{x}_{j+1}]\text{,}j=0\text{,}\dots \text{,}k-1$ , of equal distances $h=\frac{b}{k}$ by using the nodes $x_j=\mathit{jh}\text{,}j=0\text{,}1\text{,}\dots \text{,}k$ . Then the modified trapezoidal rule $$\begin{array}{cc}\hfill & T(f\text{,}h\text{,}\alpha )=\frac{h^{\alpha }}{\mathrm{\Gamma }(\alpha +2)}({(k-1)}^{\alpha +1}-(k-\alpha -1)k^{\alpha })f(0)+f(b)\hfill \\ \hfill & +\frac{h^{\alpha }}{\mathrm{\Gamma }(\alpha +2)}{\displaystyle \sum _{j=1}^{k-1}}\left({(k-j+1)}^{\alpha +1}-2{(k-j)}^{\alpha +1}+{(k-j-1)}^{\alpha +1}\right)f(x_j)\hfill \end{array}$$ is an approximation to the fractional integral ${}_0{J}_x^{\alpha }f{|}_{x=b}$ : $${}_0{J}_x^{\alpha }f{|}_{x=b}=T(f\text{,}h\text{,}\alpha )-E_T(f\text{,}h\text{,}\alpha )\text{.}$$ Furthermore, if $f\in C^2[0\text{,}b]$ , then $$|E_T(f\text{,}h\text{,}\alpha )|⩽c_{\alpha }^{\prime }\Vert f^{″}{\Vert }_{\infty }{b}^{\alpha }{h}^2=O(h^2)\text{,}$$ where $c_{\alpha }^{\prime }$ is a constant depending only on α.

Theorem 3.2

[See Odibat, 2006, 2009] Let $b>0\text{,}\alpha >0$ with $n-1<\alpha ⩽n\text{,}n\in \mathbb{N}$ , and suppose that the interval $[0\text{,}b]$ is subdivided into k subintervals $[x_j\text{,}{x}_{j+1}]\text{,}j=0\text{,}\dots \text{,}k-1$ , of equal distances $h=\frac{b}{k}$ by using the nodes $x_j=\mathit{jh}\text{,}j=0\text{,}1\text{,}\dots \text{,}k$ . Then the modified trapezoidal rule $$\begin{array}{cc}\hfill & C(f\text{,}h\text{,}\alpha )=\frac{h^{n-\alpha }}{\mathrm{\Gamma }(n+2-\alpha )}[\left({(k-1)}^{n-\alpha +1}-(k-n+\alpha -1)k^{n-\alpha }\right)f^{(n)}(0)\hfill \\ \hfill & +f^{(n)}(b)+{\displaystyle \sum _{j=1}^{k-1}}\left({(k-j+1)}^{n-\alpha +1}-2{(k-j)}^{n-\alpha +1}+{(k-j-1)}^{n-\alpha +1}\right)f^{(n)}(x_j)]\hfill \end{array}$$ is an approximation to the Caputo fractional derivative ${{}_0{}^{C}D}_x^{\alpha }f{|}_{x=b}$ : $${{}_0{}^{C}D}_x^{\alpha }f{|}_{x=b}=C(f\text{,}h\text{,}\alpha )-E_C(f\text{,}h\text{,}\alpha )\text{.}$$ Furthermore, if $f\in C^{n+2}[0\text{,}b]$ , then $$|E_C(f\text{,}h\text{,}\alpha )|⩽c_{n-\alpha }^{\prime }\Vert f^{(n+2)}{\Vert }_{\infty }{b}^{n-\alpha }{h}^2=O(h^2)\text{,}$$ where $c_{n-\alpha }^{\prime }$ is a constant depending only on α.

Theorem 4.1

The solution to problem (4.1) is

$$g(x)=\frac{\sin (\alpha \pi )}{\pi }{\int }_0^x\frac{f^{\prime }(t)}{{(x-t)}^{1-\alpha }}\mathit{dt}\text{.}$$

Proof

According to Definition 2.1, we can write (4.1) in the equivalent form $$f(x)=\mathrm{\Gamma }(1-\alpha ){}_0{J}_x^{1-\alpha }g(x)\text{.}$$ Then, using (2.1), it follows that

$${{}_0{}^{C}D}_x^{1-\alpha }f(x)=\mathrm{\Gamma }(1-\alpha )g(x)\text{.}$$ Therefore, $$g(x)=\frac{1}{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(1-\alpha )}{\int }_0^x\frac{f^{\prime }(t)}{{(x-t)}^{1-\alpha }}\mathit{dt}=\frac{\sin (\alpha \pi )}{\pi }{\int }_0^x\frac{f^{\prime }(t)}{{(x-t)}^{1-\alpha }}\mathit{dt}\text{,}$$ where we have used the identity $\pi =\sin (\alpha \pi )\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(1-\alpha )$ .

Theorem 4.2

Let $0<x<b$ and suppose that the interval $[0\text{,}x]$ is subdivided into k subintervals $[t_j\text{,}{t}_{j+1}]\text{,}j=0\text{,}\dots \text{,}k-1$ , of equal length $h=\frac{x}{k}$ by using the nodes $t_j=\mathit{jh}\text{,}j=0\text{,}\dots \text{,}k$ . An approximate solution $\stackrel{\sim }{g}$ to the solution g of the Abel integral Eq. (4.1) is given by

$$\stackrel{\sim }{g}(x)=\frac{h^{\alpha }}{\mathrm{\Gamma }(1-\alpha )\mathrm{\Gamma }(2+\alpha )}[\left({(k-1)}^{1+\alpha }-(k-1-\alpha )k^{\alpha }\right)f^{\prime }(0)+f^{\prime }(x)+{\displaystyle \sum _{j=1}^{k-1}}\left({(k-j+1)}^{1+\alpha }-2{(k-j)}^{1+\alpha }+{(k-j-1)}^{1+\alpha }\right)f^{\prime }(t_j)]\text{.}$$ Moreover, if $f\in C^3[0\text{,}x]$ , then $g(x)=\stackrel{\sim }{g}(x)-\frac{1}{\mathrm{\Gamma }(1-\alpha )}E(x)$ with $$|E(x)|⩽c_{\alpha }^{\prime }\Vert f^{‴}{\Vert }_{\infty }{x}^{\alpha }{h}^2=O(h^2)\text{,}$$ where $c_{\alpha }^{\prime }$ is a constant depending only on α and $\Vert f^{‴}{\Vert }_{\infty }={\max }_{t\in [0\text{,}x]}|f^{‴}(t)|$ .

Proof

We want to approximate the Caputo derivative in (4.3), i.e., to approximate $$g(x)=\frac{1}{\mathrm{\Gamma }(1-\alpha )}{{}_0{}^{C}D}_x^{1-\alpha }f(x)=\frac{\sin (\alpha \pi )\mathrm{\Gamma }(\alpha )}{\pi }{{}_0{}^{C}D}_x^{1-\alpha }f(x)\text{.}$$ If the Caputo fractional derivative of order $1-\alpha$ for $f\text{,}0<\alpha <1$ , is calculated at collocation nodes $t_j\text{,}j=0\text{,}\dots \text{,}k$ , then the result is a direct consequence of Theorem 3.2. □

Example 5.1

Consider the Abel integral equation

$$e^x-1={\int }_0^x\frac{g(t)}{{(x-t)}^{1/2}}\mathit{dt}\text{.}$$ The exact solution to (5.1) is given by Theorem 4.1:

(5.2)

$$g(x)=\frac{e^x}{\sqrt{\pi }}\mathit{erf}(\sqrt{x})\text{,}$$ where $\mathit{erf}(x)$ is the error function, that is, $$\mathit{erf}(x)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-t^2}\mathit{dt}\text{.}$$ In Table 1 we present the approximate values for $g(x_i)$ , $x_i=0.1\text{,}0.2\text{,}0.3$ , obtained from Theorem 4.2 with $k=1\text{,}10\text{,}100$ .

Example 5.2

Consider the following Abel integral equation of first kind:

$$x={\int }_0^x\frac{g(t)}{{(x-t)}^{4/5}}\mathit{dt}\text{.}$$ The exact solution (4.2) of (5.3) takes the form

(5.4)

$$g(x)=\frac{5}{4}\frac{\sin \left(\frac{\pi }{5}\right)}{\pi }{x}^{4/5}\text{.}$$ We can see the numerical approximations of $g(x_i)$ , $x_i=0.4\text{,}0.5\text{,}0.6$ , obtained from Theorem 4.2 with $k=1\text{,}10$ in Table 2.

Example 5.3

Consider now the Abel integral equation

$$x^{7/6}={\int }_0^x\frac{g(t)}{{(x-t)}^{1/3}}\mathit{dt}\text{.}$$ Theorem 4.2 asserts that

(5.6)

$$g(x)=\frac{7\sqrt{3}}{12\pi }{\int }_0^x\frac{t^{1/6}}{{\left(x-t\right)}^{2/3}}\mathit{dt}\text{.}$$ The numerical approximations of $g(x_i)\text{,}{x}_i=0.6\text{,}0.7\text{,}0.8$ , obtained from Theorem 4.2 with $k=10\text{,}100\text{,}1000$ , are given in Table 3.