A Spectral Collocation Method for Nonlinear Fractional Boundary Value Problems with a Caputo Derivative

Abstract

In this paper, we consider the nonlinear boundary value problems involving the Caputo fractional derivatives of order $\alpha \in (1,2)$ on the interval (0, T). We present a Legendre spectral collocation method for the Caputo fractional boundary value problems. We derive the error bounds of the Legendre collocation method under the $L^2$- and $L^\infty$-norms. Numerical experiments are included to illustrate the theoretical results.

Appendix

In this appendix, we verify the existence and uniqueness of scheme (3.7) with the Lipschitz constant $L<\frac{\Gamma (\alpha +1)}{2T^{\alpha }}$. We consider the following iteration process:

$$\begin{aligned} \begin{aligned} U^m(x)=&\,\displaystyle \frac{T^{\alpha }}{4^{\alpha }\Gamma (\alpha )}\mathcal {I}_{x,N}\left[ (x+1)^{\alpha }\displaystyle \int _{-1}^1\big (1-\theta \big )^{\alpha -1} \mathcal {I}_{\theta ,N}^{\alpha -1,0}F\Big (\xi (x,\theta ),U^{m-1}\big (\xi (x,\theta )\big )\Big )d\theta \right] \\&-\displaystyle \frac{T^{\alpha }(x+1)}{2^{\alpha +1}\Gamma (\alpha )}\int _{-1}^1\big (1-\lambda \big )^{\alpha -1}\mathcal {I}_{\lambda ,N}^{\alpha -1,0}F(\lambda ,U^{m-1}(\lambda ))d\lambda . \end{aligned}\end{aligned}$$

(5.5)

Let $\widetilde{U}^m(x)=U^m(x)-U^{m-1}(x).$ Then we have from (5.5) and (4.8) that

$$\begin{aligned} \begin{aligned} \widetilde{U}^m(x)&=\displaystyle \frac{T^{\alpha }}{2^{\alpha }\Gamma (\alpha )}\mathcal {I}_{x,N}\left[ \displaystyle \int _{-1}^{x}\big (x-\xi \big )^{\alpha -1} {}_{x}\widetilde{\mathcal {I}}_{\xi ,N}^{\alpha -1,0}\Big (F\big (\xi ,U^{m-1}(\xi )\big )-F\big (\xi ,U^{m-2}(\xi )\big )\Big )d\xi \right] \\&\quad -\displaystyle \frac{T^{\alpha }(x+1)}{2^{\alpha +1}\Gamma (\alpha )}\int _{-1}^1\big (1-\lambda \big )^{\alpha -1}\mathcal {I}_{\lambda ,N}^{\alpha -1,0}\Big (F(\lambda ,U^{m-1}(\lambda ))-F(\lambda ,U^{m-2}(\lambda ))\Big )d\lambda \\&=:A_1+A_2, \end{aligned}\end{aligned}$$

(5.6)

where

$$\begin{aligned} A_1= & {} \displaystyle \frac{T^{\alpha }}{2^{\alpha }\Gamma (\alpha )}\mathcal {I}_{x,N}\left[ \displaystyle \int _{-1}^{x}\big (x-\xi \big )^{\alpha -1} {}_{x}\widetilde{\mathcal {I}}_{\xi ,N}^{\alpha -1,0}\Big (F\big (\xi ,U^{m-1}(\xi )\big )-F\big (\xi ,U^{m-2}(\xi )\big )\Big )d\xi \right] ,\\ A_2= & {} -\displaystyle \frac{T^{\alpha }(x+1)}{2^{\alpha +1}\Gamma (\alpha )}\int _{-1}^1\big (1-\lambda \big )^{\alpha -1}\mathcal {I}_{\lambda ,N}^{\alpha -1,0}\Big (F(\lambda ,U^{m-1}(\lambda ))-F(\lambda ,U^{m-2}(\lambda ))\Big )d\lambda .\end{aligned}$$

By (3.3), (4.4) and the Cauchy–Schwarz inequality, we derive that

$$\begin{aligned}\begin{aligned}&\Vert A_1\Vert \\&\quad = \displaystyle \frac{T^{\alpha }}{2^{\alpha }\Gamma (\alpha )}\Big \Vert \mathcal {I}_{x,N}\displaystyle \int _{-1}^x(x-\xi )^{\alpha -1} {}_x\widetilde{\mathcal {I}}_{\xi ,N}^{\alpha -1,0}\big (F(\xi ,U^{m-1}(\xi ))-F(\xi ,U^{m-2}(\xi ))\big )d\xi \Big \Vert \\ {}&\quad = \displaystyle \frac{T^{\alpha }}{2^{\alpha }\Gamma (\alpha )}\left[ \displaystyle \sum _{j=0}^N\omega _j\left( \displaystyle \int _{-1}^{x_j}(x_j-\xi )^{\alpha -1} {}_{x_j}\widetilde{\mathcal {I}}_{\xi ,N}^{\alpha -1,0}\big (F(\xi ,U^{m-1}(\xi ))-F(\xi ,U^{m-2}(\xi ))\big )d\xi \right) ^2\right] ^{\frac{1}{2}}\\&\quad \le \displaystyle \frac{T^{\alpha }}{2^{\alpha }\Gamma (\alpha )}\left[ \displaystyle \sum _{j=0}^N\omega _j\displaystyle \int _{-1}^{x_j}(x_j-\xi )^{\alpha -1}d\xi \displaystyle \int _{-1}^{x_j}(x_j-\xi )^{\alpha -1}\Big |{}_{x_j}\widetilde{\mathcal {I}}_{\xi ,N}^{\alpha -1,0}\big (F(\xi ,U^{m-1}(\xi ))\right. \\&\left. \qquad -F(\xi ,U^{m-2}(\xi ))\big )\Big |^2d\xi \right] ^{\frac{1}{2}} \\ {}&\quad = \displaystyle \frac{T^{\alpha }}{2^{\alpha }\Gamma (\alpha )}\left[ \displaystyle \sum _{j=0}^N\omega _j\displaystyle \frac{(x_j+1)^{2\alpha }}{2^\alpha \alpha } \displaystyle \sum _{k=0}^{N}\Big |F(\xi _k^{\alpha -1,0},U^{m-1}(\xi _k^{\alpha -1,0}))\right. \\&\left. \qquad -F(\xi _k^{\alpha -1,0},U^{m-2}(\xi _k^{\alpha -1,0}))\Big |^2 \omega _k^{\alpha -1,0}\right] ^{\frac{1}{2}}. \end{aligned}\end{aligned}$$

Therefore, by (4.4), (4.13) and the Lipschitz condition, we further deduce that for $L<\frac{\Gamma (\alpha +1)}{2T^{\alpha }},$

$$\begin{aligned} \Vert A_1\Vert&\le \displaystyle \frac{LT^{\alpha }}{2^{\alpha }\Gamma (\alpha )}\left[ \displaystyle \sum _{j=0}^N\omega _j\displaystyle \frac{(x_j+1)^{2\alpha }}{2^\alpha \alpha } \displaystyle \sum _{k=0}^{N}\Big |U^{m-1}(\xi _k^{\alpha -1,0})-U^{m-2}(\xi _k^{\alpha -1,0})\Big |^2\omega _k^{\alpha -1,0}\right] ^{\frac{1}{2}}\nonumber \\&\le \displaystyle \frac{\alpha }{2^{\alpha +1}}\left[ \displaystyle \sum _{j=0}^N\omega _j\displaystyle \frac{(x_j+1)^{\alpha }}{\alpha }\displaystyle \int _{-1}^{x_j}(x_j-\xi )^{\alpha -1}\Big |{}_{x_j}\widetilde{\mathcal {I}}_{\xi ,N}^{\alpha -1,0}\widetilde{U}^{m-1}(\xi )\Big |^2d\xi \right] ^{\frac{1}{2}}\nonumber \\&\le \displaystyle \frac{\alpha }{2^{\alpha +1}}\left( \displaystyle \sum _{j=0}^N\omega _j\displaystyle \frac{(x_j+1)^{\alpha }}{\alpha }\right) ^{\frac{1}{2}}\max _{0 \le j \le N} \left( \displaystyle \int _{-1}^{x_j}(x_j-\xi )^{\alpha -1}\Big |{}_{x_j}\widetilde{\mathcal {I}}_{\xi ,N}^{\alpha -1,0}\widetilde{U}^{m-1}(\xi )\Big |^2d\xi \right) ^{\frac{1}{2}}\nonumber \\&= \displaystyle \frac{\alpha }{2^{\alpha +1}}\left( \displaystyle \sum _{j=0}^N\omega _j\displaystyle \frac{(x_j+1)^{\alpha }}{\alpha }\right) ^{\frac{1}{2}}\max _{0 \le j \le N} \left( \displaystyle \int _{-1}^{x_j}(x_j-\xi )^{\alpha -1}\Big |\widetilde{U}^{m-1}(\xi )\Big |^2d\xi \right) ^{\frac{1}{2}}\nonumber \\&\le \displaystyle \frac{\alpha }{2^{\alpha +1}}\sqrt{\displaystyle \frac{8}{3\alpha }}\max _{0 \le j \le N} \left( \displaystyle \int _{-1}^{x_j}(x_j-\xi )^{\alpha -1}\Big |\widetilde{U}^{m-1}(\xi )\Big |^2d\xi \right) ^{\frac{1}{2}}\nonumber \\&\le \displaystyle \frac{\alpha }{2^{\alpha +1}}\sqrt{\displaystyle \frac{2^{\alpha +2}}{3\alpha }}\left( \displaystyle \int _{-1}^1\Big |\widetilde{U}^{m-1}(\xi )\Big |^2d\xi \right) ^{\frac{1}{2}}\nonumber \\&= \sqrt{\displaystyle \frac{\alpha }{3\times 2^\alpha }}\Vert \widetilde{U}^{m-1}\Vert \nonumber . \end{aligned}$$

We next estimate the term $\Vert A_2\Vert$. By the Cauchy–Schwarz inequality, we know that

$$\begin{aligned}\begin{aligned} \Vert A_2\Vert&=\displaystyle \frac{T^{\alpha }\Vert x+1\Vert }{2^{\alpha +1}\Gamma (\alpha )} \Big |\displaystyle \int _{-1}^1\big (1-\lambda \big )^{\alpha -1}\mathcal {I}_{\lambda ,N}^{\alpha -1,0} \Big (F(\lambda ,U^{m-1}(\lambda ))-F(\lambda ,U^{m-2}(\lambda ))\Big )d\lambda \Big |\\&\le \displaystyle \frac{T^{\alpha }}{2^{\alpha +1}\Gamma (\alpha )}\sqrt{\displaystyle \frac{2^{\alpha +3}}{3\alpha }} \Big [\displaystyle \int _{-1}^1\big (1-\lambda \big )^{\alpha -1}\Big |\mathcal {I}_{\lambda ,N}^{\alpha -1,0} \Big (F(\lambda ,U^{m-1}(\lambda ))-F(\lambda ,U^{m-2}(\lambda ))\Big )\Big |^2d\lambda \Big ]^{\frac{1}{2}}. \end{aligned}\end{aligned}$$

The previous result, along with (3.3) and the Lipschitz condition, yields

$$\begin{aligned} \Vert A_2\Vert&\le \displaystyle \frac{T^{\alpha }}{2^{\alpha +1}\Gamma (\alpha )}\sqrt{\displaystyle \frac{2^{\alpha +3}}{3\alpha }} \left( \displaystyle \sum _{j=0}^{N}\omega _j^{\alpha -1,0}\Big |F\left( x_j^{\alpha -1,0},U^{m-1}(x_j^{\alpha -1,0})\right) \right. \nonumber \\&\left. \quad -F\left( x_j^{\alpha -1,0},U^{m-2}(x_j^{\alpha -1,0})\right) \Big |^2\right) ^{\frac{1}{2}}\nonumber \\&\le \displaystyle \frac{LT^{\alpha }}{2^{\alpha +1}\Gamma (\alpha )}\sqrt{\displaystyle \frac{2^{\alpha +3}}{3\alpha }} \left( \displaystyle \sum _{j=0}^{N}\omega _j^{\alpha -1,0}\Big |\widetilde{U}^{m-1}(x_j^{\alpha -1,0})\Big |^2\right) ^{\frac{1}{2}}\nonumber \\&= \displaystyle \frac{LT^{\alpha }}{2^{\alpha +1}\Gamma (\alpha )}\sqrt{\displaystyle \frac{2^{\alpha +3}}{3\alpha }} \left( \displaystyle \int ^1_{-1}\big (1-\lambda \big )^{\alpha -1}\Big |\widetilde{U}^{m-1}(\lambda )\Big |^2d\lambda \right) ^{\frac{1}{2}}\nonumber \\&\le \sqrt{\displaystyle \frac{\alpha }{12}} \Vert \widetilde{U}^{m-1}\Vert ,\qquad \forall \alpha \in (1,2). \end{aligned}$$

(5.7)

Hence

$$\begin{aligned} \Vert \widetilde{U}^m\Vert \le \left( \sqrt{\displaystyle \frac{\alpha }{3\times 2^\alpha }}+\sqrt{\displaystyle \frac{\alpha }{12}}\right) \Vert \widetilde{U}^{m-1}\Vert . \end{aligned}$$

Since

$$\begin{aligned} \sqrt{\displaystyle \frac{\alpha }{3\times 2^\alpha }}+\sqrt{\displaystyle \frac{\alpha }{12}}<1,\qquad \forall \alpha \in (1,2), \end{aligned}$$

we have $\Vert \widetilde{U}^m\Vert \rightarrow 0$ as $m\rightarrow \infty .$ This implies the existence of solution of (3.7). It is easy to prove the uniqueness of solution of (3.7).