Well Conditioned Pseudospectral Schemes with Tunable Basis for Fractional Delay Differential Equations

Abstract

The main purpose of this work is to develop spectrally accurate and well conditioned pseudospectral schemes for solving fractional delay differential equations (FDDEs). The essential idea is to recast FDDEs into fractional integral equations (FIEs) and then discretize the FIEs via generalized fractional pseudospectral integration matrices (GFPIMs). We construct GFPIMs by employing the basis of weighted Lagrange interpolating functions, and provide an exact, efficient, and stable approach to computing GFPIMs. The GFPIM schemes have two remarkable features: (i) the endpoint singularity of the solution to FDDEs can be effectively captured via the tunable basis, and (ii) the linear system resulting from pseudospectral discretization is well conditioned. We also provide a rigorous convergence analysis for the particular FPIM schemes via a linear FIE with any $\gamma >0$ where $\gamma$ is the order of fractional integrals. Numerical results on benchmark FDDEs with smooth/singular solutions demonstrate the spectral rate of convergence for the GFPIM schemes. For FDDEs with piecewise smooth solutions, the GFPIM schemes can obtain accurate solutions but converge slowly due to their essential feature of “global” approximation on the entire time interval.

Appendices

Appendix A: Proof of Theorem 1

Proof

In order to calculate the integral in Eq. (11), we first employ the change of variables

$$\begin{aligned} s=\frac{\psi (t_{k})-t_{L}}{2}\sigma +\frac{\psi (t_{k})+t_{L}}{2}, \quad \sigma \in [-1,+1], \end{aligned}$$

(A.1)

to transform the integral interval $[t_L,\psi (t_k)]$ to the standard interval $[-1,+1]$. Then, we have

$$\begin{aligned} \psi (t_{k})-s=\frac{\psi (t_{k})-t_{L}}{2}(1-\sigma ). \end{aligned}$$

(A.2)

Substituting Eqs. (A.1) and (A.2) into Eq. (11) and taking account of Eqs. (1) and (12) yields

$$\begin{aligned} {_{t_{L}}^{\psi _{t}}}\varvec{I}_{ki}^{\gamma _{\grave{\varrho }}}= & {} \, \frac{1}{\Gamma (\gamma )}\int _{t_{L}}^{\psi (t_{k})} (\psi (t_{k})-s)^{\gamma -1}{\mathcal {L}}_{i}^{\grave{\varrho }}(s)\,\mathrm {d}s\nonumber \\= & {} \,\frac{(\psi (t_{k})-t_{L})^{\gamma +\varrho }}{\Gamma (\gamma ) 2^{\gamma +\varrho }(t_{i}-t_{L})^\varrho }\int _{-1}^{+1}(1-\sigma ) ^{\gamma -1}(1+\sigma )^{\varrho }\,\cdot {\mathcal {L}}_{i}(\sigma ;t_{L},\psi (t_{k}))\,\mathrm {d}\sigma . \end{aligned}$$

(A.3)

Recalling that ${\mathcal {L}}_{i}(\sigma ;t_{L},\psi (t_{k}))\in {\mathcal {P}}_{N-1}$, the integral in the last equality of Eq. (A.3) can then be calculated exactly using the JG quadrature [33, p. 80]. Thus, Eq. (15a) follows. In the same way, Eq. (15b) can be obtained. $\square$

Appendix B: Proof of Theorem 5

Proof

Using Eqs. (45) and (43c), we have

$$\begin{aligned} \Vert e(\tau )\Vert _{\infty }\,\le \,C\,\Vert \zeta (\tau )\Vert _{\infty }\, \le \,C\sum _{j=1}^2\Vert {\mathcal {E}}_{j}(\tau )\Vert _{\infty }. \end{aligned}$$

(B.1)

Now, it follows from Lemma 2 that

$$\begin{aligned} \begin{aligned} \Vert {\mathcal {E}}_{1}(\tau )\Vert _{\infty }\,&=\,\Vert g(\tau ) -{\mathcal {T}}_{N}^{\mathcal {L}}g(\tau )\Vert _{\infty } \\&\le {\left\{ \begin{array}{ll} CN^{\frac{1}{2}-m}\log N|g(\tau )|_{H_{\omega ^{c}}^{m;N} (-1,+1)}, &{}-1<\alpha ,\beta <-\frac{1}{2} \\ CN^{1+\max (\alpha ,\beta )-m}|g(\tau )|_{H_{\omega ^{c}}^{m;N} (-1,+1)}, &{}-\frac{1}{2}\le \alpha ,\beta \le 0 \end{array}\right. }. \end{aligned} \end{aligned}$$

(B.2)

Next, we estimate $\Vert {\mathcal {E}}_{2}(\tau )\Vert _{\infty }$. It is clear from Eq. (40a) that $e(\tau )\in C[-1,+1]$ as $g(\tau )$ is assumed to be sufficiently smooth. Consequently, using Lemma 3 and Theorem 4, we have

$$\begin{aligned} \Vert {\mathcal {E}}_{2}(\tau )\Vert _{\infty }\,\le & {} \,C_{\lfloor \gamma \rfloor , \varepsilon }N^{-(\lfloor \gamma \rfloor +\varepsilon )}\Vert {_{-1}}{\mathcal {I}}_{\tau }^{\gamma } e(\sigma )\Vert _{\lfloor \gamma \rfloor ,\varepsilon }\nonumber \\\le & {} \,CN^{-(\lfloor \gamma \rfloor +\varepsilon )}\Vert e(\tau )\Vert _{\infty }, \quad \alpha ,\beta >-1. \end{aligned}$$

(B.3)

Thus, combining Eqs. (B.1)–(B.3) and taking account of N sufficiently large leads to Eq. (46). $\square$

Appendix C: Proof of Theorem 6

Proof

It follows from Eq. (45) and the generalized Hardy inequality (see, e.g., [34]) that

$$\begin{aligned} \Vert e(\tau )\Vert _{\omega ^{(\alpha ,\beta )}}\le \, C\sum _{j=1}^{2}\Vert {\mathcal {E}}_{j}(\tau )\Vert _{\omega ^{(\alpha ,\beta )}}. \end{aligned}$$

(C.1)

Now, using Lemma 2, we have

$$\begin{aligned} \Vert {\mathcal {E}}_{1}(\tau )\Vert _{\omega ^{(\alpha ,\beta )}}\le \, CN^{-m}|g(\tau )|_{H_{\omega ^{(\alpha ,\beta )}}^{m;N} (-1,+1)}, \quad \alpha ,\beta >-1. \end{aligned}$$

(C.2)

Next, it follows from Eq. (B.3) that

$$\begin{aligned} \Vert {\mathcal {E}}_{2}(\tau )\Vert _{\omega ^{(\alpha ,\beta )}}\,\le & {} \, \Vert {\mathcal {E}}_{2}(\tau )\Vert _\infty \nonumber \\\le & {} \,CN^{-(\lfloor \gamma \rfloor +\varepsilon )}\Vert e(\tau )\Vert _{\infty }, \quad \alpha ,\beta >-1. \end{aligned}$$

(C.3)

Thus, combining Eqs. (C.1)–(C.3) and recalling the estimate of $\Vert e(\tau )\Vert _{\infty }$ in Eq. (46) leads to Eq. (47) provided that N is sufficient large. $\square$