A predictor–corrector scheme for the tempered fractional differential equations with uniform and non-uniform meshes

Abstract

Tempered fractional derivatives and the corresponding tempered fractional differential equations have played a key role in physical science. In this paper, for solving the tempered fractional ordinary differential equation, the predictor–corrector (PC) methods with uniform and non-uniform meshes of Deng et al. (Numer Algorithms 74(3):717–754, 2017) are developed, by using the piecewise quadratic interpolation polynomial. The error bounds of proposed predictor–corrector schemes with uniform and equidistributing meshes are obtained. We proved that the presented numerical method has a higher-order convergence order $O(h^3)$. Also, some numerical examples are constructed to demonstrate the efficacy and usefulness of the numerical methods. Finally, the results of PC schemes with uniform and non-uniform given in Deng et al. (2017) and presented schemes (improved PC with uniform and non-uniform meshes) are compared for different values of parameters.

Appendix: Proofs of Theorems

1.1 Proof of Theorem 1

Proof

By using Lemmas 1 and 2 and the Lipschitz property of f (2.6), let $j=n+1$, for predictor formula (3.10), and we can write

$$\begin{aligned} \begin{aligned} \left| {u({t_{n + 1}}) - u_{n + 1}^P} \right|&= \frac{1}{{\varGamma (\alpha )}}\left| {\int _0^{{t_{n + 1}}} {{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - \tau )}}{{({t_{n + 1}} - \tau )}^{\alpha - 1}}f(\tau ,u(\tau )){\mathrm{d}}\tau - \sum \limits _{j = 0}^n {{c_{j,n + 1}}{f_j}} } } \right| \\&= \frac{1}{{\varGamma (\alpha )}}\left| {\int _0^{{t_{n + 1}}} {{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - \tau )}}{{({t_{n + 1}} - \tau )}^{\alpha - 1}}f(\tau ,u(\tau )){\mathrm{d}}\tau - \sum \limits _{j = 0}^n {{c_{j,n + 1}}{f_j}} } } \right. \\&\qquad \left. { - \sum \limits _{j = 0}^n {{c_{j,n + 1}}f({t_j},u({t_j}))} + \sum \limits _{j = 0}^n {{c_{j,n + 1}}f({t_j},u({t_j}))} } \right| \\&= \frac{1}{{\varGamma (\alpha )}}\left| {\int _0^{{t_{n + 1}}} {{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - \tau )}}{{({t_{n + 1}} - \tau )}^{\alpha - 1}}f(\tau ,u(\tau )){\mathrm{d}}\tau - \sum \limits _{j = 0}^n {{c_{j,n + 1}}f({t_j},u({t_j}))} } } \right. \\&\qquad \left. { + \sum \limits _{j = 0}^n {{c_{j,n + 1}}(f({t_j},u({t_j})) - {f_j})} } \right| \le \frac{1}{{\varGamma (\alpha )}}\left| {\int _0^{{t_{n + 1}}} {{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - \tau )}}{{({t_{n + 1}} - \tau )}^{\alpha - 1}}f(\tau ,u(\tau )){\mathrm{d}}\tau } } \right. \\&\qquad \left. { - \sum \limits _{j = 0}^n {{c_{j,n + 1}}f({t_j},u({t_j}))} } \right| + \frac{1}{{\varGamma (\alpha )}}\left| {\sum \limits _{j = 0}^n {{c_{j,n + 1}}(f({t_j},u({t_j})) - {f_j})} } \right| \\&\le \frac{{C{h^3}}}{{\varGamma (\alpha )}} + \frac{L}{{\varGamma (\alpha )}}\sum \limits _{j = 0}^n {\left| {{c_{j,n + 1}}} \right| } \left| {y({t_j}) - {y_j}} \right| \le {q_1}{h^3} + {q_2}\mathop {\max }\limits _{0 \le j \le n} \left| {y({t_j}) - {y_j}} \right| ; \end{aligned} \end{aligned}$$

(6.1)

then for corrector formula (3.4), we can write

$$\begin{aligned} \begin{aligned} \left| {u({t_{n + 1}}) - {u_{n + 1}}} \right|&= \frac{1}{{\varGamma (\alpha )}}\left| {\int _0^{{t_{n + 1}}} {{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - \tau )}}{{({t_{n + 1}} - \tau )}^{\alpha - 1}}f(\tau ,u(\tau )){\mathrm{d}}\tau } } \right. \\&\quad \left. -\, \left( \sum \limits _{j = 0}^n {d_{j,n + 1}}{f_j} + {d_{n + 1,n + 1}}f({t_{n + 1}},u_{n + 1}^P)\right) \right| \\&=\frac{1}{{\varGamma (\alpha )}}\left| {\int _0^{{t_{n + 1}}} {{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - \tau )}}{{({t_{n + 1}} - \tau )}^{\alpha - 1}}f(\tau ,u(\tau )){\mathrm{d}}\tau } } \right. \\&\quad -\, \left( \sum \limits _{j = 0}^n {d_{j,n + 1}}{f_j} + {d_{n + 1,n + 1}}f({t_{n + 1}},u_{n + 1}^P)\right) - \sum \limits _{j = 0}^{n + 1} {{d_{j,n + 1}}f({t_j},u({t_j}))} \\&\quad \left. { +\, \sum \limits _{j = 0}^{n + 1} {{d_{j,n + 1}}f({t_j},u({t_j}))} } \right| \\&= \frac{1}{{\varGamma (\alpha )}}\left| {\int _0^{{t_{n + 1}}} {{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - \tau )}}{{({t_{n + 1}} - \tau )}^{\alpha - 1}}f(\tau ,u(\tau )){\mathrm{d}}\tau } \, - \sum \limits _{j = 0}^{n + 1} {{d_{j,n + 1}}f({t_j},u({t_j}))} } \right. \\&\quad \left. { + \,\sum \limits _{j = 0}^n {{d_{j,n + 1}}(f({t_j},u({t_j})) - {f_j})} + {d_{n + 1,n + 1}}[f({t_{n + 1}},u_{n + 1}^P) - f({t_{n + 1}},u({t_{n + 1}}))]} \right| \\&\le \frac{1}{{\varGamma (\alpha )}}\left| {\int _0^{{t_{n + 1}}} {{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - \tau )}}{{({t_{n + 1}} - \tau )}^{\alpha - 1}}f(\tau ,u(\tau )){\mathrm{d}}\tau } } \right. \left. { - \sum \limits _{j = 0}^{n + 1} {{d_{j,n + 1}}f({t_j},u({t_j}))} } \right| \\&\quad +\, \frac{1}{{\varGamma (\alpha )}}\left| {\sum \limits _{j = 0}^n {{d_{j,n + 1}}(f({t_j},u({t_j})) - {f_j})} } \right| \\&\quad +\, \frac{1}{{\varGamma (\alpha )}}\left| {{d_{n + 1,n + 1}}[f({t_{n + 1}},u_{n + 1}^P) - f({t_{n + 1}},u({t_{n + 1}}))} \right| \\&\le \frac{{C{h^3}}}{{\varGamma (\alpha )}} + \frac{L}{{\varGamma (\alpha )}}\sum \limits _{j = 0}^n {\left| {{d_{j,n + 1}}} \right| } \left| {u({t_j}) - {y_j}} \right| + \frac{L}{{\varGamma (\alpha )}}\left| {{d_{n + 1,n + 1}}} \right| \left| {u({t_{n + 1}}) - u_{n + 1}^P} \right| \\&\le {q_3}{h^3} + {q_4}\mathop {\max }\limits _{0 \le j \le n} \left| {u({t_j}) - {u_j}} \right| + {q_5}{h^3} + {q_6}\mathop {\max }\limits _{0 \le j \le n} \left| {u({t_j}) - {u_j}} \right| \\&\quad = {C_1}{h^3} + {C_2}\mathop {\max }\limits _{0 \le j \le n} \left| {u({t_j}) - {u_j}} \right| . \end{aligned} \end{aligned}$$

(6.2)

Finally, by using the mathematical induction, for all $0 \le j \le n + 1$, we have

$$\begin{aligned} \mathop {\max }\limits _{0 \le j \le n + 1} \left| {u({t_j}) - {u_j}} \right| = \mathrm{O}({h^3}). \end{aligned}$$

(6.3)

$\square$

1.2 Proof of Lemma 3

Proof

By using the piecewise quadratic interpolation for ${{\mathrm{e}}^{ - \lambda ({t_j} - \tau )}}f(\tau ,u(\tau ))$ at the nodes ${t_{j - 2}}$, ${t_{j-1}}$ and ${t_{j}}$, we can write

$$\begin{aligned} \begin{aligned}&{\sum \limits _{j = 2}^n {\int _{{t_j}}^{{t_{j + 1}}} {{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - \tau )}}{{({t_{n + 1}} - \tau )}^{\alpha - 1}}f(\tau ,u(\tau )){\mathrm{d}}\tau } } }\\&\quad { \approx \sum \limits _{j = 2}^n {\left[ \dfrac{{{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - {t_j})}}f({t_j},u({t_j}))}}{{({t_j} - {t_{j - 1}})({t_j} - {t_{j - 2}})}}\int _{{t_j}}^{{t_{j + 1}}} {{{({t_{n + 1}} - \tau )}^{\alpha - 1}}(\tau - {t_{j - 1}})(\tau - {t_{j - 2}}){\mathrm{d}}\tau }\right] } }\\&\qquad { + \sum \limits _{j = 1}^{n - 1} {\left[ \dfrac{{{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - {t_j})}}f({t_j},u({t_j}))}}{{({t_j} - {t_{j + 1}})({t_j} - {t_{j - 1}})}}\int _{{t_{j + 1}}}^{{t_{j + 2}}} {{{({t_{n + 1}} - \tau )}^{\alpha - 1}}(\tau - {t_{j + 1}})(\tau - {t_{j - 1}}){\mathrm{d}}\tau } \right] } }\\&\qquad { + \sum \limits _{j = 0}^{n - 2} {\left[ \dfrac{{{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - {t_j})}}f({t_j},u({t_j}))}}{{({t_j} - {t_{j + 2}})({t_j} - {t_{j + 1}})}}\int _{{t_{j + 2}}}^{{t_{j + 3}}} {{{({t_{n + 1}} - \tau )}^{\alpha - 1}}(\tau - {t_{j + 2}})(\tau - {t_{j + 1}}){\mathrm{d}}\tau } \right] } }\\&\quad { = {{E}_1}\sum \limits _{j = 2}^n {\dfrac{{{h^\alpha }{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - {t_j})}}f({t_j},u({t_j}))}}{{2\alpha (\alpha + 1)(\alpha + 2)}} + {{E}_2}} \sum \limits _{j = 1}^{n - 1} {\dfrac{{{h^\alpha }{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - {t_j})}}f({t_j},u({t_j}))}}{{2\alpha (\alpha + 1)(\alpha + 2)}}} }\\&\qquad {+ {{E}_3}\sum \limits _{j = 0}^{n - 2} {\dfrac{{{h^\alpha }{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - {t_j})}}f({t_j},u({t_j}))}}{{2\alpha (\alpha + 1)(\alpha + 2)}},} } \end{aligned} \end{aligned}$$

(6.4)

where

$$\begin{aligned} \begin{aligned} {{E}_1}&= (\alpha + 1)(\alpha + 2)(j - n - 2)(j - n - 3)W(j - 1,\alpha )\\&\quad +\, \alpha (\alpha + 2)(2j - 2k - 5)W(j - 1,\alpha + 1)\\&\quad + \,\alpha (\alpha + 1)W(j - 1,\alpha + 2),\\ {{E}_2}&= - 2(\alpha + 1)(\alpha + 2)(j - n)(j - n - 2)W(j,\alpha )\\&\quad -\, 2\alpha (\alpha + 2)(2j - 2k - 2)W(j,\alpha + 1)\\&\quad -\, 2\alpha (\alpha + 1)W(j,\alpha + 2),\\ {{E}_3}&= (\alpha + 1)(\alpha + 2)(j - n + 1)(j - n)W(j + 1,\alpha )\\&\quad + \,\alpha (\alpha + 2)(2j - 2k - 1)W(j + 1,\alpha + 1)\\&\quad + \,\alpha (\alpha + 1)W(j + 1,\alpha + 2). \end{aligned} \end{aligned}$$

(6.5)

By using (6.4) and (6.5), the proof is complete. $\square$

1.3 Proof of Lemma 5

Proof

By using the piecewise quadratic interpolation for ${{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - \tau )}}f(\tau ,u(\tau ))$ at the nodes ${t_{{n_{i - 2}}}}$, ${t_{{n_{i - 1}}}}$ and ${t_{{n_{i}}}}$ in the predictor formula, we have

$$\begin{aligned} \begin{aligned}&\sum \limits _{i = 2}^{{m_n}} {\int _{{t_{{n_i}}}}^{{t_{{n_{i + 1}}}}} {{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - \tau )}}{{({t_{n + 1}} - \tau )}^{\alpha - 1}}f(\tau ,u(\tau )){\mathrm{d}}\tau } } \\&\quad \approx \sum \limits _{i = 2}^{{m_n}} {\left[ \dfrac{{{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - {t_{{n_i}}})}}f({t_{{n_i}}},u({t_{{n_i}}}))}}{{({t_{{n_i}}} - {t_{{n_{i - 1}}}})({t_{{n_i}}} - {t_{{n_{i - 2}}}})}}\int _{{t_{{n_i}}}}^{{t_{{n_{i + 1}}}}} {{{({t_{n + 1}} - \tau )}^{\alpha - 1}}(\tau - {t_{{n_{i - 1}}}})(\tau - {t_{{n_{i - 2}}}}){\mathrm{d}}\tau } \right] } \\&\qquad + \sum \limits _{i = 1}^{{m_n} - 1} {\left[ \dfrac{{{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - {t_{{n_i}}})}}f({t_{{n_i}}},u({t_{{n_i}}}))}}{{({t_{{n_i}}} - {t_{{n_{i + 1}}}})({t_{{n_i}}} - {t_{{n_{i - 1}}}})}}\int _{{t_{{n_{i + 1}}}}}^{{t_{{n_{i + 2}}}}} {{{({t_{n + 1}} - \tau )}^{\alpha - 1}}(\tau - {t_{{n_{i + 1}}}})(\tau - {t_{{n_{i - 1}}}}){\mathrm{d}}\tau } \right] }\\&\qquad + \sum \limits _{i = 0}^{{m_n} - 2} {\left[ \dfrac{{{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - {t_{{n_i}}})}}f({t_{{n_i}}},u({t_{{n_i}}}))}}{{({t_{{n_i}}} - {t_{{n_{i + 2}}}})({t_{{n_i}}} - {t_{{n_{i + 1}}}})}}\int _{{t_{{n_{i + 2}}}}}^{{t_{{n_{i + 3}}}}} {{{({t_{n + 1}} - \tau )}^{\alpha - 1}}(\tau - {t_{{n_{i + 2}}}})(\tau - {t_{{n_{i + 1}}}}){\mathrm{d}}\tau } \right] } \\&\quad = \Theta \left[ {\sum \limits _{i = 2}^{{m_n}} {\dfrac{{{{\mathrm{e}}^{ - \lambda (n + 1 - {n_i})h}}f({t_{{n_i}}},u({t_{{n_i}}}))}}{{({n_i} - {n_{i - 1}})({n_i} - {n_{i - 2}})}}{\Lambda _1} + } \sum \limits _{i = 1}^{{m_n} - 1} {\dfrac{{{{\mathrm{e}}^{ - \lambda (n + 1 - {n_i})h}}f({t_{{n_i}}},u({t_{{n_i}}}))}}{{({n_i} - {n_{i + 1}})({n_i} - {n_{i - 1}})}}{\Lambda _2}} } \right. \\&\qquad \left. { + \sum \limits _{i = 0}^{{m_n} - 2} {\dfrac{{{{\mathrm{e}}^{ - \lambda (tn + 1 - {n_i})h}}f({t_{{n_i}}},u({t_{{n_i}}}))}}{{({n_i} - {n_{i + 2}})({n_i} - {n_{i + 1}})}}{\Lambda _3}} } \right] , \end{aligned} \end{aligned}$$

(6.6)

where

$$\begin{aligned} \begin{aligned} {\Lambda _1}&= (\alpha + 1)(\alpha + 2)(n + 1 - {n_{i - 1}})(n + 1 - {n_{i - 2}})R(i,\alpha )\\&\quad +\, \alpha (\alpha + 2)(2n + 2 - {n_{i - 1}} - {n_{i - 2}})R(i,\alpha + 1) + \alpha (\alpha + 1)R(i,\alpha + 2),\\ {\Lambda _2}&= (\alpha + 1)(\alpha + 2)(n + 1 - {n_{i + 1}})(n + 1 - {n_{i - 1}})R(i + 1,\alpha )\\&\quad +\, \alpha (\alpha + 2)(2n + 2 - {n_{i - 1}} - {n_{i + 1}})R(i + 1,\alpha + 1) + \alpha (\alpha + 1)R(i + 1,\alpha + 2),\\ {\Lambda _3}&= (\alpha + 1)(\alpha + 2)(n + 1 - {n_{i + 2}})(n + 1 - {n_{i + 1}})R(i + 2,\alpha )\\&\quad +\, \alpha (\alpha + 2)(2n + 2 - {n_{i + 2}} - {n_{i + 1}})R(i + 2,\alpha + 1) + \alpha (\alpha + 1)R(i + 2,\alpha + 2); \end{aligned} \end{aligned}$$

(6.7)

by using (6.6) and (6.7), the proof is complete. $\square$

1.4 Proof of Lemma 7

Proof

Let ${{\tilde{f}}_1}$ and ${{\tilde{f}}_2}$ be the piecewise quadratic interpolation for ${{\mathrm{e}}^{\lambda t}}f(t)$ with nodes, ${t_{{n_i}}},{t_{{n_{i+1}}}},{n_{{n_{i+2}}}}$, and ${t_{j}},{t_{j+1}},{t_{j+2}}$, respectively. Since

$$\begin{aligned} \begin{aligned}&\int _{{t_{{n_i}}}}^{{t_{{n_{i + 1}}}}} {{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - \tau )}}{{({t_{n + 1}} - \tau )}^{\alpha - 1}}f(\tau ,u(\tau )){\mathrm{d}}\tau = {{\mathrm{e}}^{ - \lambda {t_{n + 1}}}}\int _{{t_{{n_i}}}}^{{t_{{n_{i + 1}}}}} {{{({t_{n + 1}} - \tau )}^{\alpha - 1}}{{\mathrm{e}}^{\lambda \tau }}f(\tau ,u(\tau )){\mathrm{d}}\tau } } \\&\quad \approx {{\mathrm{e}}^{ - \lambda {t_{n + 1}}}}\int _{{t_{{n_i}}}}^{{t_{{n_{i + 1}}}}} {{{({t_{k + 1}} - \tau )}^{\alpha - 1}}{{\tilde{f}}_1}(\tau ){\mathrm{d}}\tau = \Theta {{\mathrm{e}}^{ - \lambda {t_{n + 1}}}}\left[ {{{\mathrm{e}}^{\lambda {t_{{n_i}}}}}{\mathop {b}\limits ^{\frown }} _{i,n + 1}^Rf({t_{{n_i}}},u({t_{{n_i}}}))} \right. } \\&\qquad \left. { + {{\mathrm{e}}^{\lambda {t_{{n_{i + 1}}}}}}{\mathop {b}\limits ^{\frown }} _{i + 1,n + 1}^Mf({t_{{n_{i + 1}}}},u({t_{{n_{i + 1}}}})) + {{\mathrm{e}}^{\lambda {t_{{n_{i + 2}}}}}}{\mathop {b}\limits ^{\frown }} _{i + 2,n + 1}^Lf({t_{{n_{i + 2}}}},u({t_{{n_{i + 2}}}}))} \right] ,\\&\int _{{t_{{n_i}}}}^{{t_{{n_{i + 1}}}}} {{{\mathrm{e}}^{ - \lambda ({t_{n + 1}} - \tau )}}{{({t_{n + 1}} - \tau )}^{\alpha - 1}}f(\tau ,u(\tau )){\mathrm{d}}\tau } \approx {{\mathrm{e}}^{ - \lambda {t_{n + 1}}}}\int _{{t_{{n_i}}}}^{{t_{{n_{i + 1}}}}} {{{({t_{n + 1}} - \tau )}^{\alpha - 1}}{{\tilde{f}}_2}(\tau ){\mathrm{d}}\tau } \\&\quad = \Theta {{\mathrm{e}}^{ - \lambda {t_{n + 1}}}}\left[ {\sum \limits _{j = {n_i}}^{{n_{i + 1}} - 1} {({{\mathrm{e}}^{\lambda {t_j}}}{\mathop {d}\limits ^{\frown }} _{j,n + 1}^Rf({t_j},u({t_j})) + {{\mathrm{e}}^{\lambda {t_{j + 1}}}}{\mathop {d}\limits ^{\frown }} _{j + 1,n + 1}^Mf({t_{j + 1}},u({t_{j + 1}}))} } \right. \\&\qquad \left. { + {{\mathrm{e}}^{\lambda {t_{j + 2}}}}{\mathop {d}\limits ^{\frown }} _{j + 2,n + 1}^Lf({t_{j + 2}},u({t_{j + 2}})))} \right] , \end{aligned} \end{aligned}$$

(6.8)

if we take $f(\tau ) \equiv 1$ and $\lambda =0$, we can write

$$\begin{aligned} \int _{{t_{{n_i}}}}^{{t_{{n_{i + 1}}}}} {{{({t_{n + 1}} - \tau )}^{\alpha - 1}}f(\tau ,u(\tau )){\mathrm{d}}\tau = \frac{{{h^\alpha }}}{\alpha }} \left[ {{{(n + 1 - {n_i})}^\alpha } - {{(n + 1 - {n_{i + 1}})}^\alpha }} \right] ; \end{aligned}$$

(6.9)

by using (6.8) and (6.9), we have

$$\begin{aligned} \begin{aligned}&{{\mathop {b}\limits ^{\frown }}}_{i,n + 1}^R + {\mathop {b}\limits ^{\frown }} _{i + 1,n + 1}^M + {\mathop {b}\limits ^{\frown }} _{i + 2,n + 1}^L = \sum \limits _{j = {n_i}}^{{n_{i + 1}} - 1} {({\mathop {d}\limits ^{\frown }} _{j,n + 1}^R + {\mathop {d}\limits ^{\frown }} _{j + 1,n + 1}^M + {\mathop {d}\limits ^{\frown }} _{j + 2,n + 1}^L)} \\&\quad = (\alpha + 1)(\alpha + 2)\left[ {{{(n + 1 - {n_i})}^\alpha } - {{(n + 1 - {n_{i + 1}})}^\alpha }} \right] . \end{aligned} \end{aligned}$$

(6.10)

We let $G(t) = {{\mathrm{e}}^{\lambda t}}f(t)$, by combining (3.21, 4.29, 4.32) and (6.10), and we can write

$$\begin{aligned} \begin{aligned}&{{\mathrm{e}}^{\lambda {t_{n + 1}}}}\left| {\sum \limits _{j = 0}^n {{\mathop {d}\limits ^{\frown }}_{j,n + 1}}f({t_j},u({t_j})) - \sum \limits _{i = 0}^{{m_n}} {{\mathop {b}\limits ^{\frown }}_{i,k + 1}}f({t_{{n_i}}},u({t_{{n_i}}}))} \right| \\&\quad = \left| {\sum \limits _{j = 0}^n {({\mathop {d}\limits ^{\frown }} _{j,n + 1}^L + {\mathop {d}\limits ^{\frown }} _{j,n + 1}^M + {\mathop {d}\limits ^{\frown }} _{j,n + 1}^R)G({t_j}) - \sum \limits _{i = 0}^{{m_n}} {({\mathop {b}\limits ^{\frown }} _{i,n + 1}^L + {\mathop {b}\limits ^{\frown }} _{i,n + 1}^M + {\mathop {b}\limits ^{\frown }} _{i,n + 1}^R)G({t_{{n_i}}})} } } \right| \\&\quad = \left| {\sum \limits _{i = 0}^{{m_n} - 2} {\sum \limits _{j = {n_i}}^{{n_{i + 1}} - 1} {[{\mathop {d}\limits ^{\frown }} _{j,n + 1}^RG({t_j}) + {\mathop {d}\limits ^{\frown }} _{j + 1,n + 1}^MG({t_{j + 1}}) + {\mathop {d}\limits ^{\frown }} _{j + 2,n + 1}^LG({t_{j + 2}})]} } } \right. \\&\qquad \left. { - \sum \limits _{i = 0}^{{m_n} - 2} {({\mathop {b}\limits ^{\frown }} _{i,n + 1}^RG({t_{{n_i}}}) + {\mathop {b}\limits ^{\frown }} _{i + 1,n + 1}^MG({t_{{n_{i + 1}}}}) + {\mathop {b}\limits ^{\frown }} _{i + 2,n + 1}^LG({t_{{n_{i + 2}}}})]} } \right| \\&\quad = \left| {\sum \limits _{i = 0}^{{m_n} - 2} {\sum \limits _{j = {n_i}}^{{n_{i + 1}} - 1} {({\mathop {d}\limits ^{\frown }} _{j,n + 1}^R[G({t_{{k_i}}}) + G'({\xi _j})({t_j} - {t_{{n_i}}})] + {\mathop {d}\limits ^{\frown }} _{j + 1,n + 1}^M[G({t_{{n_i}}})} } } \right. \\&\qquad + G'({\xi _{j + 1}})({t_{j + 1}} - {t_{{n_i}}})] + {\mathop {d}\limits ^{\frown }} _{j + 2,n + 1}^L[G({t_{{n_i}}}) + G'({\xi _{j + 2}})({t_{j + 2}} - {t_{{n_i}}})])\\&\qquad - \sum \limits _{i = 0}^{{m_n} - 2} {({\mathop {b}\limits ^{\frown }} _{i,n + 1}^RG({t_{{n_i}}}) + {\mathop {b}\limits ^{\frown }} _{i + 1,n + 1}^M[G({t_{{n_i}}}) + G'({\xi _{{n_{i + 1}}}})({t_{{n_{i + 1}}}} - {t_{{n_i}}})]} \\&\qquad \left. { + {\mathop {b}\limits ^{\frown }} _{i + 2,n + 1}^L[G({t_{{n_i}}}) + G'({\xi _{{n_{i + 2}}}})({t_{{n_{i + 2}}}} - {t_{{n_i}}})])} \right| \\&\quad \le \left| {\sum \limits _{i = 0}^{{m_n} - 2} {[(\sum \limits _{j = {n_i}}^{{n_{i + 1}} - 1} {[{\mathop {d}\limits ^{\frown }} _{j,n + 1}^R + {\mathop {d}\limits ^{\frown }} _{j + 1,n + 1}^M + {\mathop {d}\limits ^{\frown }} _{j + 2,n + 1}^L]} )} } \right. \\&\qquad - ({\mathop {b}\limits ^{\frown }} _{i,n + 1}^R + {\mathop {b}\limits ^{\frown }} _{i + 1,n + 1}^M + {\mathop {b}\limits ^{\frown }} _{i + 2,n + 1}^L)\left. {]G({t_{{n_i}}})} \right| + {\left\| {G'} \right\| _\infty }\sum \limits _{i = 0}^{{m_n} - 2} {({t_{{n_{i + 1}}}} - {t_{{n_i}}})} \\&\qquad .\left| {(\sum \limits _{j = {n_i}}^{{n_{i + 1}} - 1} {[{\mathop {d}\limits ^{\frown }} _{j,n + 1}^R + {\mathop {d}\limits ^{\frown }} _{j + 1,n + 1}^M + {\mathop {d}\limits ^{\frown }} _{j + 2,n + 1}^L]} ) + ({\mathop {b}\limits ^{\frown }} _{i,n + 1}^R + {\mathop {b}\limits ^{\frown }} _{i + 1,n + 1}^M + {\mathop {b}\limits ^{\frown }} _{i + 2,n + 1}^L)} \right| \\&\quad = {\left\| {G'} \right\| _\infty }\sum \limits _{i = 0}^{{m_n} - 2} {({t_{{n_{i + 1}}}} - {t_{{n_i}}})\left| {(\sum \limits _{j = {n_i}}^{{n_{i + 1}} - 1} {[{\mathop {d}\limits ^{\frown }} _{j,n + 1}^R + {\mathop {d}\limits ^{\frown }} _{j + 1,n + 1}^M + {\mathop {d}\limits ^{\frown }} _{j + 2,n + 1}^L]} )} \right. } \\&\qquad \left. { + ({\mathop {b}\limits ^{\frown }} _{i,n + 1}^R + {\mathop {b}\limits ^{\frown }} _{i + 1,n + 1}^M + {\mathop {b}\limits ^{\frown }} _{i + 2,n + 1}^L)} \right| \le 2(\alpha + 1)(\alpha + 2){{\mathrm{e}}^{\lambda {t_{n + 1}}}}{\left\| {f' + \lambda f} \right\| _\infty }\\&\qquad .\sum \limits _{i = 0}^{{m_n} - 2} {({t_{{n_{i + 1}}}} - {t_{{n_i}}})[{{(n + 1 - {n_i})}^\alpha } - {{(n + 1 - {n_{i + 1}})}^\alpha }]} \end{aligned} \end{aligned}$$

(6.11)

where ${\xi _S} \in [{t_{{n_i}}},{t_S}]$. For equal-area distribution method, by the use of (4.10), we have

$$\begin{aligned} {({t_{n + 1}} - {t_{{n_i}}})^\alpha } - {({t_{n + 1}} - {t_{{n_{i + 1}}}})^\alpha } = \alpha ({t_{{n_{i + 1}}}} - {t_{{n_i}}}){({t_{n + 1}} - {t_{{n_i}}})^{\alpha - 1}}; \end{aligned}$$

therefore, we can write

$$\begin{aligned}{}[{(n + 1 - {n_i})^\alpha } - {(n + 1 - {n_{i + 1}})^\alpha }] \le \dfrac{{\alpha {{\mathrm{e}}^{\lambda {t_{n + 1}}}}}}{{{h^\alpha }}}\Delta S; \end{aligned}$$

(6.12)

and by using (6.11) and (6.12), finally we have

$$\begin{aligned} \frac{{{h^\alpha }}}{{\alpha (\alpha + 1)(\alpha + 2)}}\begin{array}{l} {\left| {\sum \limits _{j = 0}^n {{\mathop {d}\limits ^{\frown }}_{j,n + 1}}f({t_j},u({t_j})) - \sum \limits _{i = 0}^{{m_n}} {{\mathop {b}\limits ^{\frown }}_{i,n + 1}}f({t_{{n_i}}},u({t_{{n_i}}}))} \right| \le C{h^3}.} \end{array} \end{aligned}$$

(6.13)

For the equal-height distribution method, we assume

$$\begin{aligned} \int _{{t_{{n_{{i^*}}}}}}^{{t_{{n_{{i^*} + 1}}}}} {{{({t_{n + 1}} - \tau )}^{\alpha - 1}}{\mathrm{d}}\tau = \mathop {Max}\limits _{0 \le i \le {m_n} - 2} \int _{{t_{{n_i}}}}^{{t_{{n_{i + 1}}}}} {{{({t_{n + 1}} - \tau )}^{\alpha - 1}}{\mathrm{d}}\tau } } ; \end{aligned}$$

(6.14)

therefore, we can write

$$\begin{aligned} {\int _{{t_{{n_{{i^*}}}}}}^{{t_{{n_{{i^*} + 1}}}}} {{{({t_{n + 1}} - \tau )}^{\alpha - 1}}{\mathrm{d}}\tau \le ({t_{n + 1}} - {t_{{n_{{i^*} + 1}}}})} ^{\alpha - 1}}({t_{{n_{{i^*} + 1}}}} - {t_{{n_{{i^*}}}}}); \end{aligned}$$

by using (4.6), we have

$$\begin{aligned} \begin{aligned}&{({t_{{n_{{i^*} + 2}}}} - {t_{{n_{{i^*} + 1}}}}){{({t_{n + 1}} - {t_{{n_{{i^*} + 1}}}})}^{\alpha - 2}} \le \dfrac{{{{\mathrm{e}}^{\lambda ({t_{n + 1}} - {t_{{n_{{i^*} + 1}}}})}}\Delta u}}{{(1 - \alpha ) + \lambda ({t_{n + 1}} - {t_{{n_{{i^*} + 1}}}})}}}\\&{{{({t_{n + 1}} - {t_{{n_{{i^*} + 1}}}})}^{\alpha - 1}} \le \dfrac{{({t_{n + 1}} - {t_{{n_{{i^*} + 1}}}}){{\mathrm{e}}^{\lambda ({t_{n + 1}} - {t_{{n_{{i^*} + 1}}}})}}\Delta u}}{{({t_{{n_{{i^*} + 2}}}} - {t_{{n_{{i^*} + 1}}}})[(1 - \alpha ) + \lambda ({t_{n + 1}} - {t_{{n_{{i^*} + 1}}}})]}}}\\&\quad {\le \dfrac{{{{\mathrm{e}}^{\lambda ({t_{n + 1}} - {t_{{n_{{i^*} + 1}}}})}}\Delta u}}{{\lambda ({t_{{n_{{i^*} + 2}}}} - {t_{{n_{{i^*} + 1}}}})}};} \end{aligned} \end{aligned}$$

(6.15)

and thus,

$$\begin{aligned} {\int _{{t_{{n_{{i^*}}}}}}^{{t_{{n_{{i^*} + 1}}}}} {{{({t_{k + 1}} - \tau )}^{\alpha - 1}}{\mathrm{d}}\tau \le \frac{{({n_{{i^*} + 1}} - {n_{{i^*}}}){{\mathrm{e}}^{\lambda ({t_{n + 1}} - {t_{{n_{{i^*} + 1}}}})}}\Delta u}}{{\lambda ({n_{{i^*} + 2}} - {n_{{i^*} + 1}})}} = {H_{{n_{{i^*}}}}}\Delta u,} } \end{aligned}$$

where ${H_{{n_{{i^*}}}}} = \frac{{({n_{{i^*} + 1}} - {n_{{i^*}}}){{\mathrm{e}}^{\lambda ({t_{n + 1}} - {t_{{n_{{i^*} + 1}}}})}}}}{{\lambda ({n_{{i^*} + 2}} - {n_{{i^*} + 1}})}}$. By using (6.11) and (4.6)

$$\begin{aligned} \begin{aligned}&{\left| {\sum \limits _{j = 0}^n {{\mathop {d}\limits ^{\frown }}_{j,n + 1}}f({t_j},u({t_j})) - \sum \limits _{i = 0}^{{m_n}} {{\mathop {b}\limits ^{\frown }}_{i,n + 1}}f({t_{{n_i}}},u({t_{{n_i}}}))} \right| }\\&\quad { \le \frac{{2\alpha (\alpha + 1)(\alpha + 2)}}{{{h^\alpha }}}{{\left\| {f' + \lambda f} \right\| }_\infty }\sum \limits _{i = 0}^{{m_n} - 2} {({t_{{n_{i + 1}}}} - {t_{{n_i}}})} \int _{{t_{{n_{{i^*}}}}}}^{{t_{{n_{{i^*} + 1}}}}} {{{({t_{n + 1}} - \tau )}^{\alpha - 1}}{\mathrm{d}}\tau } }\\&\quad { \le \frac{{2\alpha (\alpha + 1)(\alpha + 2){t_n}}}{{{h^\alpha }}}{{\left\| {f' + \lambda f} \right\| }_\infty }{H_{{n_{{i^*}}}}}\Delta u}\\&\quad { \le \frac{{\alpha (\alpha + 1)(\alpha + 2)}}{{{h^\alpha }}}C\Delta u,} \end{aligned} \end{aligned}$$

(6.16)

and finally, we have

$$\begin{aligned} \frac{{{h^\alpha }}}{{\alpha (\alpha + 1)(\alpha + 2)}}\begin{array}{l} {\left| {\sum \limits _{j = 0}^n {{\mathop {d}\limits ^{\frown }}_{j,n + 1}}f({t_j},u({t_j})) - \sum \limits _{i = 0}^{{m_n}} {{\mathop {b}\limits ^{\frown }}_{i,n + 1}}f({t_{{n_i}}},u({t_{{n_i}}}))} \right| \le C\dfrac{{\Delta u}}{h}.} \end{array} \end{aligned}$$

(6.17)

$\square$