An H¹ convergence of the spectral method for the time-fractional non-linear diffusion equations

Abstract

The generalized discrete Gronwall inequality is applied to analyze the optimal H¹ error estimate of the time-stepping spectral method for the time-fractional diffusion equations, where the time-fractional derivative is discretized by the second-order fractional backward difference formula or the second-order generalized Newton-Gregory formula. The methodology is extended to analyze the fractional Crank–Nicolson spectral method and the time-stepping spectral method for the multi-term time-fractional differential equations. Numerical simulations are provided to support the theoretical analysis.

Appendices

A Appendix

A Proof of $d^{\varepsilon }_{n}\geq 0$ for FBDF-2

Proof

It is very easy to obtain b^(α)(z) = (3/2 − z/2)^α and $b^{(\alpha )}_{n}=b^{(\alpha )}_{0}3^{-n}{a}^{(\alpha )}_{n}$, where $a^{(\alpha )}_{n}=\frac {{\varGamma }(n+\alpha )}{{\varGamma }(\alpha ){\varGamma }(n+1)}$, $b^{(\alpha )}_{0} = ({3}/{2})^{\alpha }$, and

$$ \begin{array}{@{}rcl@{}} B_{0}={\sum}_{n=0}^{\infty}{b}^{(\alpha)}_{n}=b^{(\alpha)}_{0} ({2}/{3})^{\alpha}=1. \end{array} $$

(86)

Then, $d^{\varepsilon }_{n}$ can be expressed by

$$ \begin{aligned} d^{\varepsilon}_{n}=& (B_{0} -\varepsilon){a}^{(-\alpha)}_{n} +{\sum}_{j=0}^{n}b^{(\alpha)}_{j}{a}^{(-\alpha)}_{n-j}\\ =& (1+b^{(\alpha)}_{0}-\varepsilon){a}^{(-\alpha)}_{n} +b^{(\alpha)}_{0}{\sum}_{j=1}^{n}3^{-j}{a}^{(\alpha)}_{j}{a}^{(-\alpha)}_{n-j} \end{aligned} $$

(87)

Using ${a}^{(\alpha )}_{j}\leq 0$ and (??), we have

$$ \begin{aligned} {\sum}_{j=1}^{n}3^{-j}{a}^{(\alpha)}_{j}{a}^{(-\alpha)}_{n-j} \geq& \frac{1}{3}{\sum}_{j=1}^{n} {a}^{(\alpha)}_{j} {a}^{(-\alpha)}_{n-j} =-\frac{1}{3}{a}^{(-\alpha)}_{n},\quad n\geq 1. \end{aligned} $$

(88)

Combining (2)–(3) and choosing $\varepsilon =\frac {2}{3}b^{(\alpha )}_{0}=(3/2)^{\alpha -1}$ yields

$$ \begin{aligned} &d^{\varepsilon}_{0} =b^{(\alpha)}_{0}+1-\varepsilon=1+b^{(\alpha)}_{0}/3\geq 4/3,\\ &d^{\varepsilon}_{n}\geq \left( b^{(\alpha)}_{0}+1 -\varepsilon- b^{(\alpha)}_{0}/3\right){a}^{(-\alpha)}_{n} ={a}^{(-\alpha)}_{n}\geq0,\quad n \geq 1. \end{aligned} $$

(89)

The proof is complete. □

B Proof of $d^{\varepsilon }_{n}\geq 0$ for GNGF-2

Proof

For the GNGF-2, we have $b^{(\alpha )}_{0}=1+\alpha /2,b^{(\alpha )}_{1}=-\alpha /2$, $b^{(\alpha )}_{n}=0$ for n > 1, and B₀ = 1.

Letting ε = 1/2 yields $d^{\varepsilon }_{0}=1/2+b^{(\alpha )}_{0}=(3+\alpha )/2>0$. For n ≥ 1, we have

$$ \begin{aligned} d^{\varepsilon}_{n}=& (B_{0} -\varepsilon){a}^{(-\alpha)}_{n} + \left( b^{(\alpha)}_{0}{a}^{(-\alpha)}_{n} + {b}^{(\alpha)}_{1}{a}^{(-\alpha)}_{n-1}\right)\\ =&\left( \frac{3+\alpha}{2} - \frac{\alpha}{2}\frac{n}{n-1+\alpha}\right){a}^{(-\alpha)}_{n}\\ \geq&\left( \frac{3+\alpha}{2}-\frac{\alpha}{2}\frac{1}{\alpha}\right){a}^{(-\alpha)}_{n} =\frac{2+\alpha}{2}{a}^{(-\alpha)}_{n}\geq 0, \end{aligned} $$

(90)

where $\frac {{a}^{(-\alpha )}_{n}}{{a}^{(-\alpha )}_{n-1}}=\frac {n-1+\alpha }{n}$. The proof is completed. □