Let \(t_{n}=n\tau\), \(n=0,1,\ldots,N\), \(T=\tau N\), \(N\in\mathbb{Z}^{+}\), and \(x_{j}=a+jh\), \(j=-1,0,\ldots,M+1\), \(h=(b-a)/M\), \(M\in\mathbb{Z}^{+}\). On this time-space lattice, we set about deriving the desired exponential B-spline collocation method for Eqs. (1.1)-(1.3).
Discretization of Caputo derivative
We recall the definitions of fractional derivatives. Given a smooth enough \(f(x,t)\), the αth Caputo derivative is defined by
$$ {^{C}_{0}}D^{\alpha}_{t}f(x,t)= \frac{1}{\Gamma(m-\alpha)} \int^{t}_{0}\frac{\partial^{m} f(x,\xi)}{\partial\xi^{m}}\frac{d\xi}{(t-\xi )^{1+\alpha-m}}, $$
(3.1)
and the αth Riemann-Liouville type derivative is defined by
$$ {^{\mathrm{RL}}_{0}}D^{\alpha}_{t}f(x,t)= \frac{1}{\Gamma(m-\alpha)}\frac{\partial ^{m}}{\partial t^{m}} \int^{t}_{0}\frac{f(x,\xi)\,d\xi}{(t-\xi)^{1+\alpha-m}}, $$
(3.2)
where \(m-1<\alpha<m\), \(m\in\mathbb{N}\) is not less than 1. In common sense, (3.1) owns merits in handling the initial-valued problems, and thereby is utilized in time in most instances. (3.1), (3.2) interconvert into each other through
$$ {^{C}_{0}}D^{\alpha}_{t}f(x,t)={^{\mathrm{RL}}_{0}}D^{\alpha}_{t}f(x,t)- \sum^{m-1}_{l=0}\frac {f^{(l)}(x,0)t^{l-\alpha}}{\Gamma(l+1-\alpha)}. $$
(3.3)
They are equal when \(f^{(k)}(x,0)=0\), \(k=0,1,\ldots,m-1\), are fixed; we refer the readers to [15, 16] for deeper insight. An effective approximation for Caputo derivative can be derived by rewriting Eq. (3.3) and using proper schemes to discretize (3.2), i.e.,
$$ {^{C}_{0}}D^{\alpha}_{t}f(x,t_{n}) \approx\frac{1}{\tau^{\alpha}}\sum_{k=0}^{n} \omega^{q,\alpha}_{k}f(x,t_{n-k}) -\frac{1}{\tau^{\alpha}}\sum ^{m-1}_{l=0}\sum^{n}_{k=0} \frac{\omega ^{q,\alpha}_{k} f^{(l)}(x,0)t_{n-k}^{l}}{l!}, $$
(3.4)
with several sets of coefficients \(\omega^{q,\alpha}_{k}\), \(q=1,2,3,4,5\), (see [37]). Let \(\omega^{\alpha}_{k}=\omega^{1,\alpha}_{k}\). Then
$$ \omega^{\alpha}_{k}=(-1)^{k}\binom{\alpha}{k}=\frac{\Gamma{(k-\alpha)}}{\Gamma {(-\alpha)}\Gamma{(k+1)}}, \quad k=0,1,2,\ldots $$
(3.5)
in which case the scheme is the one given by Gorenflo et al. [38]. On imposing \(0<\alpha<1\), (3.4) simply reduces to
$$ {^{C}_{0}}D^{\alpha}_{t}f(x,t_{n})= \frac{1}{\tau^{\alpha}}\sum_{k=0}^{n}\omega ^{q,\alpha}_{k}f(x,t_{n-k}) -\frac{1}{\tau^{\alpha}}\sum _{k=0}^{n}\omega^{q,\alpha}_{k}f(x,0)+{R_{q}( \tau)}, $$
(3.6)
with the truncated error \(R_{q}(\tau)\) satisfying \(R_{q}(\tau)=O(\tau ^{q})\), \(q=1,2,3,4,5\).
Lemma 3.1
The coefficients
\(\omega^{\alpha}_{k}\)
defined in (3.5) fulfill
-
(a)
\(\omega^{\alpha}_{0}=1\), \(\omega^{\alpha}_{k}< 0\), \(\forall k\geq1\),
-
(b)
\(\sum_{k=0}^{\infty}\omega^{\alpha}_{k}=0\), \(\sum_{k=0}^{n-1}\omega^{\alpha}_{k}>0\).
Proof
See references [15, 39] for details. □
A fully discrete exponential B-spline based scheme
Define \(V_{M+3}=\textrm{span}\{B_{-1}(x),B_{0}(x),\ldots ,B_{M}(x),B_{M+1}(x)\}\) over the interval \([a,b]\) referred to as an \((M+3)\)-dimensional exponential spline space. Then an approximate solution to Eqs. (1.1)-(1.3) is sought on \(V_{M+3}\) in the form
$$ u_{N}(x,t)=\sum_{j=-1}^{M+1} \alpha_{j}(t)B_{j}(x), $$
(3.7)
with the unknown weights \(\{\alpha_{j}(t)\}_{j=-1}^{M+1}\) yet to be determined by some certain restrictions. Discretizing Eq. (1.1) by using (3.6) in time, we have
$$\begin{gathered} \omega^{q,\alpha}_{0}u(x,t_{n})-\tau^{\alpha}\kappa\frac{\partial^{2} u(x,t_{n})}{\partial x^{2}}\\\quad= -\sum_{k=1}^{n-1} \omega^{q,\alpha}_{k}u(x,t_{n-k})+\sum _{k=0}^{n-1}\omega ^{q,\alpha}_{k}u(x,0)+ \tau^{\alpha}f(x,t_{n})+{\tau^{\alpha}R_{q}( \tau)}.\end{gathered} $$
Let \(\alpha^{n}_{j}=\alpha_{j}(t_{n})\). On replacing \(u(x,t)\) by \(u_{N}(x,t)\) and imposing the following collocation and boundary conditions
$$\begin{gathered} \omega^{q,\alpha}_{0}u_{N}(x_{j},t_{n})- \tau^{\alpha}\kappa\frac{\partial^{2} u_{N}(x_{j},t_{n})}{\partial x^{2}} \\\quad= -\sum_{k=1}^{n-1} \omega^{q,\alpha}_{k}u_{N}(x_{j},t_{n-k})+ \sum_{k=0}^{n-1}\omega^{q,\alpha}_{k}u_{N}(x_{j},0)+ \tau^{\alpha}f(x_{j},t_{n}), \\ u_{N}(x_{0},t_{n})=g_{1}(t_{n}), \qquad u_{N}(x_{M},t_{n})=g_{2}(t_{n}),\end{gathered} $$
at each nodal point \(x_{j}\), \(j=0,1,\ldots,M\), we obtain
$$ A\alpha^{n}_{j-1}+A' \alpha^{n}_{j}+A\alpha^{n}_{j+1} =-\sum _{k=1}^{n-1}\omega^{q,\alpha}_{k}P^{n-k}_{j}+ \sum_{k=0}^{n-1}\omega ^{q,\alpha}_{k}P_{j}^{0}+R^{n}_{j}, $$
(3.8)
and the boundary sets
$$\begin{aligned}& \frac{s-ph}{2(phc-s)}\alpha^{n}_{-1}+\alpha^{n}_{0}+ \frac {s-ph}{2(phc-s)}\alpha^{n}_{1}=g_{1}^{n}, \end{aligned}$$
(3.9)
$$\begin{aligned}& \frac{s-ph}{2(phc-s)}\alpha^{n}_{M-1}+\alpha^{n}_{M}+ \frac {s-ph}{2(phc-s)}\alpha^{n}_{M+1}=g_{2}^{n}, \end{aligned}$$
(3.10)
owing to (3.7) and (2.1)-(2.3) with
$$\begin{gathered} A=-\tau^{\alpha}\kappa p^{2}s+\omega^{q,\alpha}_{0}(s-ph), \qquad A'=2\tau ^{\alpha}\kappa p^{2}s+2 \omega^{q,\alpha}_{0}(phc-s), \\ P^{m}_{j}=(s-ph)\alpha^{m}_{j-1}+2(phc-s) \alpha^{m}_{j}+(s-ph)\alpha ^{m}_{j+1}, \qquad R^{n}_{j}=2\tau^{\alpha}(phc-s)f_{j}^{n}, \end{gathered}$$
where \(m=0,1,\ldots,n-1\). As a result, using Eqs. (3.9)-(3.10) to remove the unknown variables \(\alpha^{n}_{-1}\), \(\alpha^{n}_{M+1}\) in Eq. (3.8) when \(j=0\), M, the above system admits a linear system of algebraic equations of size \((M+1)\times (M+1)\) as follows:
$$ \textbf{A}\boldsymbol{\alpha}^{n}=-\sum _{k=1}^{n-1}\omega^{q,\alpha }_{k} \textbf{B}\boldsymbol{\alpha}^{n-k} +\sum_{k=0}^{n-1} \omega^{q,\alpha}_{k}\textbf{B}\boldsymbol{\alpha }^{0}+ \textbf{F}^{n}, \quad{q=1,2,3,4,5,} $$
(3.11)
where
$$\begin{gathered} \textbf{A}=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 2\tau^{\alpha}\kappa p^{3}hs(c-1) & 0 & & & \\ A & A' & A & & \\ & \ldots& \ldots& \ldots& \\ & & \ldots& \ldots& \ldots\\ & & A & A' & A \\ & & & 0 & 2\tau^{\alpha}\kappa p^{3}hs(c-1) \ \end{array}\displaystyle \right ), \\\textbf{B}=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & 0 & & & \\ s-ph & 2(phc-s) & s-ph & & \\ & \ldots& \ldots& \ldots& \\ & & \ldots& \ldots& \ldots\\ & & s-ph & 2(phc-s) & s-ph \\ & & & 0 & 0 \end{array}\displaystyle \right ), \\\boldsymbol{\alpha}^{m}=\left ( \textstyle\begin{array}{c} \alpha^{m}_{0} \\ \alpha^{m}_{1} \\ \vdots\\ \alpha^{m}_{M-1} \\ \alpha^{m}_{M} \end{array}\displaystyle \right ), \qquad\textbf{F}^{n}=(phc-s)\left ( \textstyle\begin{array}{c} 2\tau^{\alpha}(s-ph)f^{n}_{0}+d^{n}_{0} \\ 2\tau^{\alpha}f^{n}_{1} \\ \vdots\\ 2\tau^{\alpha}f^{n}_{M-1} \\ 2\tau^{\alpha}(s-ph)f^{n}_{M}+d^{n}_{M} \end{array}\displaystyle \right ), \end{gathered}$$
in which \(m=0,1,\ldots,n\) and \(d^{n}_{0}\), \(d^{n}_{M}\) are as follows:
$$\begin{gathered} d^{n}_{0}=-2(s-ph)\sum_{k=0}^{n-1} \omega^{q,\alpha}_{k}g_{1}^{n-k}+2(s-ph)\sum _{k=0}^{n-1}\omega^{q,\alpha}_{k} \varphi_{0}+2\tau^{\alpha}\kappa p^{2}sg^{n}_{1}, \\ d^{n}_{M}=-2(s-ph)\sum_{k=0}^{n-1} \omega^{q,\alpha}_{k}g_{2}^{n-k}+2(s-ph)\sum _{k=0}^{n-1}\omega^{q,\alpha}_{k} \varphi_{M}+2\tau^{\alpha}\kappa p^{2}sg^{n}_{2}. \end{gathered}$$
The unknown weights \(\boldsymbol{\alpha}^{n}\) depend on \(\boldsymbol {\alpha}^{n-k}\), \(k=0,1,\ldots,n\), at their previous time levels and are found via a recursive style; once \(\boldsymbol{\alpha}^{n}\) is obtained, \(\alpha ^{n}_{-1}\), \(\alpha^{n}_{M+1}\) can further be determined with the help of Eqs. (3.9)-(3.10). On the other hand, A is an \((M+1)\times(M+1)\) tri-diagonal matrix, therefore the system can be performed by the well-known Thomas algorithm, which simply needs the arithmetic operation cost \(O(M+1)\).