A local meshless method to approximate the time-fractional telegraph equation

Abstract

In the present work, we investigate the numerical solution of time-fractional telegraph equation by a local meshless method. The fractional-order derivative is defined in the Caputo’s sense. The time semi-discretization was carried out using finite difference method followed by radial basis function-based spatial discretization. The theoretical convergence analysis and stability analysis of time semi-discrete scheme are also proved. Several test problems with regular and irregular domains with uniform and non-uniform points are considered. To demonstrate the accuracy and efficiency of the proposed method, we compared the analytical and numerical solution of the proposed problem.

1 Introduction

In recent decades, fractional calculus has become a powerful tool because many complex problems can be easily and successfully modeled in various fields by fractional calculus. Fractional calculus has many applications in the fields of science, engineering, and finance [2, 4, 6, 8, 24, 48, 57]. Fractional partial differential equations are obtained by replacing integer-order derivatives of partial differential equations with fractional-order derivatives. A lot of publications related to fractional partial differential equations are presented; for example, see [10, 19, 47, 52, 62, 64].

In this paper, we are dealing with following time-fractional telegraph equation

$$\begin{aligned} \left\{ \begin{array}{llc} _{0}^{c}{\mathscr {D}}_{t}^{\alpha }u({\mathbf{x}},t)+_{0}^{c}{\mathscr {D}}_{t}^{\alpha -1}u({\mathbf{x}},t)+u({\mathbf{x}},t)=\Delta u({\mathbf{x}},t)+f({\mathbf{x}},t), &{} &{}({\mathbf{x}},t) \in \Omega \times (0,T], \\ u({\mathbf{x}},0)=\xi ({\mathbf{x}}),~~\frac{\partial u({\mathbf{x}},0)}{\partial t}=\psi ({\mathbf{x}}) ,&{} &{} {\mathbf{x}} \in \Omega \\ u({\mathbf{x}},t)=\zeta ({\mathbf{x}},t) , &{} &{}{\mathbf{x}} \in \partial \Omega , \end{array} \right. \end{aligned}$$

(1.1)

where $1< \alpha < 2$, $T>0$ and $\xi ({\mathbf{x}})$, $\psi ({\mathbf{x}})$, $\zeta ({\mathbf{x}},t)$ and $f({\mathbf{x}},t)$ are sufficiently smooth functions on closed and bounded domain $\Omega \subset {\mathbb {R}}^{2}$, with boundary $\partial \Omega$. Moreover, for any given positive integer, the Caputo’s differential operator k, $_{0}^{c}{\mathscr {D}}_{t}^{\alpha }u({\mathbf{x}},t)$ is defined as

$$\begin{aligned} _{0}^{c}{\mathscr {D}}_{t}^{\alpha }u({\mathbf{x}},t)=\left\{ \begin{array}{lc} \frac{1}{\Gamma (k-\alpha )} \int \limits _{0}^{t}\frac{\partial ^{k} u({\mathbf{x}},s)}{\partial s^{k}}\frac{\mathrm{d}s}{(t-s)^{\alpha -(k-1)}}, &{} k-1<\alpha <k, \\ \frac{\partial ^{k} u({\mathbf{x}},t)}{\partial t^{k}}, &{} \alpha =k. \end{array} \right. \end{aligned}$$

(1.2)

It is found that the classical telegraph equation has many applications in neutron transport [59], random walk of suspension flows [3], signal analysis, propagation and transmission of electrical signals [36], etc. Recently, fractional telegraph equation has been solved by many authors. Mittal and Bhatia [37] developed differential quadrature method based on cubic B-spline basis function to solve hyperbolic telegraph equation. Ömer Oruç [41] considered two-dimensional hyperbolic telegraph equation using finite difference method in time and Hermite wavelets approach for space. In [28] Jiang and Lin solved time-fractional telegraph equation using reproducing kernel theorem. Kumar et al. [31] considered a generalized time-fractional telegraph-type equation using a numerical scheme based on the finite difference method. Liang et al. [33] discussed time-fractional telegraph equation using a fast high-order difference scheme. In [17] Dehghan et al. solved four different types of linear telegraph equations using He’s variational iteration method. Li and Cao [32] used finite difference method to solve linear time-fractional telegraph equation. In [39] Adomian decomposition method is derived by Momani to solve space- and time-fractional telegraph equation. Chen et al. [5] derived the analytical solution of time-fractional telegraph equation and applied the method of separating variables. Yildirim [61] proposed homotopy perturbation method (HPM) to solve space- and time-fractional telegraph equation. Wang and Mei [60] presented the time-fractional telegraph equation using a generalized finite difference method in time and Legendre spectral Galerkin method in space. In [7] homotopy analysis method (HAM) is presented by Das et al. to solve the time-fractional telegraph equation. A finite difference method in time and Galerkin finite element method in space are used to solve the time–space-fractional telegraph equation by Zhao and Li [63].

Recently, radial basis function (RBF)-based meshfree methods are increasingly attracted the attention of the researchers for solving fractional partial differential equations. Several problems [6, 12, 18, 27, 35, 46, 54] of fractional partial differential equations have been solved by many researchers and references therein. Abbaszadeh and Dehghan [1] considered the distributed order time-fractional diffusion-wave equation using interpolating element-free Galerkin method. Recently, Kumar et al. [29, 30] considered time-fractional linear and nonlinear diffusion-wave equation using RBF-based meshless local collocation method, respectively. For further applications of meshless methods, we refer Nikan et al. [40], Dehghan et al. [9, 11, 13, 15], Ghehsareh et al. [20,21,22,23], Liu et al. [34], Salehi et al. [49] Shivanian and Jafarabadi [54, 55] and references therein. Recently, Oruç [45] developed two meshless methods, in which one is based on local radial basis function and other is based on barycentric rational interpolation for solving 2D viscoelastic wave equation, and also the same author in [42,43,44] proposed a meshless method based on multiple-scale Pascal polynomials for different problems. This method is easily implemented for the problems of irregular domain as it is mesh independent. Hosseini et al. in [25, 26] developed radial basis function-based method and meshless local radial point interpolation (MLRPI) method to solve time-fractional telegraph equation, respectively. Shivanian [51] considered a time-fractional telegraph equation using spectral meshless radial point interpolation (SMRPI) method. In [14] a meshless local weak-strong (MLWS) method is derived by Dehghan and Ghesmati to solve a hyperbolic telegraph equation. In [16] a hyperbolic telegraph equation is solved using RBF collocation method by Dehghan and Shokri. The nonlinear time-fractional telegraph equation is solved by Sepehrian and Shamohammadi [50] with collocation method. Mohebbi et al. [38] proposed a RBF-based meshless method to solve time-fractional telegraph equation. Shivanian et al. [56] solved telegraph equation using meshless local radial point interpolation (MLRPI) method. Shivanian et al. [53] considered a time-fractional telegraph equation using meshless local Petrov–Galerkin (MLPG) method in which Galerkin weak form and moving least-squares (MLS) approximation is applied.

This manuscript is organized as follows: Time semi-discretization scheme is described in Sect. 2; furthermore, in this section stability and convergence analysis of the time discrete numerical scheme is also described. In Sect.3, we have briefly discussed the local collocation method and how the proposed method is numerically implemented. We examined some numerical experiments to demonstrate the computational efficiency and accuracy of the proposed method in Sect. 4. In Sect. 5, we end this paper finally, with the help of some concluding remark.

2 The time semi-discretization

In the present section, we will develop and analyze the time semi-discrete scheme of the proposed Eq. (1.1), and the Caputo’s fractional derivative $_{0}^{c}{\mathscr {D}}_{t}^{\alpha }u({\mathbf{x}},t)$ could be rewritten as follows

$$\begin{aligned} _{0}^{c}{\mathscr {D}}_{t}^{\alpha }u({\mathbf{x}},t)=\left\{ \begin{array}{lc} \frac{1}{\Gamma (1-\alpha )} \int \limits _{0}^{t}\frac{\partial u({\mathbf{x}},s)}{\partial s}\frac{\mathrm{d}s}{(t-s)^{\alpha }}, &{} 0<\alpha<1, \\ \frac{1}{\Gamma (2-\alpha )} \int \limits _{0}^{t}\frac{\partial ^{2} u({\mathbf{x}},s)}{\partial s^{2}}\frac{\mathrm{d}s}{(t-s)^{\alpha -1}}, &{} 1<\alpha <2. \end{array} \right. \end{aligned}$$

(2.1)

If $1<\alpha <2$, then $0<\alpha -1<1$, so

$$\begin{aligned} _{0}^{c}{\mathscr {D}}_{t}^{\alpha -1}u({\mathbf{x}},t)= & {} \frac{1}{\Gamma (1-(\alpha -1))} \int \limits _{0}^{t} \frac{\partial u({\mathbf{x}},s)}{\partial s}\frac{\mathrm{d}s}{(t-s)^{\alpha -1}}\nonumber \\= & {} \frac{1}{\Gamma (2-\alpha )} \int \limits _{0}^{t} \frac{\partial u({\mathbf{x}},s)}{\partial s}\frac{\mathrm{d}s}{(t-s)^{\alpha -1}}. \end{aligned}$$

(2.2)

For any positive integer N, we let $\delta t= \frac{T}{N}$, be the step size in time, and $t_{n}=n \delta t, ~ n \in \mathbb {N^{+}}$ are the temporal discretization points. Now, we define the notation as $u^{n-\frac{1}{2}}=\frac{1}{2}(u^{n}+u^{n-1})$, and $\delta _{t}u^{n-\frac{1}{2}}=\frac{1}{\delta t}(u^{n}-u^{n-1})$, together with $u^{n}$ being the abbreviation of the $u({\mathbf{x}},t_{n})$.

Lemma 1

Let us suppose $\eta (t) \in C^{2}[0,T]$ and $1<\alpha <2$, it holds that

$$\begin{aligned}&\int \limits _{0}^{t_{n}}\eta '(s)(t_{n}-s)^{1-\alpha }\mathrm{d}s\nonumber \\&\quad = \sum _{k=1}^{n}\frac{\eta (t_{k})-\eta (t_{k-1})}{\delta t} \int \limits _{t_{k-1}}^{t_{k}}(t_{n}-s)^{1-\alpha }\mathrm{d}s+R^{n},~1\le n \le N \end{aligned}$$

(2.3)

and

$$\begin{aligned} |R^{n}|\le \left( \frac{1}{2(2-\alpha )}+\frac{1}{2}\right) \delta t^{3-\alpha }\max \limits _{0\le t \le t_{n}}|\eta ''(t)|. \end{aligned}$$

(2.4)

Proof

See Sun et al. [58]. $\square$

Lemma 2

Let $1< \alpha < 2$, $a_{0}=\frac{1}{\delta t \Gamma (2-\alpha )}$ and $b_{k}=\frac{\delta t^{2-\alpha }}{ (2-\alpha )}\left[ (k+1)^{2-\alpha }-(k)^{2-\alpha }\right]$, then

$$\begin{aligned}&\left| \frac{1}{\Gamma (2-\alpha )} \int \limits _{0}^{t_{n}}\frac{\eta '(s)}{(t_{n}-s)^{\alpha -1}}\mathrm{d}s\right. \nonumber \\&\qquad \left. - a_{0}\left[ b_{0}\eta (t_{n})-\sum _{k=1}^{n-1}(b_{n-k-1}-b_{n-k})\eta (t_{k}) -b_{n-1}\eta (0)\right] \right| \nonumber \\&\quad \le \frac{1}{2\Gamma (2-\alpha )} \left( 1 +\frac{1}{(2-\alpha )}\right) \delta t^{3-\alpha }\max \limits _{0\le t \le t_{n}}|\eta ''(t)| \end{aligned}$$

(2.5)

Proof

Directly follows from Lemma 1. $\square$

Lemma 3

Let $b_{k}=\frac{\delta t^{2-\alpha }}{ (2-\alpha )}\left[ (k+1)^{2-\alpha }-(k)^{2-\alpha }\right]$, where $1< \alpha < 2$, $k=0,1,2,\ldots$, then

$$\begin{aligned} b_{0}>b_{1}>b_{2}>\ldots >b_{k}\rightarrow 0,~ \text {as}~ k \rightarrow \infty . \end{aligned}$$

Proof

See Sun et al. [58]. $\square$

For convention of the theory let us define,

$$\begin{aligned} v({\mathbf{x}},t)= & {} \frac{\partial u ({\mathbf{x}},t)}{\partial t} \end{aligned}$$

(2.6)

$$\begin{aligned} w({\mathbf{x}},t)= & {} \frac{1}{\Gamma (2-\alpha )} \int \limits _{0}^{t}\frac{\partial v ({\mathbf{x}},s)}{\partial s} \frac{\mathrm{d}s}{(t-s)^{\alpha -1}} \end{aligned}$$

(2.7)

$$\begin{aligned} z({\mathbf{x}},t)= & {} \frac{1}{\Gamma (2-\alpha )} \int \limits _{0}^{t}\frac{\partial u ({\mathbf{x}},s)}{\partial s} \frac{\mathrm{d}s}{(t-s)^{\alpha -1}} \end{aligned}$$

(2.8)

Now applying Taylor expansion on (2.6), we have

$$\begin{aligned} v^{n-\frac{1}{2}}=\delta _{t}u^{n-\frac{1}{2}}+r_{1}^{n-\frac{1}{2}} \end{aligned}$$

(2.9)

and the numerical scheme is

$$\begin{aligned}&w^{n-\frac{1}{2}}+z^{n-\frac{1}{2}}+u^{n-\frac{1}{2}}\nonumber \\&\quad =\Delta u^{n-\frac{1}{2}}+f^{n-\frac{1}{2}}+r_{2}^{n-\frac{1}{2}},~~n \ge 1, \end{aligned}$$

(2.10)

where $r_{1}^{n-\frac{1}{2}}$ and $r_{2}^{n-\frac{1}{2}}$ are the local truncation errors which is bounded by

$$\begin{aligned} |r_{1}^{n-\frac{1}{2}}| \le C_{1}\delta t ^{2},~~~~~|r_{2}^{n-\frac{1}{2}}| \le C_{2}\delta t^{2}. \end{aligned}$$

(2.11)

Discretizing Eqs. (2.7) and (2.8), we have

$$\begin{aligned} w({\mathbf{x}},t_{n})= & {} \frac{1}{\Gamma (2-\alpha )} \int \limits _{0}^{t_{n}}\frac{\partial v ({\mathbf{x}},t)}{\partial t} \frac{\mathrm{d}t}{(t_{n}-t)^{\alpha -1}} \\ z({\mathbf{x}},t_{n})= & {} \frac{1}{\Gamma (2-\alpha )} \int \limits _{0}^{t_{n}}\frac{\partial u ({\mathbf{x}},t)}{\partial t} \frac{\mathrm{d}t}{(t_{n}-t)^{\alpha -1}}, \end{aligned}$$

using Lemma 2, we have

$$\begin{aligned} w^{n}= & {} a_{0} \left[ b_{0}v^{n}-\sum _{k=1}^{n-1}(b_{n-k-1}-b_{n-k})v^{k} -b_{n-1}v^{0}\right] \nonumber \\&+{\mathcal {O}}(\delta t^{3-\alpha }), \end{aligned}$$

(2.12)

$$\begin{aligned} z^{n}= & {} a_{0} \left[ b_{0}u^{n}-\sum _{k=1}^{n-1}(b_{n-k-1}-b_{n-k})u^{k} -b_{n-1}u^{0}\right] \nonumber \\&+{\mathcal {O}}(\delta t^{3-\alpha }). \end{aligned}$$

(2.13)

Now define the operator [58]

$$\begin{aligned} {\mathcal {P}}(v^{n},q)=\left[ b_{0}v^{n}-\sum _{k=1}^{n-1}(b_{n-k-1}-b_{n-k})v^{k} -b_{n-1}q\right] \end{aligned},$$

and using both the initial condition $v^{0}=v({\mathbf{x}},0)=\psi$ and $u^{0}=u({\mathbf{x}},0)=\xi,$ we have

$$\begin{aligned} w^{n-\frac{1}{2}}= & {} a_{0} {\mathcal {P}}(v^{n-\frac{1}{2}},\psi )+\left( r_{3}\right) ^{n-\frac{1}{2}}, \end{aligned}$$

(2.14)

$$\begin{aligned} z^{n-\frac{1}{2}}= & {} a_{0} {\mathcal {P}}(u^{n-\frac{1}{2}},\xi )+\left( r_{4}\right) ^{n-\frac{1}{2}}, \end{aligned}$$

(2.15)

where

$$\begin{aligned} |\left( r_{3}\right) ^{n-\frac{1}{2}}| \le C_{3}\delta t ^{3-\alpha } ~~\text {and}~~ |\left( r_{4}\right) ^{n-\frac{1}{2}}| \le C_{4}\delta t ^{3-\alpha } \end{aligned}$$

(2.16)

Now substituting (2.9) into (2.15), we have

$$\begin{aligned} w^{n-\frac{1}{2}}=a_{0} {\mathcal {P}}(\delta _{t}u^{n-\frac{1}{2}},\psi )+a_{0} {\mathcal {P}}(r_{1}^{n-\frac{1}{2}},0)+\left( r_{3}\right) ^{n-\frac{1}{2}}, \end{aligned}$$

(2.17)

now substituting (2.15) and above expression in (2.10), we have

$$\begin{aligned}&a_{0} {\mathcal {P}}(\delta _{t}u^{n-\frac{1}{2}},\psi )+a_{0} {\mathcal {P}}(r_{1}^{n-\frac{1}{2}},0)\nonumber \\&\qquad +r_{3}^{n-\frac{1}{2}} +a_{0}{\mathcal {P}}(u^{n-\frac{1}{2}},\xi )+r_{4}^{n-\frac{1}{2}} = \Delta u^{n-\frac{1}{2}}+f^{n-\frac{1}{2}}\nonumber \\&\qquad + r_{2}^{n-\frac{1}{2}} \nonumber \\&a_{0} {\mathcal {P}}(\delta _{t}u^{n-\frac{1}{2}}, \psi )+a_{0}{\mathcal {P}}(u^{n-\frac{1}{2}},\xi ) \nonumber \\&\quad =\Delta u^{n-\frac{1}{2}}+f^{n-\frac{1}{2}}+R^{n-\frac{1}{2}} \end{aligned}$$

(2.18)

where

$$\begin{aligned} R^{n-\frac{1}{2}}= & {} -\left\{ a_{0}{\mathcal {P}}(r_{1}^{n-\frac{1}{2}},0) +r_{3}^{n-\frac{1}{2}}+r_{4}^{n-\frac{1}{2}} \right\} +r_{2}^{n-\frac{1}{2}} \\ |R^{n-\frac{1}{2}}|= & {} \big |-\left\{ a_{0}{\mathcal {P}} (r_{1}^{n-\frac{1}{2}},0)+r_{3}^{n-\frac{1}{2}}+r_{4}^{n-\frac{1}{2}} \right\} +r_{2}^{n-\frac{1}{2}}\big | \\\le & {} \left\{ a_{0}\left[ b_{0}r_{1}^{n-\frac{1}{2}}+\sum _{k=1}^{n-1}(b_{n-k-1}-b_{n-k}) r_{1}^{k-\frac{1}{2}}\right] \right. \\&\left. +r_{3}^{n-\frac{1}{2}}+r_{4}^{n-\frac{1}{2}} \right\} +r_{2}^{n-\frac{1}{2}} \\\le & {} \left\{ a_{0}\left[ b_{0}C_{1}\delta t^{2}+\sum _{k=1}^{n-1}(b_{n-k-1}-b_{n-k})C_{1}\delta t^{2}\right] \right. \\&\left. +C_{3}\delta t^{3-\alpha }+C_{4}\delta t^{3-\alpha } \right\} +C_{2}\delta t^{2}\\= & {} \left\{ a_{0}\left[ b_{0}C_{1}\delta t^{2}+(b_{0}-b_{n-1})C_{1}\delta t^{2}\right] \right. \\&\left. +C_{3}\delta t^{3-\alpha }+C_{4}\delta t^{3-\alpha } \right\} +C_{2}\delta t^{2}\\\le & {} \left\{ a_{0}\left[ 2 b_{0}C_{1}\delta t^{2}\right] \right. \\&\left. +C_{3}\delta t^{3-\alpha }+C_{4}\delta t^{3-\alpha } \right\} +C_{2}\delta t^{2} \\= & {} \left\{ \frac{1}{ \delta t \Gamma (2-\alpha )}\left[ \frac{2\delta t^{2-\alpha }C_{1}\delta t^{2} }{(2-\alpha )}\right] \right. \\&\left. +C_{3}\delta t^{3-\alpha }+C_{4}\delta t^{3-\alpha } \right\} +C_{2}\delta t^{2} \\\le & {} C \delta t^{3-\alpha }. \end{aligned}$$

where $C=\left\{ \left[ { \frac{2C_{1}}{(2-\alpha )\Gamma (2-\alpha )}+C_{3}+C_{4}}\right] + C_{2} \right\}$. Now omitting the truncation error term $R^{n-\frac{1}{2}}$ from Eq. (2.18), with exact value $u{^{n}}$ is approximated by its numerical approximation $U{^{n}}$, The resulted numerical scheme is as follows:

$$\begin{aligned}&a_{0}{\mathcal {P}}(\delta _{t}U^{n-\frac{1}{2}} + U^{n-\frac{1}{2}},\psi + \xi ) + U^{n-\frac{1}{2}} \nonumber \\&\quad = \Delta U^{n-\frac{1}{2}}+f^{n-\frac{1}{2}},~~~1 \le n \le N. \end{aligned}$$

(2.19)

The above equation can be written in more precise form as

$$\begin{aligned} {\mathcal {L}}U^{n}=F, \end{aligned}$$

(2.20)

where ${\mathcal {L}}$ and F are given as:

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {L}}U^{n} &{}= \frac{1}{\delta t\Gamma (2-\alpha )} \frac{b_{0}}{\delta t} U^{n} + \frac{1}{\delta t\Gamma (2-\alpha )}\frac{b_{0}}{2} U^{n} + \frac{1}{2}U^{n} - \frac{1}{2}\Delta U^{n} \\ F &{}= \frac{1}{\delta t\Gamma (2-\alpha )} \frac{b_{0}}{\delta t} U^{n-1} - \frac{1}{\delta t\Gamma (2-\alpha )}\frac{b_{0}}{2} U^{n-1} - \frac{1}{2} U^{n-1}+ \frac{1}{2}\Delta U^{n-1}+ \\ &{} \frac{1}{\delta t\Gamma (2-\alpha )} \sum _{k=1}^{n-1}(b_{n-k-1}-b_{n-k})(\delta _{t}U^{k-\frac{1}{2}}+U^{k-\frac{1}{2}}) +\frac{1}{\delta t \Gamma (2-\alpha )}b_{n-1}(\psi +\xi )\\ &{}+ f^{n-\frac{1}{2}}. \end{array}\right. } \end{aligned}$$

2.1 Convergence and stability analysis

This section devoted to discuss the stability of the time semi-discrete scheme and also to prove that the time discrete scheme is convergent with convergence order $\delta t^{3-\alpha }$ in $\mathrm {L}_{2}$ norm.

Lemma 4

For any function $\eta =\{\eta _{1},\eta _{2},\ldots \}$, $\theta$ with $1<\alpha <2$, we have

$$\begin{aligned} \sum _{i=1}^{n}{\mathcal {P}}(\eta _{i},\theta )\eta _{i}\ge \frac{t_{n}^{1-\alpha }}{2}\delta t \sum _{i=1}^{n} \eta _{i}^{2}-\frac{t_{n}^{2-\alpha }}{2(2-\alpha )}\theta ^{2} \end{aligned}$$

Proof

See [58]. $\square$

As the considered problem (1.1) is linear, it is sufficient to do analysis for homogeneous boundary conditions, i.e., $\zeta ({\mathbf{x}},t)=0$.

Theorem 1

Let $U^{n} \in H _{0}^{1}$, the time discrete scheme (2.19) is unconditionally stable and we have the following inequality:

$$\begin{aligned} \Vert U^{n}\Vert ^{2} \le C\left( \Vert \xi \Vert ^{2}+\Vert \nabla \xi \Vert ^{2}+ \Vert \psi +\xi \Vert ^{2}+\max \limits _{1 \le j\le n} \Vert f^{j-\frac{1}{2}}\Vert ^{2}\right) . \end{aligned}$$

Proof

Multiplying Eq. (2.19) by $(\delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}})$ and integrating over $\Omega$ give

$$\begin{aligned}&a_{0}\Big \{b_{0}\left( \delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}, \delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\right) \nonumber \\&\qquad -\sum _{k=1}^{n-1}\left( b_{n-k-1}-b_{n-k}\right) \left( \delta _{t}U^{k-\frac{1}{2}} +U^{k-\frac{1}{2}},\delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\right) \nonumber \\&\qquad - b_{n-1}\left( \psi +\xi ,\delta _{t}U^{n-\frac{1}{2}}+ U^{n-\frac{1}{2}}\right) \Big \} + \left( U^{n-\frac{1}{2}}, \delta _{t}U^{n-\frac{1}{2}} +U^{n-\frac{1}{2}}\right) \nonumber \\&\quad =\left( \Delta U^{n-\frac{1}{2}},\delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\right) + \left( f^{n-\frac{1}{2}},\delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\right) , \end{aligned}$$

(2.21)

where $\left( \cdot ,\cdot \right)$ is used for inner product. Now we are using the following fact

$$\begin{aligned}&\left( U^{n-\frac{1}{2}}, \delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\right) \nonumber \\&\quad = \left( U^{n-\frac{1}{2}}, \delta _{t}U^{n-\frac{1}{2}}\right) +\left( U^{n-\frac{1}{2}},U^{n-\frac{1}{2}}\right) \\&\quad = \int _{\Omega }\left( \frac{U^{n}+ U^{n-1}}{2} \right) \left( \frac{U^{n}-U^{n-1}}{\delta t} \right) {\hbox {d}}\Omega + \int _{\Omega } \left( U^{n-\frac{1}{2}}\right) ^{2} {\hbox {d}}\Omega \\&\quad = \frac{1}{2 \delta t} \int _{\Omega } \left[ (U^{n})^{2} -( U^{n-1})^{2}\right] {\hbox {d}}\Omega +\int _{\Omega }\left( U^{n-\frac{1}{2}}\right) ^{2} {\hbox {d}}\Omega \\&\quad = \frac{1}{2 \delta t} \left( \Vert U^{n} \Vert ^{2}-\Vert U^{n-1} \Vert ^{2}\right) +\Vert U^{n-\frac{1}{2}} \Vert ^{2} \end{aligned}$$

and

$$\begin{aligned}&\left( \Delta U^{n-\frac{1}{2}}, \delta _{t}U^{n-\frac{1}{2}} +U^{n-\frac{1}{2}}\right) \nonumber \\&\quad = - \left( \nabla U^{n-\frac{1}{2}}, \nabla \left( \delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\right) \right) \\&\quad = -\left( \nabla U^{n-\frac{1}{2}}, \nabla \delta _{t}U^{n-\frac{1}{2}} +\nabla U^{n-\frac{1}{2}}\right) \\&\quad = -\left[ \left( \nabla U^{n-\frac{1}{2}},\nabla \delta _{t}U^{n -\frac{1}{2}}\right) +\left( \nabla U^{n-\frac{1}{2}},\nabla U^{n -\frac{1}{2}}\right) \right] \\&\quad = -\int _{\Omega }\left( \frac{\nabla U^{n}+\nabla U^{n-1}}{2} \right) \left( \frac{\nabla U^{n}-\nabla U^{n-1}}{\delta t} \right) {\hbox {d}}\Omega \\&\qquad - \int _{\Omega }\left( \nabla U^{n-\frac{1}{2}}\right) ^{2} {\hbox {d}}\Omega \\&\quad = - \frac{1}{2 \delta t} \int _{\Omega } \left[ (\nabla U^{n})^{2} -(\nabla U^{n-1})^{2}\right] {\hbox {d}}\Omega -\int _{\Omega }\left( \nabla U^{n -\frac{1}{2}}\right) ^{2} {\hbox {d}}\Omega \\&\quad = - \frac{1}{2 \delta t} \left( \Vert \nabla U^{n} \Vert ^{2}-\Vert \nabla U^{n-1} \Vert ^{2}\right) -\Vert \nabla U^{n-\frac{1}{2}} \Vert ^{2} \end{aligned}$$

we have

$$\begin{aligned}&a_{0} \Big \{ b_{0}\Vert \delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\Vert ^{2}\\&\qquad -\sum _{k=1}^{n-1}\left( b_{n-k-1}-b_{n-k} \right) \Vert \delta _{t}U^{k-\frac{1}{2}}\\&\qquad +U^{k-\frac{1}{2}}\Vert \Vert \delta _{t}U^{n-\frac{1}{2}} +U^{n-\frac{1}{2}}\Vert \\&\qquad -b_{n-1}\Vert \psi +\xi \Vert \Vert \delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\Vert \Big \}\\&\qquad +\frac{1}{2 \delta t} \left( \Vert U^{n}\Vert ^{2}-\Vert U^{n-1}\Vert ^{2}\right) + \Vert U^{n-\frac{1}{2}}\Vert ^{2}\\&\quad \le -\Vert \nabla U^{n-\frac{1}{2}} \Vert ^{2} - \frac{1}{2 \delta t} \left( \Vert \nabla U^{n} \Vert ^{2}-\Vert \nabla U^{n-1} \Vert ^{2}\right) \\&\qquad +\Vert f^{n-\frac{1}{2}}\Vert \Vert \delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\Vert ; \end{aligned}$$

now taking the summation from $n=1$ to $n=m$ on both sides of the above inequality, we have

$$\begin{aligned}&a_{0}\sum _{n=1}^{m}\left\{ b_{0}\Vert \delta _{t}U^{n-\frac{1}{2}} +U^{n-\frac{1}{2}}\Vert \right. \nonumber \\&\qquad -\sum _{k=1}^{n-1}\left( b_{n-k-1}-b_{n-k} \right) \Vert \delta _{t}U^{k-\frac{1}{2}}\nonumber \\&\qquad \left. +U^{k-\frac{1}{2}}\Vert -b_{n-1}\Vert \psi +\xi \Vert \right\} \Vert \delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\Vert \nonumber \\&\qquad + \frac{1}{2 \delta t} \left( \Vert U^{m}\Vert ^{2}-\Vert U^{0}\Vert ^{2}\right) + \sum _{n=1}^{m} \Vert U^{n-\frac{1}{2}}\Vert ^{2} \nonumber \\&\quad \le - \frac{1}{2 \delta t} \left( \Vert \nabla U^{m} \Vert ^{2}-\Vert \nabla U^{0} \Vert ^{2}\right) -\sum _{n=1}^{m}\Vert \nabla U^{n-\frac{1}{2}}\Vert ^{2}\nonumber \\&\qquad +\sum _{n=1}^{m} \Vert f^{n-\frac{1}{2}}\Vert \Vert \delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\Vert . \end{aligned}$$

(2.22)

Also using inequality $|xy| \le \frac{1}{2 \theta } x^{2}+ \frac{\theta }{2}y^{2}$, together with $\theta = \frac{t_{m}^{1-\alpha }}{\Gamma (2-\alpha )}$, we get

$$\begin{aligned}&\sum _{n=1}^{m} \Vert f^{n-\frac{1}{2}}\Vert \Vert \delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\Vert \\&\quad \le \frac{\Gamma (2-\alpha )}{2 t_{m}^{1-\alpha } } \sum _{n=1}^{m} \Vert f^{n-\frac{1}{2}}\Vert ^{2}\\&\qquad +\frac{t_{m}^{1-\alpha } }{2 \Gamma (2-\alpha )} \sum _{n=1}^{m} \Vert \delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\Vert ^{2}. \end{aligned}$$

Now using above relation together with Lemma 4, we have

$$\begin{aligned}&\frac{t_{m}^{1-\alpha }}{2\delta t\Gamma (2-\alpha )} \delta t \sum _{n=1}^{m} \Vert \delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\Vert ^{2}\\&\qquad -\frac{t_{m}^{2-\alpha }}{2 \delta t \Gamma (3-\alpha )} \Vert \psi +\xi \Vert ^{2} + \frac{1}{2 \delta t}\left( \Vert U^{m}\Vert ^{2}-\Vert U^{0}\Vert ^{2}\right) \\&\qquad + \sum _{n=1}^{m} \Vert U^{n-\frac{1}{2}}\Vert ^{2}\\&\quad \le - \frac{1}{2 \delta t} \left( \Vert \nabla U^{m} \Vert ^{2}-\Vert \nabla U^{0} \Vert ^{2}\right) -\sum _{n=1}^{m}\Vert \nabla U^{n-\frac{1}{2}} \Vert ^{2}\\&\qquad +\frac{\Gamma (2-\alpha )}{2 t_{m}^{1-\alpha } } \sum _{n=1}^{m} \Vert f^{n-\frac{1}{2}}\Vert ^{2}+ \frac{t_{m}^{1-\alpha } }{2 \Gamma (2-\alpha )} \sum _{n=1}^{m} \Vert \delta _{t}U^{n-\frac{1}{2}}+U^{n-\frac{1}{2}}\Vert ^{2}. \end{aligned}$$

Now simplifying above relation and switching index from m to n, we have

$$\begin{aligned}&\Vert U^{n}\Vert ^{2} +\Vert \nabla U^{n} \Vert ^{2}+2 \delta t \sum _{k=1}^{n}\Vert U^{k-\frac{1}{2}}\Vert ^{2}+2 \delta t \sum _{k=1}^{n}\Vert \nabla U^{k-\frac{1}{2}}\Vert ^{2}\nonumber \\&\quad \le \left( \Vert U^{0}\Vert ^{2}+\Vert \nabla U^{0} \Vert ^{2}\right) + \frac{t_{n}^{2-\alpha }}{\Gamma (3-\alpha )}\Vert \psi +\xi \Vert ^{2} \nonumber \\&\qquad + \Gamma (2-\alpha )t_{n}^{\alpha -1} \delta t \sum _{j=1}^{n} \Vert f^{j-\frac{1}{2}}\Vert ^{2}, \end{aligned}$$

(2.23)

$$\begin{aligned}&\Vert U^{n} \Vert ^{2} \le \left( \Vert \nabla U^{0} \Vert ^{2} +\Vert U^{0}\Vert ^{2}\right) + \frac{t_{n}^{2-\alpha }}{\Gamma (3-\alpha )} \Vert \psi +\xi \Vert ^{2}\nonumber \\&\qquad + \Gamma (2-\alpha )t_{n}^{\alpha -1} n \delta t \max \limits _{1 \le j \le n } \Vert f^{j-\frac{1}{2}}\Vert ^{2}, \end{aligned}$$

(2.24)

$$\begin{aligned}&\Vert U^{n} \Vert ^{2} \le \left( \Vert \nabla U^{0} \Vert ^{2} +\Vert U^{0}\Vert ^{2}\right) + \frac{T^{2-\alpha }}{\Gamma (3-\alpha )} \Vert \psi +\xi \Vert ^{2}\nonumber \\&\qquad + \Gamma (2-\alpha )T^{\alpha } \max \limits _{1 \le j \le n } \Vert f^{j-\frac{1}{2}}\Vert ^{2}, \nonumber \\&\quad = \left( \Vert \nabla \xi \Vert ^{2} +\Vert \xi \Vert ^{2}\right) + \frac{T^{2-\alpha }}{\Gamma (3-\alpha )} \Vert \psi +\xi \Vert ^{2}\nonumber \\&\qquad +\Gamma (2-\alpha )T^{\alpha } \max \limits _{1 \le j \le n } \Vert f^{j-\frac{1}{2}}\Vert ^{2}, \end{aligned}$$

(2.25)

where $\psi =U_{t}^{0}$ and $\xi =U^{0}$. Therefore, we have

$$\begin{aligned} \Vert U^{n}\Vert ^{2} \le C\left( \Vert \nabla \xi \Vert ^{2} +\Vert \xi \Vert ^{2}+ \Vert \psi +\xi \Vert ^{2} + \max \limits _{1 \le j \le n }\Vert f^{j-\frac{1}{2}}\Vert ^{2}\right) , \end{aligned}$$

where $C=\left( 1+\frac{T^{2-\alpha }}{\Gamma (3-\alpha )}+T^{\alpha }\Gamma (2-\alpha )\right) .$ $\square$

Theorem 2

Let $u^{n}$ and $U^{n}$ both belonging to $H^{1}$ be the analytical and numerical solution of (2.18) and (2.19), respectively, then time semi-discrete scheme defined by (2.19) have ${\mathcal {O}}(\delta t^{3-\alpha })$ convergence order .

Proof

Let us define ${\mathcal {E}}^{n}=u^{n}-U^{n}$ with $n \ge 1$, and also ${\mathcal {E}}^{0}=0$. From Eqs. (2.18) and (2.19), we get

$$\begin{aligned}&a_{0}\left\{ b_{0} \left( \delta _{t}{\mathcal {E}}^{n-\frac{1}{2}}+{\mathcal {E}}^{n-\frac{1}{2}}\right) \right. \nonumber \\&\qquad \left. -\sum _{k=1}^{n-1} (b_{n-k-1}-b_{n-k})\left( \delta _{t}{\mathcal {E}}^{k-\frac{1}{2}}+{\mathcal {E}}^{k-\frac{1}{2}}\right) \right\} +{\mathcal {E}}^{n-\frac{1}{2}}\nonumber \\&\quad = \Delta {\mathcal {E}}^{n-\frac{1}{2}}+R^{n-\frac{1}{2}}, \end{aligned}$$

(2.26)

Multiplying the above equation by $(\delta _{t}{\mathcal {E}}^{n-\frac{1}{2}}+{\mathcal {E}}^{n-\frac{1}{2}})$, and integrating over $\Omega$, gives

$$\begin{aligned}&a_{0} \left\{ b_{0} \Vert \delta _{t}{\mathcal {E}}^{n-\frac{1}{2}}+{\mathcal {E}}^{n-\frac{1}{2}}\Vert \right. \\&\qquad \left. -\sum _{k=1}^{n-1} (b_{n-k-1}-b_{n-k})\Vert \delta _{t}{\mathcal {E}}^{k-\frac{1}{2}}+{\mathcal {E}}^{k-\frac{1}{2}}\Vert \right\} \\&\qquad \Vert \delta _{t}{\mathcal {E}}^{n-\frac{1}{2}}+{\mathcal {E}}^{n-\frac{1}{2}}\Vert \\&\qquad + \frac{1}{2 \delta t}\left( \Vert {\mathcal {E}}^{n}\Vert ^{2}-\Vert {\mathcal {E}}^{n-1}\Vert ^{2}\right) +\Vert {\mathcal {E}}^{n-\frac{1}{2}}\Vert ^{2}\\&\quad = -\frac{1}{2 \delta t} \left( \Vert \nabla {\mathcal {E}}^{n}\Vert ^{2}-\Vert \nabla {\mathcal {E}}^{n-1}\Vert ^{2} \right) -\Vert \nabla {\mathcal {E}}^{n-\frac{1}{2}}\Vert ^{2} \\&\qquad + ( R^{n-\frac{1}{2}}, \delta _{t}{\mathcal {E}}^{n-\frac{1}{2}}+{\mathcal {E}}^{n-\frac{1}{2}} ) \end{aligned}$$

Now summing the above relation from $n=1$ to m, we have

$$\begin{aligned}&\sum _{n=1}^{m}\frac{1}{\delta t\Gamma (2-\alpha )} \left\{ b_{0} \Vert \delta _{t}{\mathcal {E}}^{n-\frac{1}{2}}+{\mathcal {E}}^{n-\frac{1}{2}}\Vert \right. \\&\quad \left. -\sum _{k=1}^{n-1} (b_{n-k-1}-b_{n-k})\Vert \delta _{t}{\mathcal {E}}^{k-\frac{1}{2}}+{\mathcal {E}}^{k-\frac{1}{2}}\Vert \right\} \\&\qquad \Vert \delta _{t}{\mathcal {E}}^{n-\frac{1}{2}}+{\mathcal {E}}^{n-\frac{1}{2}}\Vert \\&\quad + \frac{1}{2 \delta t} \left( \Vert {\mathcal {E}}^{m}\Vert ^{2}-\Vert {\mathcal {E}}^{0}\Vert ^{2}\right) \\&\qquad \le - \frac{1}{2 \delta t} \left( \Vert \nabla {\mathcal {E}}^{m}\Vert ^{2}-\Vert \nabla {\mathcal {E}}^{0}\Vert ^{2} \right) -\sum _{n=1}^{m}\Vert \nabla {\mathcal {E}}^{n-\frac{1}{2}}\Vert ^{2} \\&\qquad + \sum _{n=1}^{m}\Vert R^{n-\frac{1}{2}}\Vert \Vert \delta _{t}{\mathcal {E}}^{n-\frac{1}{2}}+{\mathcal {E}}^{n-\frac{1}{2}}\Vert \end{aligned}$$

Now application of Lemma 4 yields

$$\begin{aligned}&\frac{t_{m}^{1-\alpha }}{2 \delta t\Gamma (2-\alpha )}\delta t \sum _{n=1}^{m} \Vert \delta _{t}{\mathcal {E}}^{n-\frac{1}{2}}+{\mathcal {E}}^{n-\frac{1}{2}} \Vert ^{2}\nonumber \\&\qquad +\frac{1}{2 \delta t} \Vert {\mathcal {E}}^{m}\Vert ^{2}+\sum _{n=1}^{m}\Vert {\mathcal {E}}^{n-\frac{1}{2}}\Vert ^{2}\nonumber \\&+\frac{1}{2 \delta t} \Vert \nabla {\mathcal {E}}^{m}\Vert ^{2}+\sum _{n=1}^{m}\Vert \nabla {\mathcal {E}}^{n-\frac{1}{2}}\Vert ^{2} \nonumber \\&\quad \le \sum _{n=1}^{m} \Vert R^{n-\frac{1}{2}}\Vert \Vert \delta _{t}{\mathcal {E}}^{n-\frac{1}{2}}+{\mathcal {E}}^{n-\frac{1}{2}} \Vert . \end{aligned}$$

(2.27)

Using inequality $|xy| \le \frac{1}{2 \theta } x^{2}+ \frac{\theta }{2}y^{2}$, together with $\theta = \frac{t_{m}^{1-\alpha }}{\Gamma (2-\alpha )}$, we have

$$\begin{aligned}&\sum _{n=1}^{m} \Vert R^{n-\frac{1}{2}}\Vert \Vert \delta _{t}{\mathcal {E}}^{n-\frac{1}{2}}+{\mathcal {E}}^{n-\frac{1}{2}} \Vert \\&\quad \le \frac{\Gamma (2-\alpha )}{2 t_{m}^{1-\alpha }}\sum _{n=1}^{m} \Vert R^{n-\frac{1}{2}}\Vert ^{2} \\&\qquad + \frac{t_{m}^{1-\alpha }}{2 \Gamma (2-\alpha )} \sum _{n=1}^{m} \Vert \delta _{t}{\mathcal {E}}^{n-\frac{1}{2}}+{\mathcal {E}}^{n-\frac{1}{2}}\Vert ^{2}. \end{aligned}$$

Using the above relation into Eq. (2.27), we have

$$\begin{aligned}&\frac{t_{m}^{1-\alpha }}{2\Gamma (2-\alpha )} \sum _{n=1}^{m} \Vert \delta _{t}{\mathcal {E}}^{n-\frac{1}{2}}+{\mathcal {E}}^{n-\frac{1}{2}} \Vert ^{2}\nonumber \\&\qquad +\frac{1}{2 \delta t} \Vert {\mathcal {E}}^{m}\Vert ^{2}+\sum _{n=1}^{m}\Vert {\mathcal {E}}^{n-\frac{1}{2}}\Vert ^{2}\nonumber \\&\qquad +\frac{1}{2 \delta t} \Vert \nabla {\mathcal {E}}^{m}\Vert ^{2}+\sum _{n=1}^{m}\Vert \nabla {\mathcal {E}}^{n-\frac{1}{2}}\Vert ^{2} \nonumber \\&\quad \le \frac{\Gamma (2-\alpha )}{2 t_{m}^{1-\alpha }} \sum _{n=1}^{m} \Vert R^{n-\frac{1}{2}}\Vert ^{2} \nonumber \\&+ \frac{t_{m}^{1-\alpha }}{2 \Gamma (2-\alpha )} \sum _{n=1}^{m} \Vert \delta _{t}{\mathcal {E}}^{n-\frac{1}{2}}+{\mathcal {E}}^{n-\frac{1}{2}}\Vert ^{2}. \end{aligned}$$

(2.28)

Multiplying both sides of the above inequality by $2 \delta t$ and switching index from m to n, with some calculation we get

$$\begin{aligned}&\Vert {\mathcal {E}}^{n}\Vert ^{2} +2\delta t\sum _{k=1}^{n} \Vert {\mathcal {E}}^{k-\frac{1}{2}}\Vert ^{2}+\Vert \nabla {\mathcal {E}}^{n}\Vert ^{2}\\&\qquad +2\delta t\sum _{k=1}^{n}\Vert \nabla {\mathcal {E}}^{k-\frac{1}{2}}\Vert ^{2} \le \delta t \Gamma (2-\alpha ) t_{n}^{\alpha -1} \sum _{j=1}^{n} \Vert R^{j-\frac{1}{2}}\Vert ^{2} \\&\quad \le n\delta t \Gamma (2-\alpha ) t_{n}^{\alpha -1} \max \limits _{1 \le j \le n } \Vert R^{j-\frac{1}{2}}\Vert ^{2}\\&\Vert {\mathcal {E}}^{n}\Vert ^{2} \le n\delta t \Gamma (2-\alpha ) t_{n}^{\alpha -1} \max \limits _{1 \le j \le n } \Vert R^{j-\frac{1}{2}}\Vert ^{2} \\&\quad \le T^{2} \Gamma (2-\alpha ) C^{2} \left( \delta t^{3-\alpha }\right) ^{2} \end{aligned}$$

Therefore, we have

$$\begin{aligned} \Vert {\mathcal {E}}^{n}\Vert \le C^{*}\delta t^{3-\alpha }, \end{aligned}$$

where $C^{*}=\sqrt{T^{2}\Gamma (2-\alpha )C^{2}}$, which completes the proof. $\square$

3 The local collocation method for Spatial discretization

The local collocation method has been developed by taking the global domain $\Omega$ that contains the M discretization points. For each discretization point ${\mathbf{x}}_{i}$, $i=1,2,\ldots ,M$, there is a local sub-domain $\Omega _{i}=$ $\{{\mathbf{x}}_{j}\}_{j=1}^{m_{i}}$, where $m_{i}$ are the closest points of the discretized point ${\mathbf{x}}_{i}$ in sub-domain $\Omega _{i}$. In the local interpolation form, the $u({\mathbf{x}},t_{n})$ can be numerically approximated as

$$\begin{aligned} {\hat{u}}({\mathbf{x}},t_{n})=\sum _{j=1}^{m_{i}}\lambda _{j}\phi (\Vert {\mathbf{x}} -{\mathbf{x}}_{j}\Vert )+\sum _{j=1}^{l}\gamma _{j} p_{j}({\mathbf{x}}), \end{aligned}$$

(3.1)

where $\{\lambda _{j}\}$ and $\{\gamma _{j}\}$ are coefficients at nth time level that need to be determined, $\phi$ is any considered radial basis function, considered norm is defined in Euclidean sense and $\{p_{j}(x)\}_{j=1}^{l}$ is the basis of the l-dimensional polynomial space of total degree $\le m-1$. The application of the interpolation condition on each sub-domain $\Omega _{i}$ give us

$$\begin{aligned} {\hat{u}}({\mathbf{x}}_{i},t_{n})= u({\mathbf{x}}_{i},t_{n}),~{i=1,2,\ldots ,m_{i}}, \end{aligned}$$

(3.2)

with l homogeneous conditions

$$\begin{aligned} \sum _{j=1}^{m_{i}}\lambda _{j} p_{k}({\mathbf{x}}_{j}) = 0,~{k=1,2,\ldots ,l}. \end{aligned}$$

(3.3)

Equations (3.2–3.3) can be written in the matrix form as

$$\begin{aligned} \begin{bmatrix} \Phi &{} P \\ P^{t} &{} O \\ \end{bmatrix} \begin{bmatrix} \lambda \\ \gamma \\ \end{bmatrix}= \begin{bmatrix} u^{n}\mid _{\Omega _{i}} \\ O \\ \end{bmatrix} \end{aligned}$$

(3.4)

where $\Phi :=[\phi \Vert {\mathbf{x}}_{j}-{\mathbf{x}}_{k}\Vert ]_{1\le j,k \le m_{i}}$, $P:= [p_{k}({\mathbf{x}}_{j})]_{1\le j \le m_{i}, 1 \le k \le l}$. The system (3.4) can be rewritten as

$$\begin{aligned} \Lambda _{\Omega _{i}}=A_{\Omega _{i}}^{-1}U_{\Omega _{i}}^{n}, \end{aligned}$$

(3.5)

where $\Lambda _{\Omega _{i}}=[\lambda _{1},\ldots ,\lambda _{m_{i}},\gamma _{1},\ldots ,\gamma _{l}]^{\intercal }$, $U_{\Omega _{i}}^{n}=[u({\mathbf{x}}_{1},t_{n}),\ldots ,u({\mathbf{x}}_{m_{i}},t_{n}),0,\ldots ,0]^{\intercal }$, and $A_{\Omega _{i}}$ is coefficient matrix defined in (3.4). At each stencil ${\mathbf{x}}_{i} \in {\Omega _{i}}$, we have approximation for ${\mathscr {D}}u({\mathbf{x}},t_{n})$ as;

$$\begin{aligned} {\mathscr {D}}{\hat{u}}({\mathbf{x}}_{i},t_{n})= & {} \sum _{j=1}^{m_{i}}\lambda _{j}{\mathscr {D}}\phi (\Vert {\mathbf{x}}_{i}-{\mathbf{x}}_{j}\Vert )+\sum _{j=1}^{l}\gamma _{j}{\mathscr {D}}p_{j}({\mathbf{x}}_{i}), \nonumber \\= & {} [{\mathscr {D}} \phi (\Vert {\mathbf{x}}_{i}-{\mathbf{x}}_{1}\Vert ),\ldots , {\mathscr {D}}\phi (\Vert {\mathbf{x}}_{i}-{\mathbf{x}}_{m_{i}}\Vert ), \nonumber \\&{\mathscr {D}} p_{1}({\mathbf{x}}_{i}), \ldots {\mathscr {D}} p_{l}({\mathbf{x}}_{i})]\Lambda _{\Omega _{i}} \nonumber \\= & {} {\mathscr {D}}\Psi _{\Omega _{i}} A_{\Omega _{i}}^{-1}U_{\Omega _{i}}^{n}, \end{aligned}$$

(3.6)

where $\Psi _{\Omega _{i}}=[ \phi (\Vert {\mathbf{x}}_{i}-{\mathbf{x}}_{1}\Vert ),\ldots , \phi (\Vert {\mathbf{x}}_{i}-{\mathbf{x}}_{m_{i}}\Vert ), p_{1}({\mathbf{x}}_{i}), \ldots p_{l}({\mathbf{x}}_{i})]$. For each i, the local operator ${\mathscr {D}}\Psi _{\Omega _{i}} A_{\Omega _{i}}^{-1}$ is a 1 $\times$ $m_{i}$ row vector. For the application of the local collocation method as defined in (3.6), for each collocation point ${\mathbf{x}}_{i} \in \Omega$, to the linear operator ${\mathscr {L}}$ as defined in Eq. (2.20), we have

$$\begin{aligned} {\mathscr {L}}\Psi _{\Omega _{i}} A_{\Omega _{i}}^{-1}U_{\Omega _{i}}^{n}= F_{i}, {\mathbf{x}}_{i}\in \Omega \end{aligned}$$

(3.7)

For each row we have only $m_{i}$ nonzero entry that will be stored in $M^{2}$ global coefficient matrix, and extra spaces will be filled by zeros. Then, we get the following linear system

$$\begin{aligned} \mathbf{LU }^{n}=\mathbf{F }. \end{aligned}$$

(3.8)

The resulting system is sparse because in each row we have only $m_{i}$ nonzero entries that can be calculated very efficiently and easily.

4 Numerical simulation and discussion

In this section, we present some numerical results for confirmation of the validity and efficiency of the present numerical method. To measure the accuracy of the method, we used maximum absolute error $L_{\infty }$ and root mean square error $L_\mathrm{rms}$ which are defined by using the definition

$$\begin{aligned} L_\mathrm{rms} = \sqrt{\frac{1}{M}\sum _{i=1}^{M}|u(x_{i},T) - U(x_{i},T)|^{2}}, ~~~~~~L_{\infty } = \max \limits _{1 \le i \le M}|u(x_{i},T)-U(x_{i},T)|, \end{aligned}$$

where M denotes the number of collocation points and $u(x_{i},T)$ and $U(x_{i},T)$ represent an exact and numerical solution of the considered problem. We calculated the convergence rate of the proposed method by using the formula $\frac{\log (E_{1}/E_{2})}{\log (\delta t_{1}/\delta t_{2})}$ where $E_{1}$ and $E_{2}$ are errors corresponding to temporal mesh size $\delta t_{1}$ and $\delta t_{2}$, respectively. To avoid the effect of the shape parameter $\epsilon$ over the numerical solution, the thin plate spline $r^{4}\ln (r)$ RBF is used for computation purpose. In all sub-domain, the number of collocation points is constant.

Example 1

Consider the following one-dimensional test problem

$$\begin{aligned} _{0}^{c}{\mathscr {D}}_{t}^{\alpha }u({x},t)+_{0}^{c}{\mathscr {D}}_{t}^{\alpha -1}u({x},t) + u({x},t)=\Delta u({x},t)+f({x},t). \end{aligned}$$

The initial conditions and boundary conditions are calculated using analytic solution

$$\begin{aligned} u(x,t) = t^{3}({\sin {x}})^{2}. \end{aligned}$$

The linear source term read $f(x,t)=\left( \frac{6t^{3-\alpha }}{\Gamma (4-\alpha )}+\frac{6t^{4-\alpha }}{\Gamma (5-\alpha )}\right) ({\sin {x}})^{2} -2t^{3}\cos {2x}+t^{3}({\sin {x}})^{2}$.

Table 1 The different errors along with cpu time for different values of $\delta t$ and $\alpha$ for Example 1

Considered problem is solved with the present method for different values of $\alpha$ and $\delta t$ on computational domain [0, 1]. The values of the root mean square error and absolute error for $M=501$ spatial points, $m=3$ at $T=1$s are reported in Table 1. From the data given in Table 1, we can easily observe that the numerical rate has a good agreement with theoretical rate of convergence, i.e., $O(\delta t^{3-\alpha })$. The graph of numerical solution and absolute error in the numerical solution for $\alpha =1.5$ and $N=640$ is plotted in Fig. 1.

Fig. 1

Graph of numerical solution and absolute error with $\alpha =1.5$ for Example 1

Table 2 The different errors along with cpu time for $\alpha =1.9$ and different values of m for Example 1

The values of different errors for $M=1000$, $N=1000$, $\alpha =1.9$ and different values of local points m at $T=1$ s are reported in Table 2. From the table, we observed that in most cases the improvement in the error is very small with respect to a increase in the number of local collocation points, while the computational time increases as m gets larger.

Example 2

Now consider the following one-dimensional test problem

$$\begin{aligned} _{0}^{c}{\mathscr {D}}_{t}^{\alpha }u({x},t)+_{0}^{c}{\mathscr {D}}_{t}^{\alpha -1}u({x},t)+u({x},t)=\Delta u({x},t)+f({x},t). \end{aligned}$$

The linear source term f(x, t) along with initial conditions and boundary conditions is calculated using analytic solution

$$\begin{aligned} u(x,t) = x~{\cos ({x}^2+{t}^2)}. \end{aligned}$$

It is not easy to find out the linear source term explicitly. Therefore, we used MATLAB symbolic calculation procedure for the same. We adopted this test problem from Hosseini et al. [25]; they solved the considered problem using RBF-based collocation method on the computational domain [0, 1]. The proposed method is compared with Hosseini et al. [25] at $T=1.0$ s, and the results are given in Table 3. From Table 3, we can see that the present method gives better accuracy than [25]. Finally, the graph of numerical approximation and absolute error for $\alpha =1.5$, $M=201$ spatial points, $m=3$ local points and $N=200$ is reported in Fig.2.

Table 3 Comparison of the present method with method [25] for different values of $\alpha$ and $\delta t$ for Example 2

Fig. 2

Graph of numerical approximation of the solution and absolute error for Example 2

Example 3

In this example, we consider two-dimensional test problem

$$\begin{aligned}&_{0}^{c}{\mathscr {D}}_{t}^{\alpha }u(x,y,t)+_{0}^{c}{\mathscr {D}}_{t}^{\alpha -1}u(x,y,t) +u(x,y,t)\\&\quad =\Delta u(x,y,t)+f(x,y,t). \end{aligned}$$

The initial conditions and boundary conditions are calculated using analytic solution

$$\begin{aligned} u(x,y,t) = t^{4}{\sin ({\pi x}+{\pi y})}. \end{aligned}$$

The linear source term read $f(x,t)=\left( \frac{24t^{4-\alpha }}{\Gamma (5-\alpha )}+\frac{24t^{5-\alpha }}{\Gamma (6-\alpha )}+2t^{4}\pi ^{2}\right) \sin ({\pi x}+{\pi y})+t^{4}{\sin ({\pi x}+{\pi y})}$.

In this example, we consider a rectangular domain $\Omega =[0,1]\times [0,1]$ with 2025 uniform points and 2060 non-uniform points as shown in Fig. 3. The computational errors with $m=5$ uniform and non-uniform points in local domain at $T=1.0$ s are listed in Table 4 and Table 5, respectively. This table ensures that the computational convergence order is close to the theoretical convergence order. The behavior of the numerical solution and absolute error for $\alpha =1.5$ and $N=160$ is plotted in Fig. 4 for uniform points and in Fig. 5 for non-uniform points.

Fig. 3

Rectangular domain with uniform and non-uniform points for Example 3

Table 4 The value of errors and cpu time with $\alpha =1.7, 1.9$ and different $\delta t$ on rectangular domain with uniform points at time $T=1.0$ s for Example 3

Fig. 4

Graph of numerical solution and absolute error on rectangular domain with uniform points for Example 3

Table 5 The value of errors and cpu time with $\alpha =1.7, 1.9$ and for different $\delta t$ on rectangular domain with non-uniform points at time $T=1.0$ s for Example 3

Fig. 5

Behavior of numerical solution and absolute error on the rectangular domain with non-uniform points for Example 3

Example 4

Finally, we consider the two-dimensional test problem

$$\begin{aligned} _{0}^{c}{\mathscr {D}}_{t}^{\alpha }u(x,y,t)+_{0}^{c}{\mathscr {D}}_{t}^{\alpha -1}u(x,y,t)+u(x,y,t)=\Delta u(x,y,t)+f(x,y,t). \end{aligned}$$

The initial conditions and boundary conditions are calculated using analytic solution

$$\begin{aligned} u(x,y,t) = x^{2}+y^{2}+t^{4}. \end{aligned}$$

The linear source term read $f(x,t)= \frac{24t^{4-\alpha }}{\Gamma (5-\alpha )}+\frac{24t^{5-\alpha }}{\Gamma (6-\alpha )}+(x^{2}+y^{2}+t^{4})-4$.

In this test problem, we consider four different complex-shape computational domains $\Omega _{i}$, $= 1,2,3,4$ as shown in Fig. 6. The first one is circular domain as shown in the first part of Fig. 6 with center (0.5, 0.5) and radius $r=0.5$. Table 6 shows errors on circular domain with $M=2395$ spatial points, $m=5$ at $T=1.0$ s. Finally, Fig. 7 shows graph of numerical solution and also the graph of absolute error for $\alpha =1.75$ and $N=400$. After that, we consider irregular polar domain with uniform points as shown in the second part of Fig. 6 whose boundary is given by parametric equation $\{(r \cos \theta ,r \sin \theta ): r=\frac{n+1}{3n^2}[n+1-\cos n \theta ]\}$ with $n=6$. The value of errors corresponding to $L^\infty$ and RMS are listed in Table 7 with $M=1982$ spatial points on irregular polar domain at $T=1.0$ s. Figure 8 represents the graph of numerical approximation and also the graph of absolute error for $\alpha =1.85$ and $N=400$.

Fig. 6

The domains ${\Omega _{i},i=1,2,3,4},$ for Example 4 with uniform and non-uniform points

Table 6 The value of both the errors and cpu time with $\alpha =1.75, 1.95$ and for different $\delta t$ on circular domain $\Omega _{1}$ at time $T=1.0$ s for Example 4

Fig. 7

Graph of approximated solution and absolute error on circular domain $\Omega _{1}$ for Example 4

Table 7 The numerical errors along with cpu time with $\alpha =1.75, 1.95$ and for different $\delta t$ on irregular polar domain $\Omega _{2}$ at time $T=1.0$ s for Example 4

Fig. 8

Graph of numerical solution and absolute error on irregular polar domain $\Omega _{2}$ for Example 4

Next, in this test problem we consider a multi-connected domain which is designed by two circles which are non-concentric as shown in the third part of Fig. 6. The center and radius of internal circle and external circle are (0.7, 0.6), $r=0.1$ and (0.5, 0.5), $r=0.5$, respectively. We used Dirichlet boundary conditions for both the inner and outer circles. The errors are computed with $M=1590$ spatial points at $T=1.0$ s and reported in Table 8. From Table 8, we can easily see that the present method is very efficient and accurate. Finally, the graph of numerical solution and also the graph of absolute error on multi-connected domain for $\alpha =1.95$ and $N=1600$ are plotted in Fig. 9. Lastly, we consider irregular shape domain with boundary $\{r(\theta ) = 0.4+0.05(\sin 6\theta +\sin 3 \theta )\}$ and $M=1645$ Halton non-uniform points as shown in the fourth part of Fig. 6. The numerical results on these non-uniform Halton points at $T=1.0$ s are given in Table 9. The graph of numerical approximation and absolute error on domain $\Omega _{4}$ for $N=400$ and $\alpha =1.55$ is given in Fig. 10.

Table 8 The computational errors along with cpu time with different values of $\alpha$ and $\delta t$ on multi-connected domain $\Omega _{3}$ at time $T=1.0$ s for Example 4

Fig. 9

Graph of approximated solution and absolute error on multi-connected domain $\Omega _{3}$ for Example 4

Table 9 The value of both the errors and cpu time with $\alpha =1.55, 1.95$ and for different $\delta t$ on irregular polar domain with non-uniform Halton points $\Omega _{4}$ at time $T=1.0$ s for Example 4

Fig. 10

Graph of approximated solution along with absolute error on domain $\Omega _{4}$ for Example 4

5 Conclusion

In the present work, we have developed a local meshless method based on RBF for solving the time-fractional telegraph equation with the fractional derivative as defined in the Caputo sense. The time semi-discretization was done by finite difference method to obtain the semi-discrete scheme, and spatial discretization was done by RBF-based meshless method to obtain a fully discrete scheme. With the help of numerical examples, we shows that the computational convergence order in time is nicely close to theoretical order. To examine the efficiency of the proposed method, being suitable for irregular domain a numerical experiment on several complex domain was carried out. It was found that the method is robust and very pleasant to deal with regular as well as irregular domains.