Numerical Approximation of Fractional Telegraph Equation via Legendre Collocation Technique

Abstract

The present study concerns with the numerical solution of space-time fractional-order telegraph equations (FOTE) on the transmission line. The Legendre collocation method is used to approximate the FOTE in space and time direction simultaneously with the help of the Kronecker product. Numerical solutions for some examples of fractional-order telegraph equations are obtained using the proposed scheme and are compared with the existing methods in the literature. The comparison shows the effectiveness and reliability of the proposed scheme. The scheme’s main feature is that fewer nodes are required in space and time direction both to achieve reasonable accuracy. Thus it is time-efficient, much easier to apply and requires less computational cost. Also, the error analysis is presented to show the stability and convergence of the proposed technique.

Appendix

If a matrix $P \in {R}^{m\times n }$ and $Q \in {R}^{s\times t }$ then the Kronecker product of matrix P and Q can be defined as [30]:

$$\begin{aligned} {P\otimes Q} = \begin{bmatrix} a_{11}Q &{} \quad {a}_{12}Q &{} \quad . &{} \quad . &{} \quad .&{} \quad &{} \quad &{} \quad {a}_{1N }Q \\ {a}_{21}Q&{} \quad {a}_{22}Q &{} \quad . &{} \quad . &{} \quad .&{} \quad &{} \quad &{} {a}_{2N }Q\\ . &{} \quad . &{} \quad . &{} \quad .&{} \quad . \\ {a}_{n1}Q &{} \quad {a}_{n2}Q &{} \quad . &{} \quad . &{} \quad .&{} \quad &{} \quad &{} \quad {a}_{nN }Q \end{bmatrix}. \end{aligned}$$

The important characteristics of the Kronecker product are that for any matrices P, Q and R which fulfill the condition of matrix multiplication,we will have :

$$\begin{aligned} vec(PQR)= (R^{T}\otimes P)vec(Q) \end{aligned}$$

vec(Q) is obtained by stacking the columns of Q on top of one another [30].

Table 4 $L_{\infty }$ error for different value of $\beta$, $\gamma$, node points in space N and n

Table 5 $L_\infty$ and $L_{rms}$ errors along with CPU time for $\beta = 1.9$ and different values of m [32]

Table 6 $L_\infty$ and $L_{rms}$ error at different value $\beta$, at $\gamma =2$ for different nodes in time and space direction for Eq. (7.4)