An efficient cubic trigonometric B-spline collocation scheme for the time-fractional telegraph equation

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An efficient cubic trigonometric B-spline collocation scheme for the time-fractional telegraph equation Muhammad Yaseen

Muhammad Abbas∗

Abstract. In this paper, a proficient numerical technique for the time-fractional telegraph equation (TFTE) is proposed. The chief aim of this paper is to utilize a relatively new type of B-spline called the cubic trigonometric B-spline for the proposed scheme. This technique is based on finite difference formulation for the Caputo time-fractional derivative and cubic trigonometric B-splines based technique for the derivatives in space. A stability analysis of the scheme is presented to confirm that the errors do not amplify. A convergence analysis is also presented. Computational experiments are carried out in addition to verify the theoretical analysis. Numerical results are contrasted with a few present techniques and it is concluded that the presented scheme is progressively right and more compelling.

§1

Introduction

In recent years, the tools of fractional calculus have been effectively used to portray numerous physical phenomena in science and engineering [12, 17, 23]. Recently, there have been reporting of many applications typically expressed by fractional partial differential equations (FPDEs). The importance of FPDEs lies in the way that the solutions offered by FPDEs have descriptions that well approximate the chemical, physical and biological phenomena than their integer order counterparts. Accordingly, FPDEs have accomplished special status among researchers and engineers. A number of phenomenon such as propagation of electric signals [14], transport of neutron in a nuclear reactor [28] and random walks [5] are described by a class of hyperbolic partial differential equations called the fractional telegraph equations [8]. The general form of the TFTE is given by ∂ γ−1 u(x, t) ∂ 2 u(x, t) ∂ γ u(x, t) + γ1 + γ2 u(x, t) = γ3 + f (x, t), (1) γ γ−1 ∂t ∂t ∂x2 Received: 2019-08-17. Revised: 2020-03-20. MR Subject Classification: 65M70, 65Z05, 65D05, 65D07, 35B35. Keywords: Time-fractional telegraph equation, finite difference method, Cubic trigonometric B-splines collocation method, Stability, Convergence. Digital Object Identifier(DOI): https://doi.org/10.1007/s11766-020-3883-y. ∗ Corresponding author.

360

Appl. Math. J. Chinese Univ.

Vol. 35, No. 3

with initial conditions u(x, 0) = ϕ1 (x), ut (x, 0) = ϕ2 (x),

a ≤ x ≤ b,

(2)

and the boundary conditions u(a, t) = ψ1 (t), u(b, t) = ψ2 (t),

0 ≤ t ≤ T,

(3)

∂γ ∂tγ u(x, t)

where 1 < γ < 2, a, b, ϕ1 (x), ϕ2 (x), ψ1 (t) and ψ2 (t) are given and represents the Caputo fractional derivative of order γ given by [2, 4, 12, 15–17, 23, 31]  ∫t ∂ n u(x,s)   1 γ (t − s)n−γ−1 ds, n−1 b1 > b2 > · · · > bn and bn → 0 as n → ∞. •

n−1 ∑

(bl − bl+1 ) + bn = 1.

l=0

It is shown in [10] that r1n+1 ≤ Λτ and r2n+1 ≤ Λτ 3−γ , where Λ is constant dependent on u, γ and T . To obtain temporal discretization, we substitute (9) and (10) into (1) to get n n ∑ ∑ ∂ 2 un+1 α0 bl (un+1−l − 2un−l + un−1−l ) + γ1 τ α0 bl (un+1−l − un−l ) + γ2 un+1 − γ3 ∂x