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ORIGINAL RESEARCH article

Front. Phys., 23 September 2020
Sec. Statistical and Computational Physics
This article is part of the Research Topic New Trends in Fractional Differential Equations with Real-World Applications in Physics View all 16 articles

Numerical Treatment of Time-Fractional Klein–Gordon Equation Using Redefined Extended Cubic B-Spline Functions

  • 1Department of Mathematics, National College of Business Administration & Economics, Lahore, Pakistan
  • 2Department of Mathematics, University of Sargodha, Sargodha, Pakistan
  • 3Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
  • 4Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
  • 5Department of Mathematics, Government College University, Faisalabad, Pakistan
  • 6Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara, Turkey
  • 7Department of Medical Research, China Medical University, Taichung, Taiwan
  • 8Institute of Space Sciences, Bucharest, Romania

In this article we develop a numerical algorithm based on redefined extended cubic B-spline functions to explore the approximate solution of the time-fractional Klein–Gordon equation. The proposed technique employs the finite difference formulation to discretize the Caputo fractional time derivative of order α ∈ (1, 2] and uses redefined extended cubic B-spline functions to interpolate the solution curve over a spatial grid. A stability analysis of the scheme is conducted, which confirms that the errors do not amplify during execution of the numerical procedure. The derivation of a uniform convergence result reveals that the scheme is O(h2 + Δt2−α) accurate. Some computational experiments are carried out to verify the theoretical results. Numerical simulations comparing the proposed method with existing techniques demonstrate that our scheme yields superior outcomes.

1. Introduction

The subject of fractional-order differential equations has attracted considerable interest due to its applications in a wide range of fields, such as traffic flow, earthquakes and other physical phenomena, signal processing, finance, control theory, fractional dynamics, and mathematical modeling [110]. In recent years, the analytical and numerical study of fractional-order differential equations has become a dynamic area of research. Several numerical and analytical techniques have been developed to handle these types of equations [1122]. There are a number of different definitions of fractional-order derivatives, with different applications. An excellent overview can be found in the works [2331]. This article is concerned with the following time-fractional non-linear Klein–Gordon equation (KGE):

αtαv(x,t)+ρ2x2v(x,t)+ρ1v(x,t)+ρ2vσ(x,t)=f(x,t),     0<xL,t0<tT,    (1)
v(x,t0)=φ1(x),     vt(x,t0)=φ2(x),    (2)
v(0,t)=φ3(t),     v(L,t)=φ4(t),    (3)

where αtα represents the Caputo fractional time derivative, v = v(x, t) denotes the displacement of the wave at (x, t), α ∈ (1, 2] is the fractional order of the time derivative, f(x, t) is the source term, ρ, ρ1 and ρ2 are real numbers, and σ = 2 or 3.

The fractional KGE plays a significant role in quantum mechanics, the study of solitons, and condensed matter physics. Many approaches have been adopted to solve equations of Klein/sine–Gordon type efficiently, including the Adomian decomposition method, the variational iteration method [3234], and the homotopy analysis method [35]; see also the references cited in these works. Jafari et al. proposed using fractional B-splines for approximate solution of fractional differential equations [36]. In Vong and Wang [37, 38] space compact difference schemes were applied to one- and two-dimensional time-fractional Klein–Gordon-type equations, and stability and convergence of the proposed numerical approaches were established with the aid of an energy method. In Dehghan et al. [39] the authors used a meshless method based on radial basis functions to develop an unconditionally stable numerical scheme for fractional Klein/sine–Gordon equations. The Adomian decomposition method and an iterative method were applied in Jafari [40] to solve Klein–Gordon-type equations involving fractional time derivatives. A fully spectral approach was employed in Chen et al. [41] that uses finite differences for time discretization and Legendre spectral approximation in the spatial direction to construct numerical solutions of non-linear partial differential equations involving fractional derivatives. A sinc–Chebyshev collocation method (SCCM) was developed in Nagy [42] for numerical treatment of the time-fractional non-linear KGE. Recently, in Kanwal et al. [43], Genocchi polynomials were employed together with the Ritz–Galerkin scheme to solve fractional KGEs and diffusion wave equations. A linearized second-order scheme was introduced in Lyu and Vong [44] to solve non-linear time-fractional Klein–Gordon-type equations. Later on, in Doha et al. [45], a space–time spectral approximation was proposed for solving non-linear variable-order fractional Klein/sine–Gordon differential equations.

In this article we propose using redefined extended cubic B-spline (RECBS) functions for numerical solution of the time-fractional KGE. RECBS functions are basically a generalization of typical cubic B-spline functions that involve a free parameter which provides the flexibility to fine-tune the solution curve. We employ the usual finite central difference approach to discretize the Caputo fractional time derivative and use RECBS functions for spatial integration.

This article is organized as follows. The Caputo definition of fractional time derivative and the finite difference formulation for temporal discretization are reviewed in section 2; this section also includes a brief introduction to extended cubic B-spline and RECBS functions and their applications to space discretization. The stability analysis of the proposed algorithm is presented in section 3, and the description of theoretical convergence is given in section 4. The approximate results are reported and discussed in section 5. Finally, concluding remarks are given in section 6.

2. Description of Numerical Technique

2.1. Time Discretization

Let the time domain [0, T] be divided into R subintervals of equal length Δt=TR with endpoints 0 = t0 < t1 < ⋯ < tR = T, where tr = rΔt and r = 0:1:R. We first discretize the Caputo fractional derivative at t = tr+1 as [46]

αv(x,tr+1)tα=1Γ(2-α)0tk2v(x,w)w2(tr+1-w)-α+1dw(1<α2)=1Γ(2-α)k=0rtktk+12v(x,w)w2(tr+1-w)-α+1dw.=1Γ(2-α)k=0rv(x,tk+1)-2v(x,tk)+v(x,tk-1)Δt2tktk+1(tr+1-w)-α+1dw+lΔtr+1=1Γ(2-α)k=0rv(x,tk+1)-2v(x,tk)+v(x,tk-1)Δt2tr-ktr-k+1(ϵ)-α+1dϵ+lΔtr+1=1Γ(2-α)k=0rv(x,tr-k+1)-2v(x,tr-k)+v(x,tr-k-1)Δt2tktk+1(ϵ)-α+1dϵ+lΔtr+1=1Γ(3-α)k=0rv(x,tr-k+1)-2v(x,tr-k)+v(x,tr-k-1)Δtα((k+1)2-α-k2-α)+lΔtr+1=1Γ(3-α)k=0rpkv(x,tr-k+1)-2v(x,tr-k)+v(x,tr-k-1)Δtα+lΔtr+1,    (4)

where pk=(k+1)2-α-k2-α, ϵ = (tr+1w), and lΔtr+1 is the truncation error. The truncation error is bounded, i.e.,

|lΔtr+1|ψ(Δt)2-α,    (5)

where ψ is a constant. The coefficients pk in (4) possess the following attributes:

• the pk's are non-negative for k = 0, 1, 2, …, r;

• 1 = p0 > p1 > p2 > p3 > ⋯ > pn, and pn → 0 as n → ∞;

(2p0-p1)+k=1r-1(-pk+1+2pk-pk-1)+(2pr-pr-1)-pr=1.

Substituting Equation (4) into Equation (1), we get

1Γ(3-α)(Δt)αk=0rpk[v(x,tr-k+1)-2v(x,tr-k)+v(x,tr-k-1)]     +ρvxx(x,t)+ρ1v(x,t)+ρ2vσ(x,t)=f(x,t)     (r=0,1,2,,R-1).    (6)

Suppose β=1Γ(3-α)(Δt)α and v(x,tr+1)=vr+1. Applying a θ-weighted scheme, Equation (6) takes the form

βp0(vr+12vr+vr1)+βk=1rpk(vrk+12vrk+vrk1)                                                        +θ(ρvxxr+1         +ρ1vr+1)=fr+1(1θ)(ρvxxr+ρ1vr)ρ2(vσ)r                                                                            (r=0,1,2,,R1).    (7)

For θ = 1, we obtain the following semi-discretized numerical scheme:

      (βp0+ρ1)vr+1+ρvxxr+1=2βp0vr+βk=1rpk(vrk+12vrk+vrk1)ρ2(vσ)rβp0vr1+fr+1(r=0,1,2,,R1).    (8)

2.2. Extended Cubic B-Spline Functions

Let the spatial domain [a, b] be partitioned into M parts of equal length h=b-aM with boundary points a = x0 < x1 < ⋯ < xM = b, where xm = x0 + mh for m = 0:1:M. For a sufficiently continuous function v(x, t), there always exists a unique extended cubic B-spline (ECBS) approximation V*(x, t):

V*(x,t)=m=-1M+1ξm(t)Sm(x,λ),    (9)

where the ξm(t) are to be calculated and the fourth-degree ECBS blending functions Sm(x, λ) are defined as [47]

Sm(x,λ)=124h4{4h(x-xm-2)3(1-λ)+3(x-xm-2)4λif x[xm-2,xm-1),h4(4-λ)+12h3(x-xm-1)+6h2(x-xm-1)2(2+λ)     -12h(x-xm-1)3-3(x-xm-1)4λif x[xm-1,xm),h4(4-λ)-12h3(x-xm+1)-6h2(x-xm+1)2(2+λ)     +12h(x-xm+1)3+3(x-xm-1)4λif x[xm,xm+1),-4h(x-xm+2)3(1-λ)-3(x-xm+2)4λif x[xm+1,xm+2),0otherwise.    (10)

Here λ, with −n(n − 2) ≤ λ ≤ 1, is a real number responsible for fine-tuning the curve, and n gives the degree of the ECBS used to generate different forms of ECBS functions. The approximate solution (V*)mr=V*(xm,tr) and its first two derivatives with respect to the spatial variable x at the rth time step can be expressed in terms of ξm as [48]

{(V*)mr=b1ξm-1r+b2ξmr+b1ξm+1r,(Vx*)mr=b3ξm-1r-b3ξm+1r,(Vxx*)mr=b4ξm-1r+b5ξmr+b4ξm+1r,    (11)

where b1=4-λ24, b2=16+2λ24, b3=-12h, b4=2+λ2h2, and b5=-4-2λ2h2.

2.3. Redefined Extended Cubic B-Spline Functions

In the typical ECBS collocation method, the basis functions S−1, S0, …, SM+1 do not vanish at the boundaries of the spatial domain when Dirichlet-type end conditions are imposed. Therefore, we need to redefine them so that the resulting set of basis functions will vanish at the boundaries. For this, a weight function Φ(x, t) is introduced to eliminate ξ−1 and ξM+1 from Equation (9) in the following manner [49]:

V(x,t)=Φ(x,t)+m=0Mξm(t)S~m(x,λ),    (12)

where the weight function Φ(x, t) and the redefined ECBS (RECBS) functions are given by

Φ(x,t)=S-1(x,λ)S-1(x0,λ)φ3(t)+SM+1(x,λ)SM+1(xM,λ)φ4(t)    (13)

and.

{S~m(x,λ)=Sm(x,λ)-Sm(x0,λ)S-1(x0,λ)S-1(x,λ)for m=0,1,S~m(x,λ)=Sm(x,λ)for m=2:1:M-2,   S~m(x,λ)=Sm(x,λ)-Sm(xM,λ)SM+1(xM,λ)SM+1(x,λ)for m=M-1,M.    (14)

2.4. Space Discretization

Using Equation (12) in Equation (8) at t = tr+1, we obtain

(βp0+ρ1)Vr+1+ρVxxr+1=2βp0Vr+βk=1rpk(Vrk+12Vrk+Vrk1)ρ2(Vσ)rβp0Vr1+fr+1.    (15)

Discretizing at x = xj, we get

(β+ρ1)Vjr+1+ρ(Vxx)jr+1=2βVjr+βk=1rpk(Vjrk+12Vjrk+Vjrk1)ρ2(Vσ)jrβVjr1+fjr+1(j=0,1,2,,M).    (16)

Using (12), the last expression takes the form

(β+ρ1)[Φjr+1+m=0Mξmr+1S˜m(xj,λ)]+ρ[(Φxx)jr+1                                          +m=0Mξmr+1S˜m(xj,λ)]          =2βVjr+βk=1rpk(Vjrk+12Vjrk+Vjrk1)                              ρ2(Vσ)jrβVjr1+fjr+1                                                                          (j=0,1,2,,M).    (17)

Consequently, we get the following system of M + 1 equations in M + 1 unknowns:

(a1*a1a2a1a1a2a1a1a2a1a1a2a1a1*)(ξ0r+1ξ1r+1ξM1r+1ξMr+1)=(y0y1yM1yM),    (18)

where

a1*=12ρ(λ+2)h2(λ-4),     a1=h2(β+ρ1)(λ-4)+12ρ(λ+2)24h2,a2=h2(β+ρ1)(λ+8)-12ρ(λ+2)12h2,yj=2βVjr+βk=1rpk(Vjr-k+1-2Vjr-k+Vjr-k-1)      -ρ2(Vσ)jr-βVjr-1+Ψjr+1,Ψjr=fjr-(β+ρ1)Φjr-ρ(Φxx)jr.

To start the numerical procedure, we use the given initial conditions to obtain the set of equations

{(V)m0=φ1(xm)     for m=0,(V)m0=φ1(xm)     for m=1:1:M-1,(V)m0=φ1(xm)     for m=M.    (19)

The matrix representation of (19) is

(b1*b2*b1b2b1b1b2b1b1b2b1b1b2b1b2*b1*)(ξ00ξ10ξM10ξM0)                                                                     =((φ1)0(Φ)00(φ1)1Φ10(φ1)M1ΦM10(φ1)M(Φ)M0),    (20)

where b1*=8+λh(4λ) and b2*=1h. We solve (20) to obtain [ξ00,ξ10,,ξM0]T. The ξj values are then substituted into (12) to get V0. Now we can use (18) for r = 0, 1, 2, …, R − 1. However, for r = 0 the term involving V−1 appears in Equation (18). This issue is resolved by using the following substitution derived from the velocity condition given in (2):

V-1=V0-Δtϕ2(x).

3. Stability Analysis

We use the Fourier method to study the stability of the proposed numerical method. Let εmr and ε~mr denote, respectively, the exact and approximate growth factors of the Fourier modes. The error, ϱmr, is given by

ϱmr=εmr-ε~mr,      m=1:1:M-1,  r=0:1:R,    (21)

where ϱr=[ε1r,ε2r,,εM-1r]T.

For the sake of simplicity, we shall investigate the stability of the proposed scheme with f = 0. The equation for the round-off error is derived from Equations (8) and (21) as

(βb1+ρ1b1+ρb4)ϱm1r+1+(βb2+ρ1b2+ρb5)ϱmr+1+(βb1+ρ1b1+ρb4)ϱm+1r+1=2β(b1ϱm1r+b2ϱmr+b1ϱm+1r)β(b1ϱm1r1+b2ϱmr1+b1ϱm+1r1)    βk=1rpk[b1(ϱm1rk+12ϱm1rk+ϱm1rk1)+b2(ϱmrk+12ϱmrk+ϱmrk1)                                    +b1(ϱm+1rk+12ϱm+1rk+ϱm+1rk1)].    (22)

The error equation satisfies the end conditions

ϱm0=φ1(xm),     m=1:1:M,    (23)

and

ϱ0r=φ3(tr),     ϱMr=φ4(tr),     r=0:1:R.    (24)

We define the grid function as

ϱr={ϱmrif xm-h2<xxm+h2, for m=1:1:M-1,0if ax2a+h2or2b-h2xb.    (25)

Now, ϱr(x) can be written in the form of a Fourier series as follows:

ϱr(x)=r=-εr(n)e2πιnxb-a,     r=1:1:R,    (26)

where

εr(n)=1b-aabϱr(x)e-2πιnxb-adx.    (27)

Taking the ‖·‖2 norm, we get

ϱr2=(n=1R-1h|ϱnr|2)12               =(aa+h2|ϱr|2dx+n=1R-1xn-h2xn+h2|ϱr|2dx+b-h2b|ϱr|2dx)12               =(ab|ϱr|2dx)12.

From Parseval's equality we have ab|ϱr(n)|2dx=-|εn(m)|2, so the above expression can be written as

ϱr22=r=|εr(n)|2.     (28)

Next, we consider the solution in terms of Fourier series,

ϱkr=εreινkh,    (29)

where ι=-1 and ν=2πnb-a. Using Equation (29) in Equation (22) and then dividing by eiνkh gives

(βb1+ρ1b1+ρb4)εr+1eινh+(βb2+ρ1b2+ρb5)εr+1+(βb1+ρ1b1+ρb4)εr+1eινh=2β(b1εreινh+b2εr+b1εreiνh)β(b1εr1eινh+b2εr1+b1εr1eiνh) βk=1rpk[b1(εrk+1eινh2εrk+εrk1eινh)                           +b2(εrk+12εrk+εrk1)+b1(εrk+1eινh2εrkeινh+εrk1eινh)].    (30)

We know that eiνh + eiνh = 2 cos(νh), so after collecting like terms, the following useful relation is obtained:

εr+1=1η[2εr-εr-1-k=1rpk(εr-k+1-2(b1+b2)εr-k+εr-k-1)],    (31)

where η=1+ρ1β+12ρ(2+ν)sin2(νh/2)βh2{-6+(4-ν)sin2(νh/2)}. Now it is obvious that η ≥ 1 for ν > −2.

Lemma 3.1. Let εr be the solution of Equation (31). Thenr| ≤ |ε0| for r = 0:1:R.

Proof: For r = 0 in (31), we have

|ε1|=1η|ε0||ε0|     for η1.

Suppose that the result is true for r = 1:1:R. Then, from Equation (31) we get

|εr+1|1η|εr|-1ηk=1rpk(|εr-k+1|-2|εr-k|+|εr-k-1|)                1η|ε0|-1η|ε0|-k=1rpk(|ε0|-|ε0|)                |ε0|.

Theorem 1. The implic it collocation technique presented in Equation (13) is unconditionally stable.

Proof: Using Lemma (3.1) and Equation (28), we obtain

ϱr2|ϱ0|2,   r=0:1:R.

4. Convergence of the Scheme

To investigate the convergence of the proposed scheme, we follow the approach in Khalid et al. [50]. Before proceeding, we state the following useful theorems [51, 52].

Theorem 2. Let Π = {a = x0, x1, …, xM = b} be a partition of [a, b] with xm = mh for m = 0, …, M, and let vC4[a, b] and fC2[a, b]. Suppose V~(x, t) is the spline that interpolates the solution curve of this problem at the knots xm ∈ Π. Then there exist constants Ϝm, not depending on h, such that

ξj(v(x,t)-V~(x,t))Ϝjh4-j     t0,  j=0,1,2.    (32)

Lemma 4.1. The extended B-splines in (10) satisfy the inequality

m=0M|Sm(x,λ)|1.75     for 0x1.    (33)

Proof: By the triangle inequality we have

|m=0MSm(x,λ)|m=0M|Sm(x,λ)|.

For any knot xm, we have

m=0M|Sm(x,λ)|=|Sm-1(xm,λ)|+|Sm(xm,λ)|+|Sm+1(xm,λ)|=1<74.

From (11) we obtain

Sm(xm,λ)=112(8+λ),       Sm-1(xm-1,λ)=112(8+λ),          Sm+1(xm,λ)=124(4-λ),       Sm-2(xm-1,λ)=124(4-λ).                  

Then, for x ∈ [xm−1, xm], Sm(x, λ) and Sm−1(x, λ) are bounded above by 112(8+λ).

Similarly, Sm+1(x, λ) and Sm−2(x, λ) are bounded above by 124(4-λ)

For any point xm−1xxm, we obtain

m=0M|Sm(x,λ)|=|Sm-1(x,λ)|+|Sm(x,λ)|+|Sm+1(x,λ)|                                                                        +|Sm-2(x,λ)|=112(λ+20).

Since λ ∈ [−8, 1], we have 153+λ1.75. Hence,

m=0M|Sm(x,λ)|1.75.

Theorem 3. The extended cubic B-spline approximation V(x, t) for the analytical exact solution v(x, t) of problem (1)–(3) exists, and if fC2[0, 1] then

v(x,t)-V(x,t)Ϝ~h2     t0,    (34)

where h is reasonably small and Ϝ~>0 is a constant not depending on h.

Proof: Let V~(x,t)=m=0Mdm(t)ηm(x) be the calculated spline for the approximate solution V(x, t) and the exact solution v(x, t).

Let Lv(xm, t) = LV(xm, t) = y~(xm, t), with m = 0:1:M, be the collocation conditions. Then

LV~(x,t)=y~(xm,t),     m=0:1:M.

Now, at any time step, the problem can be expressed in the form of a difference equation L(V~(xm, t) − V(xm, t)) as

(βb1+ρ1b1+ρb4)ζm1r+1+(βb2+ρ1b2+ρb5)ζmr+1+(βb1+ρ1b1+ρb4)ζm+1r+1=2β(b1ζm1r+b2ζmr+b1ζm+1r)β(b1ζm1r1+b2ζmr1 +b1ζm+1r1)βk=1rpk[b1(ζm1rk+12ζm1rk+ζm1rk1)+b2(ζmrk+12ζmrk+ζmrk1)                                  +b1(ζm+1rk+12ζm+1rk+ζm+1rk1)]1h2ηmr+1.    (35)

The boundary conditions can be rewritten as

b1ζm-1r+1+b2ζmr+1+b1ζm+1r+1=0,     m=0,M,

where

ζmr=ξmr-dmr,     m=0:1:M,

and

ηmr=h2[ymr-y~mr],     m=0:1:M.

From (32) we have

|ηmr|=h2|ymr-y~mr|Ϝh4.

We define ηr=max{|ηmr|:0mM}, e~mr=|ζmr| and e~r=max{|emr|:0mM}.

For r = 0, Equation (35) transforms into the following relation:

(βb1+ρ1b1+ρb4)ζm-11+(βb2+ρ1b2+ρb5)ζm1+(βb1+ρ1b1+ρb4)ζm+11=(β+ρ1)(b1ζm-10+b2ζm0+b1ζm+10)+1h2ηm1.

Using the initial condition e0 = 0, we obtain

(βb2+ρ1b2+ρb5)ζm1=(βb1+ρb4)(ζm+11-ζm-11)+ρ1b1(ζm+11-ζm-11)+1h2ηm1.

Taking absolute values of ηmr and ζmr and with adequately small h, we have

e~m16Ϝh4βh2(λ+2)+12(-2-λ)ρ+ρ1h2(2+λ)

using the boundary conditions, from which we conclude that

e~1Ϝ1h2,    (36)

where Ϝ1 is independent of the spatial grid spacing.

Using the induction technique, we assume that e~mkϜkh2 is true for k = 1:1:r.

Let Ϝ = max{Ϝk:0 ≤ kr}; then Equation (35) becomes

(βb1+ρ1b1+ρb4)ζm1r+1+(βb2+ρ1b2+ρb5)ζmr+1+(βb1+ρ1b1+ρb4)ζm+1r+1=2β(b1ζm1r+b2ζmr+b1ζm+1r)β(b1ζm1r1+b2ζmr1+b1ζm+1r1)    +β[(p02p1+p2)(b1ζm1r+b2ζmr+b1ζm+1r)+(p12p2+p3)(b1ζm1r1+b2ζmr1+b1ζm+1r1)            ++(pr42pr3+pr2)(b1ζm11+b2ζm1+b1ζm+11)+pr1(b1ζm10+b2ζm0+b1ζm+10)]     +1h2ηmr+1.

Again, taking absolute values of ηmr and ζmr, we have

e˜mr+16Ϝh2βh2(2+λ)+12(2λ)ρ+ρ1h2(2+λ)                                  [2β(b1ζm1r+b2ζmr+b1ζmr)                                        βr1k=0(pk+12pkpk1)Ϝh2+Ϝh2].

Using the boundary conditions, we have

e~mr+1Ϝh2.

Hence, for all values of n,

e~mr+1Ϝh2.    (37)

Now,

V~(x,t)-V(x,t)=m=0M(dm(t)-ξm(t))Sm(x).

Taking the infinity norm and applying Lemma (3.1), we obtain

V~(x,t)-V(x,t)1.75Ϝh2.    (38)

Making use of the triangle inequality, we get

v(x,t)-V(x,t)v(x,t)-V~(x,t)+V~(x,t)-V(x,t).    (39)

Using the inequalities (32) and (38) in (39), we obtain

v(x,t)-V(x,t)Ϝ0h4+1.75Ϝh2=Ϝ~h2,

where Ϝ~=Ϝ0h2+1.75Ϝ.

Using the above theorem with expression (5), it is easy to conclude that the numerical approach converges unconditionally. Therefore,

v(x,t)-V(x,t)Ϝ~h2+ψ(Δt)2-α,

where Ϝ~ is a constant and α ∈ (1, 2]. Hence, theoretically, the proposed scheme is O(h2 + Δt2−α) accurate.

5. Numerical Results and Discussion

To examine the accuracy of the proposed method, we conduct a numerical study of some test problems. The L and L2 error norms are calculated as [53]

L=max0mM|V(xm,t)-v(xm,t)|,L2=hm=0M|V(xm,t)-v(xm,t)|2.

Also, the experimental order of convergence (EOC) is computed by the following important formula [54]:

EOC=1log 2log[L(2m)L(m)].

All numerical computations were performed using Mathematica 9.0.

Example 5.1. Consider the non-linear time-fractional KGE [42]

αvtα-2vx2+v2(x,t)=f(x,t),      0<t1,  0<x1,    (40)

where f(x,t)=Γ(52)Γ(52-α)(1-x)52t32-α-154(1-x)12t32+(1-x)5t3. The initial/end conditions can be extracted from the analytical exact solution (1-x)52t32-α.

For Example 5.1, the piecewise-defined approximate solution obtained using the proposed method with α = 1.25, 0 ≤ x ≤ 1, n = 100, and Δt = 0.01 is given by

V(x)={0.+x(297.276+x(-29930.4+x(993222.+225927.x)))if x[0.00,0.01],0.999999+x(-2.49738+x(1.82587+(1.38305-27.8749x)x))if x[0.01,0.02],0.99999+x(-2.49605+x(1.75961+(2.48215-27.7432x)x))if x[0.02,0.03],0.99996+x(-2.49308+x(1.66094+(3.57055-27.6103x)x))if x[0.03,0.04],-0.118298+x(6.72761+x(-26.6775+(38.9565-20.3042x)x))if x[0.49,0.50],-0.201484+x(7.21369+x(-27.5747+(39.3734-20.1068x)x))if x[0.50,0.51],-2.7339+x(13.6165+x(-24.3154+(18.715-5.28228x)x))if x[0.96,0.97],-1.89304+x(10.2593+x(-19.2941+(15.3811-4.45319x)x))if x[0.97,0.98],-0.518579+x(5.07656+x(-12.0155+(10.8746-3.41708x)x))if x[0.98,0.99],4.86293+x(-13.1733+x(10.3424+(-0.616646-1.41541x)x))if x[0.99,1.00].

The absolute numerical errors at different grid points of the RECBS solution for Example 5.1 using Δt = 0.001 and M = 100 are reported in Table 1. It can easily be seen that our scheme is more accurate than the SCCM [42]. In Table 2 the absolute and relative numerical errors are listed for our method with M = 100, Δt = 0.001, and α = 1.6 at x = 0.4, 0.6, 0.8 when t = 0.4, 0.8. We can see that the computational results are superior to those obtained from the SCCM [42]. Table 3 compares the absolute errors of the proposed method, the variational iteration method (VIM) [34], and the SCCM [42] under different values of α. Figure 1 shows the behavior at different time stages of numerical solutions obtained using α = 1.5, M = 100, and Δt = 0.001. The 3D visuals of exact and numerical solutions with α = 1.5 and M = 100 are shown in Figure 2. The comparison between the exact and approximate solutions using M = 100 is plotted in Figure 3. Figure 4 depicts the absolute error between the exact and numerical solutions when α = 1.3, M = 100, and Δt = 0.001. The values of the EOC along the spatial grid, using Δt = 0.001 and α = 1.5, are given in Table 4. The experimental rate of convergence of the proposed method is found to be in line with the theoretical results.

TABLE 1
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Table 1. Absolute errors for Example 5.1 with M = 100, Δt = 0.001, and different values of α.

TABLE 2
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Table 2. Absolute and relative errors for Example 5.1 with M = 100, Δt = 0.001, and α = 1.6.

TABLE 3
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Table 3. Comparison of absolute errors for Example 5.1 using three different methods with M = 100, Δt = 0.001, and α = 1.4 or 1.6.

FIGURE 1
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Figure 1. Numerical solution of Example 5.1 with Δt = 0.001, M = 100, and α = 1.5 at different time stages.

FIGURE 2
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Figure 2. Exact and approximate solutions of Example 5.1 with M = 100, Δt = 0.001, and α = 1.50. (A) Exact. (B) Numerical.

FIGURE 3
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Figure 3. Absolute error for Example 5.1 when M = 100, α = 1.50, and Δt = 0.001.

FIGURE 4
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Figure 4. Approximate solution of Example 5.1 with M = 100, t = 0.5, and different values of α.

TABLE 4
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Table 4. Experimental order of convergence (EOC) for Example 5.1 with α = 1.3 and Δt = 0.001.

Example 5.2. Consider the fractional KGE [34, 42]

αtαv(x,t)-2x2v(x,t)+v(x,t)+32v3(x,t)=f(x,t),                                                              0<x1,  0<t1,    (41)

where the forcing term f(x, t) on right-hand side is given by

f(x,t)=12Γ(3+α)sin(πx)t2+(1+π2)t2+αsin(πx)             +32[sin(πx)t2+α]3,

For Example 5.2, the piecewise-defined numerical solution obtained using the proposed method with α = 1.5, 0 ≤ x ≤ 1, n = 100, and Δt = 0.01 is given by

V(x)={8.71156×10-19+x(3.13867+x(2.8549×10-14+(-4.97167-11.4015x)x))if x[0.00,0.01],-1.14461×10-6+x(3.13904+x(-0.041176+(-3.14329-34.194x)x))if x[0.01,0.02],-0.0000194466+x(3.14196+x(-0.205754+(0.51013-56.9551x)x))if x[0.02,0.03],-0.000112001+x(3.15183+x(-0.575584+(5.98188-79.6639x)x))if x[0.03,0.04],-40.7681+x(339.328+x(-1039.38+(1422.21-733.23x)x))if x[0.49,0.50],-44.2829+x(360.934+x(-1083.83+(1453.18-733.97x)x))if x[0.50,0.51],-71.1059+x(298.709+x(-460.613+(312.674-79.6639x)x))if x[0.96,0.97],-53.5088+x(223.56+x(-340.406+(227.31-56.9551x)x))if x[0.97,0.98],-34.2394+x(143.149+x(-214.635+(139.919-34.194x)x))if x[0.98,0.99],-13.2345+x(57.3823+x(-83.3239+(50.5776-11.4015x)x))if x[0.99,1.00].

The initial/boundary conditions can be extracted from the analytical exact solution v(x, t) = sin(πx)t2+α. The absolute numerical errors at different grid points of the RECBS solution for Example 5.2 using Δt = 0.001 and M = 100 are listed in Table 5. Again it can be observed that our scheme is more accurate than the SCCM [42]. Table 6 reports the absolute and relative errors in our numerical computation with M = 100, Δt = 0.001, and α = 1.6 at x = 0.4, 0.6, 0.8 when t = 0.4, 0.8. It is clear that the results are better than those obtained by the SCCM [42]. Table 7 compares the absolute errors of the proposed method, VIM [34], and SCCM [42] under different values of α. The EOC in the spatial direction, using Δt = 0.001 and α = 1.50, is tabulated in Table 8. The experimental rate of convergence of the proposed scheme is found to be in line with the theoretical prediction. Figure 5 shows the behavior at different time stages of numerical solutions obtained using α = 1.5, M = 100, and Δt = 0.001. The 3D plots of exact and numerical solutions with α = 1.5 and M = 100 are displayed in Figure 6. The absolute error between the exact and approximate solutions using α = 1.3, M = 100, and Δt = 0.001 is plotted in Figure 7.

TABLE 5
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Table 5. Absolute errors for Example 5.2 when M = 100, Δt = 0.001 using different values of α.

TABLE 6
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Table 6. Absolute and relative errors for Example 5.2 when M = 100, Δt = 0.001 and α = 1.6.

TABLE 7
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Table 7. Absolute errors for Example 5.2 when M = 100 and Δt = 0.001.

TABLE 8
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Table 8. Experimental order of convergence (EOC) for Example 5.2 with α = 1.5 and Δt = 0.001.

FIGURE 5
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Figure 5. Numerical solution for Example 5.2 with Δt = 0.001, M = 100, and α = 1.5 at different time stages.

FIGURE 6
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Figure 6. Exact and numerical solutions of Example 5.2 with M = 100, Δt = 0.001, and α = 1.5. (A) Exact. (B) Numerical.

FIGURE 7
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Figure 7. Absolute error for Example 5.2 when M = 100, α = 1.5, and Δt = 0.001.

6. Conclusion

In this work we have conducted a numerical investigation of the time-fractional Klein–Gordon equation by applying the redefined extended cubic B-spline collocation method. A finite central difference formulation is employed for temporal discretization, while a set of redefined extended cubic B-spline functions is used to interpolate the solution curve in the spatial direction. The unconditional stability of the proposed scheme is established, and the orders of convergence along the space and time grids are shown to be O(h2) and Ot)2−α, respectively. The computational outcomes of the proposed algorithm show that the order of convergence agrees with the theoretical results. The numerical scheme has been tested on different problems, and comparison of the results reveals our method's advantage over VIM [34] and SCCM [42].

Author Contributions

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: redefined extended cubic B-spline, time fractional Klein-Gorden equation, Caputo fractional derivative, finite difference method, convergence analysis

Citation: Amin M, Abbas M, Iqbal MK and Baleanu D (2020) Numerical Treatment of Time-Fractional Klein–Gordon Equation Using Redefined Extended Cubic B-Spline Functions. Front. Phys. 8:288. doi: 10.3389/fphy.2020.00288

Received: 02 March 2020; Accepted: 25 August 2020;
Published: 23 September 2020.

Edited by:

Jordan Yankov Hristov, University of Chemical Technology and Metallurgy, Bulgaria

Reviewed by:

Praveen Agarwal, Anand International College of Engineering, India
Hossein Jaferi, University of South Africa, South Africa

Copyright © 2020 Amin, Abbas, Iqbal and Baleanu. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Muhammad Abbas, muhammadabbas@tdtu.edu.vn

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