Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations

M.A. Abdelkawy; Engy A. Ahmed; Rubayyi T. Alqahtani

doi:10.1515/phys-2016-0031

Article Open Access

Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations

M.A. Abdelkawy , Engy A. Ahmed and Rubayyi T. Alqahtani

Published/Copyright: August 15, 2016

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From the journal Open Physics Volume 14 Issue 1

Abstract

We introduce a new numerical algorithm for solving one-dimensional time-fractional Tricomi-type equations (T-FTTEs). We used the shifted Jacobi polynomials as basis functions and the derivatives of fractional is evaluated by the Caputo definition. The shifted Jacobi Gauss-Lobatt algorithm is used for the spatial discretization, while the shifted Jacobi Gauss-Radau algorithmis applied for temporal approximation. Substituting these approximations in the problem leads to a system of algebraic equations that greatly simplifies the problem. The proposed algorithm is successfully extended to solve the two-dimensional T-FTTEs. Extensive numerical tests illustrate the capability and high accuracy of the proposed methodologies.

Keywords: Caputo fractional derivatives; Collocation method; Gauss-Radau quadrature; Gauss-Lobatto quadrature; Time-fractional Tricomi-type equations

PACS: 02.30.Gp; 02.60.-x; 47.27.er

1 Introduction

The fractional calculus is a mathematical tool was firstly mentioned by Riemann and Liouville about 300 years ago, and has been developed successively up to now. In different modeling, such as bioengineering [1], anomalous transport [2], economics [4], chemistry [3] and others [5, 6], fractional equations was introduced as a value tool in modeling various phenomena. The exact solutions of almost fractional differential equations can not be obtained, therefore, the development of numerical methods for solving fractional equations has received considerable attention in recently times [7–15].

Recently, spectral methods [16−20] are efficient and highly accurate schemes when compared with the local methods. The exponential rates of convergence and high level of accuracy are the more advantages of spectral methods. They have been used as powerful techniques to numerically solve several types of differential equations [21, 22] and fractional differential equations, see [23, 24]. The major step of all techniques of spectral methods is to write the approximate solution as a finite sum of specific basis functions, which may be orthogonal polynomials or combination of them, and then select the coefficients with a view to decay the difference between the approximate and exact solutions as possible as we can. The spectral collocation method is one of the most important of spectral methods, that is more applicable and frequently applied to obtain the numerical solution of the different types of the fractional integro-differential equations [25] and fractional partial differential equations [26, 27].

In early times, Tricomi [28] started the work on the linear partial differential equations of variable type with boundary condition. Frankl [29] mentioned that the gas flows with nearly sonic speeds can be modeled using the Tricomi problem. For more applications of the Tricomi equation, see [31–35]. The local discontinuous Galerkin finite element (LDG-FE) method has been used by Zhang et al. [30] for numerically solve the one-dimensional linear T-FTTE. The authors in [36], discussed the numerical solution of two-dimensional time-fractional tricomi-type equations using the finite element method.

The main goal of this paper is to introduce a new efficient spectral technique to numerically solve the T-FTTEs. The shifted Jacobi spectral collocation method by means of the Jacobi Gauss-Lobatto and Jacobi Gauss-Radau quadratures has been constructed in this paper. The proposed algorithm is successfully extended to solve the two-dimensional T-FTTEs. Finally, we apply this technique to numerically solve numerous examples to prove that this method is accurate and efficient compared with alternative methods.

Our paper is arranged as follows. Few facts of fractional calculus and shifted Jacobi polynomials are listed in the coming section. In Sections 3 and 4 the spectral collocation method is applied to solve the one- and two-dimensional T-FTTEs. In Section 5, several examples have been solved to show the accuracy and efficiency of the proposed method. Finally, Section 5 outlines the conclusions.

2 Mathematical preliminaries

2.1 Fractional calculus

The fractional integration definition of order ν > 0, can be expressed by several formulas and in general they are not equal to each other. The most used definitions are Caputo and Riemann-Liouville definitions.

Definition 2.1

The operator J^ν of Riemann-Liouville fractional integral is defined as

Jνf(x)=1Γ(ν)∫0x(x−ζ)ν−1f(ζ)dζ,ν>0,x>0,J0f(x)=f(x),(1)

where

Γ(ν)=∫0∞xν−1e−xdx.

The properties listed below are satisfied for the operator J^ν

JνJμf(x)=Jν+μf(x),JνJμf(x)=JμJνf(x),Jνxϑ=Γ(ϑ+1)Γ(ϑ+1+ν)xϑ+ν.(2)

Definition 2.2

The next equation define Riemann-Liouville fractional derivative D^ν of order ν

Dνf(x)=1Γ(m−ν)dmdxm∫0x(x−t)m−ν−1f(t)dt,m−1<ν≤m,x>0,(3)

where m is the ceiling function of ν.

Definition 2.3

The Caputo fractional derivatives of order ν is defined as

cDνf(x)=1Γ(m−ν)∫0x(x−ζ)m−ν−1dmdtmf(ζ)dζ,m−1<ν≤m,x>0,(4)

where m is the ceiling function of ν.

2.2 Properties of shifted Jacobi polynomials

Some few properties of shifted Jacobi polynomials are presented in this subsection. In the following, few relations related to Jacobi polynomials are listed:

Pk+1(θ,ϑ)(x)=(ak(θ,ϑ)x−bk(θ,ϑ))Pk(θ,ϑ)(x)−ck(θ,ϑ)Pk−1(θ,ϑ)(x),k≥1,P0(θ,ϑ)(x)=1,P1(θ,ϑ)(x)=12(θ+ϑ+2)x+12(θ−ϑ),Pk(θ,ϑ)(−x)=(−1)kPk(θ,ϑ)(x),Pk(θ,ϑ)(−1)=(−1)kΓ(k+ϑ+1)k!Γ(ϑ+1),(5)

where θ, ϑ > −1, x ∈ [−1, 1] and

ak(θ,ϑ)=(2k+θ+ϑ+1)(2k+θ+ϑ+2)2(k+1)(k+θ+ϑ+1),bk(θ,ϑ)=(ϑ2−θ2)(2k+θ+ϑ+1)2(k+1)(k+θ+ϑ+1)(2k+θ+ϑ),ck(θ,ϑ)=(k+θ)(k+ϑ)(2k+θ+ϑ+2)(k+1)(k+θ+ϑ+1)(2k+θ+ϑ).

Moreover, the rth derivative (r is an intger) of Pj(θ,ϑ)(x), may be obtained from

DrPj(θ,ϑ)(x)=Γ(j+θ+ϑ+q+1)2rΓ(j+θ+ϑ+1)Pj−r(θ+r,ϑ+r)(x).(6)

For the shifted Jacobi polynomial PL,k(θ,ϑ)(x)=Pk(θ,ϑ)(2xL−1),L>0, the explicit analytic form is written as

PL,k(θ,ϑ)(x)=∑j=0k(−1)k−jΓ(k+ϑ+1)Γ(j+k+θ+ϑ+1)Γ(j+ϑ+1)Γ(k+θ+ϑ+1)(k−j)!j!Ljxj=∑j=0kΓ(k+θ+1)Γ(k+j+θ+ϑ+1)j!(k−j)!Γ(j+θ+1)Γ(k+θ+ϑ+1)Lj(x−L)j.(7)

Thus, we can derive the following properties for any integer r

PL,k(θ,ϑ)(0)=(−1)kΓ(k+ϑ+1)Γ(ϑ+1)k!,PL,k(θ,ϑ)(L)=Γ(k+θ+1)Γ(θ+1)k!,(8)

DrPL,k(θ,ϑ)(0)=(−1)k−rΓ(k+ϑ+1)(k+θ+ϑ+1)rLrΓ(k−r+1)Γ(r+ϑ+1),(9)

DrPL,k(θ,ϑ)(L)=Γ(k+θ+1)(k+θ+ϑ+1)rLrΓ(k−r+1)Γ(r+θ+1),(10)

DrPL,k(θ,ϑ)(x)=Γ(r+k+θ+ϑ+1)LrΓ(k+θ+ϑ+1)PL,k−r(θ+r,ϑ+r)(x).(11)

Assuming that wL(θ,ϑ)(x)=(L−x)θxϑ, we can define the norm and inner product for the weighted space LwL(θ,ϑ)2[0,L] as

(u,v)wL(θ,ϑ)=∫0Lu(x)v(x)wL(θ,ϑ)(x)dx,vwL(θ,ϑ)=(v,v)wL(θ,ϑ)12.(12)

The set of shifted Jacobi polynomials forms a complete LwL(θ,ϑ)2[0,L]-orthogonal system. Moreover, and due to (12), we have

PL,k(θ,ϑ)wL(θ,ϑ)2=L2θ+ϑ+1hk(θ,ϑ)=hL,k(θ,ϑ).(13)

We used xN,j(θ,ϑ),andϖN,j(θ,ϑ),0⩽j⩽N, as the nodes and Christoffel numbers on the interval [−1, 1].

The corresponding nodes and corresponding Christoffel numbers of the shifted Jacobi on the interval [0, L] can be given by

xL,N,j(θ,ϑ)=L2(xN,j(θ,ϑ)+1),ϖL,N,j(θ,ϑ)=(L2)θ+ϑ+1ϖN,j(θ,ϑ),0⩽j⩽N.

For any ϕ ∈ S_{2N + 1}[0, L] and using quadrature property, we have

∫0L(L−x)θxϑϕ(x)dx=L2θ+ϑ+1∫−11(1−x)θ(1+x)ϑ⋅ϕL2(x+1)dx=L2θ+ϑ+1∑j=0NϖN,j(θ,ϑ)⋅ϕL2(xN,j(θ,ϑ)+1)=∑j=0NϖL,N,j(θ,ϑ)ϕxL,N,j(θ,ϑ).(14)

3 One-dimensional T-FTTEs

By means of the shifted Jacobi Gauss-Lobatto and shifted Jacobi Gauss-Radau quadrature formulaes, the shifted Jacobi spectral collocation method is applied to solve the T-FTTEs.

Consider the T-FTTEs in the following form

cDtνu(x,t)−t2γΔu(x,t)=f(x,t),1≤ν≤2,(15)

with the initial condition

u(x,0)=u0(x),∂u(x,t)∂t∣t=0=u1(x),x∈[0,L],(16)

and boundary conditions

u(0,t)=ϕ0(t),u(L,t)=ϕ1(t),t∈[0,T],(17)

where γ is real non-negative number and f(x, t), ϕ₀(t), ϕ₁(t), u₀(x), u₁(x) are given functions. Here, △ is the differential operator

Δu(x,t)=∂2u(x,t)∂x2=uxx.

We are ready to use the SJ-GL-C and SJ-GR-C methods to transform the above T-FTTEs into a system of algebraic equations. To do so, we approximate the independent space variable x using the SJ-GL-C method, while the independent temporal variable t was approximated by the SJ-GR-C method.

Now, we list the major steps of the mixed SJ-GL-C and SJ-GR-C methods for solving the T-FTTEs. The solution of Eq. (15) is approximated as

uN,M(x,t)=∑i=0N∑j=0Mai,jPL,i(θ1,ϑ1)(x)PT,j(θ2,ϑ2)(t)=∑i=0N∑j=0Mai,jf0i,j(x,t),(18)

where f0i,j(x,t)=PL,i(θ1,ϑ1)(x)PT,j(θ2,ϑ2)(t). Then the spatial partial derivatives ∂uN,M(x,t)∂x and ∂2uN,M(x,t)∂x2 may be written as

∂uN,M(x,t)∂x=∑i=0N∑j=0Mai,jDx(PL,i(θ1,ϑ1)(x))PT,j(θ2,ϑ2)(t)=∑i=0N∑j=0Mai,jf1i,j(x,t),(19)

∂2uN,M(x,t)∂x2=∑i=0N∑j=0Mai,jDxx(PL,i(θ1,ϑ1)(x))PT,j(θ2,ϑ2)(t)=∑i=0N∑j=0Mai,jf2i,j(x,t),(20)

where f1i,j(x,t)=Dx(PL,i(θ1,ϑ1)(x))PT,j(θ2,ϑ2)(t) and f2i,j(x,t)=Dxx(PL,i(θ1,ϑ1)(x))PT,j(θ2,ϑ2)(t). Furthermore, the time fractional derivative cDtν can be evaluated as

cDtνuN,M(x,t)=∑i=0N∑j=0Mai,jPL,i(θ1,ϑ1)(x)cDtν(PT,j(θ2,ϑ2)(t))=∑i=0N∑j=0Mai,jf3i,j(x,t),(21)

where

cDtνPT,i(θ2,ϑ2)(t)=∑k=2i(−1)i−kΓ(i+ϑ2+1)Γ(i+k+θ2+ϑ2+1)Γ(i−k+1)Γ(k+ϑ2+1)Γ(k−ν+1)Γ(i+θ2+ϑ2+1)Tk⋅tk−ν,f3i,j(x,t)=PL,i(θ1,ϑ1)(x)cDtν(PT,j(θ2,ϑ2)(t)).

Consequently, we get

∑i=0N∑j=0Mai,jf3i,j(x,t)−t2γ∑i=0N∑j=0Mai,jf2i,j(x,t)=f(x,t),(x,t)∈[0,L]×[0,T],(22)

while the numerical treatment of initial and boundary conditions are

uN,M(x,0)=∑i=0N∑j=0Mai,jPL,i(θ1,ϑ1)(x)PT,j(θ2,ϑ2)(0)=u0(x),∂uN,M(x,t)∂t∣t=0=∑i=0N∑j=0Mai,jPL,i(θ1,ϑ1)(x)DtPT,j(θ2,ϑ2)(0)=u1(x),uN,M(0,t)=∑i=0N∑j=0Mai,jPL,i(θ1,ϑ1)(0)PT,j(θ2,ϑ2)(t)=ϕ0(t),uN,M(L,t)=∑i=0N∑j=0Mai,jPL,i(θ1,ϑ1)(L)PT,j(θ2,ϑ2)(t)=ϕ1(t).(23)

In the novel spectral collocation algorithm, the residual of (15) is set to zero at (N − 1) × (M − 1) of collocation points. Consequently, we find

∑i=0N∑j=0Mai,jf3i,j(xL,N,r(θ1,ϑ2),tT,M,s(θ2,ϑ2))−(tT,M,s(θ2,ϑ2))2γ⋅∑i=0N∑j=0Mai,jf2i,j(xL,N,r(θ1,ϑ2),tT,M,s(θ2,ϑ2))=f(xL,N,r(θ1,ϑ2),tT,M,s(θ2,ϑ2)),r=1,⋯,N−1;s=1,⋯,M−1.(24)

This create a system of (M − 1) × (N − 1) algebraic equations in the unknown coefficients, a_ij, and the remainder of this system is acquired by the initial (16) and boundary (16) conditions, as

∑i=0N∑j=0Mai,jPL,i(θ1,ϑ1)(xL,N,r(θ1,ϑ1))PT,j(θ2,ϑ2)(0)=u0(xL,N,r(θ1,ϑ1)),∑i=0N∑j=0Mai,jPL,i(θ1,ϑ1)(xL,N,r(θ1,ϑ1))DtPT,j(θ2,ϑ2)(0)=u1(xL,N,r(θ1,ϑ1)),r=1,⋯,N−1,(25)

∑i=0N∑j=0Mai,jPL,i(θ1,ϑ1)(0)PT,j(θ2,ϑ2)(tT,M,s(θ2,ϑ2))=ϕ0(tT,M,s(θ2,ϑ2)),∑i=0N∑j=0Mai,jPL,i(θ1,ϑ1)(L)PT,j(θ2,ϑ2)(tT,M,s(θ2,ϑ2))=ϕ1(tT,M,s(θ2,ϑ2)),s=0,⋯,M,(26)

where tT,M,s(θ2,ϑ2),s=0,1,⋯,M are the roots of PT,M+1(θ2,ϑ2)(t), while xL,N,r(θ1,ϑ1),r=1,2,⋯,N−1 are the roots of PL,N−1(θ1,ϑ1)(x).

The combination of Eqs (24), (25) and (26) generate a system of (M + 1) × (N + 1) algebraic equations in the unknown coefficients a_{i, j}, that can be easily solved. After the coefficients a_i,j are determined, it is straight forward to compute the approximate solution u_{N, M}(x, t) at any value of (x, t) in the given domain.

4 Two-dimensional T-FTTEs

In the current section, we extend the previous algorithm to numerically solve the two-dimensional T-FTTEs in the following form

cDtνu(x,y,t)−t2γΔu(x,y,t)=H(x,y,t),1≤ν≤2,(x,y,t)∈[0,L1]×[0,L2]×[0,T],(27)

related to the initial and boundary conditions

u(x,y,0)=g0(x,y),∂u(x,y,t)∂tt=0=g1(x,y),(x,y)∈[0,L1]×[0,L2],u(0,y,t)=g2(y,t),u(L1,y,t)=g3(y,t),(y,t)∈[0,L2]×[0,T],u(x,0,t)=g4(x,t),u(x,L2,t)=g5(x,t),(x,t)∈[0,L1]×[0,T],(28)

where H(x, y, t), g₀(x, y), g₁(x, y), g₂(y, t), g₃(y, t), g₄(x, t) and g₅(x, t) are given functions.

While, the differential operator △ is given by

Δu(x,y,t)=∂2u(x,y,t)∂x2+∂2u(x,y,t)∂y2=uxx+uyy.

Similar steps to that given in the previous section, enable one to write

uN,M,K(x,y,t)=∑i=0N∑j=0M∑k=0Kai,j,kPL1,i(θ0,ϑ0)(x)PL2,j(θ1,ϑ1)(y)⋅PT,k(θ2,ϑ2)(t)=∑i=0N∑j=0M∑j=0Kai,j,kf0i,j,k(x,y,t),(29)

where f0i,j,k(x,y,t)=PL2,j(θ1,ϑ1)(y)PL2,j(θ1,ϑ1)(y)PT,k(θ2,ϑ2)(t). Then the first spatial and temporal partial derivatives ∂uN,M,K(x,y,t)∂x,∂uN,M,K(x,y,t)∂yand∂uN,M,K(x,y,t)∂t can be computed as

∂uN,M,K(x,y,t)∂x=∑i=0N∑j=0M∑k=0Kai,j,k∂x(PL1,i(θ0,ϑ0)(x))PL2,j(θ1,ϑ1)(y)⋅PT,k(θ2,ϑ2)(t)=∑i=0N∑j=0M∑j=0Kai,j,kf1i,j,k(x,y,t),(30)

∂uN,M,K(x,y,t)∂y=∑i=0N∑j=0M∑k=0Kai,j,kPL1,i(θ0,ϑ0)(x)∂y(PL2,j(θ1,ϑ1)(y))⋅PT,k(θ2,ϑ2)(t)=∑i=0N∑j=0M∑j=0Kai,j,kf2i,j,k(x,y,t),(31)

∂uN,M,K(x,y,t)∂t=∑i=0N∑j=0M∑k=0Kai,j,kPL1,i(θ0,ϑ0)(x)PL2,j(θ1,ϑ1)(y)∂y⋅(PT,k(θ2,ϑ2)(t))=∑i=0N∑j=0M∑j=0Kai,j,kf3i,j,k(x,y,t),(32)

where

f1i,j,k(x,y,t)=∂x(PL1,i(θ0,ϑ0)(x))PL2,j(θ1,ϑ1)(y)PT,k(θ2,ϑ2)(t),f2i,j,k(x,y,t)=PL1,i(θ0,ϑ0)(x)∂y(PL2,j(θ1,ϑ1)(y))PT,k(θ2,ϑ2)(t),f3i,j,k(x,y,t)=PL1,i(θ0,ϑ0)(x))PL2,j(θ1,ϑ1)(y)∂t(PT,k(θ2,ϑ2)(t)).

Also, the second spatial partial derivatives ∂2uN,M,K(x,y,t)∂x2 and ∂2uN,M,K(x,y,t)∂y2 can be computed as

∂2uN,M,K(x,y,t)∂x2=∑i=0N∑j=0M∑k=0Kai,j,k∂x2(PL1,i(θ0,ϑ0)(x))⋅PL2,j(θ1,ϑ1)(y)PT,k(θ2,ϑ2)(t)=∑i=0N∑j=0M∑j=0Kai,j,kf4i,j,k(x,y,t),(33)

∂2uN,M,K(x,y,t)∂y2=∑i=0N∑j=0M∑k=0Kai,j,kPL1,i(θ0,ϑ0)(x)∂y2⋅(PL2,j(θ1,ϑ1)(y))PT,k(θ2,ϑ2)(t)=∑i=0N∑j=0M∑j=0Kai,j,kf5i,j,k(x,y,t),(34)

where

f4i,j,k(x,y,t)=∂x2(PL1,i(θ0,ϑ0)(x))PL2,j(θ1,ϑ1)(y)PT,k(θ2,ϑ2)(t),f5i,j,k(x,y,t)=PL1,i(θ0,ϑ0)(x)∂y2(PL2,j(θ1,ϑ1)(y))PT,k(θ2,ϑ2)(t).

Moreover, the time fractional derivative cDtν can be obtainted as

cDtνuN,M,K(x,y,t)=∑i=0N∑j=0M∑k=0Kai,j,kPL1,i(θ0,ϑ0)(x)PL2,j(θ1,ϑ1)(y)⋅cDtνPT,k(θ2,ϑ2)(t)=∑i=0N∑j=0M∑k=0Kai,j,kf6i,j,k(x,y,t),(35)

where

f6i,j,k(x,y,t)=PL1,i(θ0,ϑ0)(x)PL2,j(θ1,ϑ1)(y)cDtνPT,k(θ2,ϑ2)(t).

Therefore, adopting (29)-(35), enable one to write (27)-(28) in the form:

∑i=0N∑j=0M∑k=0Kai,j,k(f6i,j,k(x,y,t)−t2γf4i,j,k(x,y,t)+f5i,j,k(x,y,t))=H(x,y,t),(x,y,t)∈[0,L1]×[0,L2]×[0,T].(36)

Moreover, the collocation treatment of the initial-boundary conditions immediately, gives

uN,M,K(x,y,0)=∑i=0N∑j=0M∑k=0Kai,j,kf0i,j,k(x,y,0)=g0(x,y),∂uN,M,K(x,y,t)∂tt=0=∑i=0N∑j=0M∑k=0Kai,j,kf3i,j,k(x,y,0)=g1(x,y),uN,M,K(0,y,t)=∑i=0N∑j=0M∑k=0Kai,j,kf0i,j,k(0,y,t)=g2(y,t),uN,M,K(L1,y,t)=∑i=0N∑j=0M∑k=0Kai,j,kf0i,j,k(L1,y,t)=g3(y,t),uN,M,K(x,0,t)=∑i=0N∑j=0M∑k=0Kai,j,kf0i,j,k(x,0,t)=g4(x,t),uN,M,K(x,L2,t)=∑i=0N∑j=0M∑k=0Kai,j,kf0i,j,k(x,L2,t)=g5(x,t).(37)

Setting the residual of (27) to be zero at (N − 1) × (M − 1) × (K − 1) of collocation points. We have (N − 1) × (M − 1) × (K − 1) algebraic equations for (M + 1) × (N + 1) × (K + 1) unknown expansion coefficients,

∑i=0N∑j=0M∑k=0KFr,s,ςi,j,kai,j,k=H(xL1,N,r(θ0,ϑ0),yL2,M,s(θ1,ϑ1),tT,K,ς(θ2,ϑ2)),r=1,⋯,N−1;s=1,⋯,M−1;ς=1,⋯,K−1,(38)

where

Fr,s,ςi,j,k=−tT,M,sθ2,ϑ22γf4i,j,k(xL1,N,rθ0,ϑ0,yL2,N,rθ1,ϑ1,tT,M,sθ2,ϑ2+f5i,j,kxL1,N,rθ0,ϑ0,yL2,N,rθ1,ϑ1,tT,M,sθ2,ϑ2+f6i,j,kxL1,N,rθ0,ϑ0,yL2,N,rθ1,ϑ1,tT,M,sθ2,ϑ2,(39)

and from the initial conditions, we obtain (2(K(M + N) + MN + 1)) algebraic equations

∑i=0N∑j=0M∑k=0Kai,j,kf0i,j,k(xL1,N,r(θ0,ϑ0),yL2,M,s(θ1,ϑ1),0)=g0(xL1,N,r(θ0,ϑ0),yL2,M,s(θ1,ϑ1)),r=1,⋯,N−1;s=0,⋯,M,∑i=0N∑j=0M∑k=0Kai,j,kf3i,j,k(xL1,N,r(θ0,ϑ0),yL2,M,s(θ1,ϑ1),0)=g1(xL1,N,r(θ0,ϑ0),yL2,M,s(θ1,ϑ1)),r=1,⋯,N−1;s=1,⋯,M−1,∑i=0N∑j=0M∑k=0Kai,j,ki,jf0i,j,k(0,yL2,M,s(θ1,ϑ1),tT,K,ς(θ2,ϑ2))=g2(yL2,M,s(θ1,ϑ1),tT,K,ς(θ2,ϑ2)),s=0,⋯,M;ς=0,⋯,K,∑i=0N∑j=0M∑k=0Kai,j,ki,jf0i,j,k(0,yL2,M,s(θ1,ϑ1),tT,K,ς(θ2,ϑ2))=g3(yL2,M,s(θ1,ϑ1),tT,K,ς(θ2,ϑ2)),s=0,⋯,M;ς=0,⋯,K,∑i=0N∑j=0M∑k=0Kai,j,ki,jf0i,j,k(xL1,N,r(θ0,ϑ0),0,tT,K,ς(θ2,ϑ2))=g4(xL1,N,r(θ0,ϑ0),tT,K,ς(θ2,ϑ2)),r=1,⋯,N−1;ς=1,⋯,K,∑i=0N∑j=0M∑k=0Kai,j,ki,jf0i,j,k(xL1,N,r(θ0,ϑ0),0,tT,K,ς(θ2,ϑ2))=g5(xL1,N,r(θ0,ϑ0),tT,K,ς(θ2,ϑ2)),r=1,⋯,N−1;ς=1,⋯,K,(40)

The combination of (39) and (40) provides (M + 1) × (N + 1) × (K + 1) algebraic equations

∑i=0N∑j=0M∑k=0KFr,s,ςi,j,kai,j,k=H(xL1,N,r(θ0,ϑ0),yL2,M,s(θ1,ϑ1),tT,K,ς(θ2,ϑ2)),r=1,⋯,N−1,s=1,⋯,M−1,ς=1,⋯,K−1,∑i=0N∑j=0M∑k=0Kai,j,kf0i,j,k(xL1,N,r(θ0,ϑ0),yL2,M,s(θ1,ϑ1),0)=g0(xL1,N,r(θ0,ϑ0),yL2,M,s(θ1,ϑ1)),r=1,⋯,N−1,s=0,⋯,M,∑i=0N∑j=0M∑k=0Kai,j,kf3i,j,k(xL1,N,r(θ0,ϑ0),yL2,M,s(θ1,ϑ1),0)=g1(xL1,N,r(θ0,ϑ0),yL2,M,s(θ1,ϑ1)),r=1,⋯,N−1,s=1,⋯,M−1,∑i=0N∑j=0M∑k=0Kai,j,ki,jf0i,j,k(0,yL2,M,s(θ1,ϑ1),tT,K,ς(θ2,ϑ2))=g2(yL2,M,s(θ1,ϑ1),tT,K,ς(θ2,ϑ2)),s=0,⋯,M,ς=0,⋯,K,∑i=0N∑j=0M∑k=0Kai,j,ki,jf0i,j,k(0,yL2,M,s(θ1,ϑ1),tT,K,ς(θ2,ϑ2))=g3(yL2,M,s(θ1,ϑ1),tT,K,ς(θ2,ϑ2)),s=0,⋯,M,ς=0,⋯,K,∑i=0N∑j=0M∑k=0Kai,j,ki,jf0i,j,k(xL1,N,r(θ0,ϑ0),0,tT,K,ς(θ2,ϑ2))=g4(xL1,N,r(θ0,ϑ0),tT,K,ς(θ2,ϑ2)),r=1,⋯,N−1,ς=1,⋯,K,∑i=0N∑j=0M∑k=0Kai,j,ki,jf0i,j,k(xL1,N,r(θ0,ϑ0),0,tT,K,ς(θ2,ϑ2))=g5(xL1,N,r(θ0,ϑ0),tT,K,ς(θ2,ϑ2)),r=1,⋯,N−1,ς=1,⋯,K,(41)

The resulting system (24) can be easily solved.

5 Numerical results

In this section, we compare the new results with those obtained in the literature for revealing that our results are more accurate and effective.

The absolute difference between the approximate and exact solutions, namely absolute error, is given by

E(x,y,t)=|uN,M,K(x,y,t)−u(x,y,t)|,(x,y,t)∈[0,L1]×[0,L2]×[0,T],(42)

where u(x, y, t) and u_{N, M, K}(x, y, t) are the exact and numerical solutions at the point (x, y, t), respectively. Moreover, the maximum absolute errors (MAEs) is computed by

MAEs=MaxE(x,y,t):∀(x,y,t)∈[0,L1]×[0,L2]×[0,T]=L∞.(43)

Also we can denote to L² by

L2=∑i=0N∑j=0M∑k=0K(u(xi,yj,tk)−uN,M,K(xi,yj,tk))2(N+1)(M+1)(K+1)(44)

Example 1

We start with the following problem

cDtνu(x,t)−tΔu(x,t)=f(x,t),(x,t)∈[0,1]×[0,1],u(0,t)=u(1,t)=0,u(x,t)∣t=0=0,∂u(x,t)∂t∣t=0=0,(45)

with f(x,t)=6Γ(4−ν)t(3−ν)(x−x2)5−t4(20x3−150x4+420x5−560x6+360x7−90x8).

The exact solution is given by u(x, t) = t³(x − x²)⁵.

Zhang et al. [30] used the LDG-FE method to numerically solve the previous problem. To show that the novel algorithm is more accurate than the LDG-FE [30], in Table 1, we give the L²-error with several choices of ν, N, M, θ₁, ϑ₁, θ₂, and ϑ₂ and compare the achieved results with those obtained using the LDG-FE [30]. While, Table 2 lists the L^∞-error at ν, N, M, θ₁, ϑ₁, θ₂, and ϑ₂ and compare the achieved results with those obtained using the LDG-FE [30]. Moreover, we list the CPU time of problem (45) using the novel algorithm at θ₁ = ϑ₁ = θ₂ = ϑ₂ = 0 with several choices of ν, N, M, in Table 3.

Table 1

Comparing the values of L²-error in Example 1.

ν	N = M	Our method			LDG-FE in [ref30]

		θ₁ = ϑ₁ = 0, θ₂ = ϑ₂ = 0,	θ₁ = ϑ₁ = 0, θ2=ϑ2=12,	θ1=ϑ1=12 θ₂ = ϑ₂ = 0

1.2	10	3.097 × 10⁻¹⁷	3.944 × 10⁻¹⁷	2.168 × 10⁻¹⁶	7.180 × 10⁻⁵
	15	2.615 × 10⁻¹⁷	2.109 × 10⁻¹⁶	8.051 × 10⁻¹⁷	4.749 × 10⁻⁵
	20	8.386 × 10⁻¹⁷	4.357 × 10⁻¹⁶	2.404 × 10⁻¹⁶	3.552 × 10⁻⁵

1.6	10	1.361 × 10⁻¹⁷	1.973 × 10⁻¹⁷	1.934 × 10⁻¹⁶	7.174 × 10⁻⁵
	15	2.271 × 10⁻¹⁷	3.501 × 10⁻¹⁷	5.623 × 10⁻¹⁷	4.748 × 10⁻⁵
	20	3.021 × 10⁻¹⁷	2.271 × 10⁻¹⁷	2.129 × 10⁻¹⁶	3.552 × 10⁻⁵

1.9	10	6.760 × 10⁻¹⁸	1.085 × 10⁻¹⁷	1.867 × 10⁻¹⁶	7.172 × 10⁻⁵
	15	1.474 × 10⁻¹⁷	1.832 × 10⁻¹⁷	5.661 × 10⁻¹⁷	4.748 × 10⁻⁵
	20	3.133 × 10⁻¹⁷	6.136 × 10⁻¹⁷	2.073 × 10⁻¹⁶	3.553 × 10⁻⁵

Table 2

Comparing the values of L^∞-error in Example 1.

ν	N = M	Our method			LDG-FE in [30]

		θ₁ = ϑ₁ = 0, θ₂ = ϑ₂ = 0,	θ₁ = ϑ₁ = 0, θ2=ϑ2=12,	θ1=ϑ1=12 θ₂ = ϑ₂ = 0

1.2	10	1.844 × 10⁻¹⁶	2.111 × 10⁻¹⁶	6.800 × 10⁻¹⁶	2.022 × 10⁻⁴
	15	1.341 × 10⁻¹⁶	1.183 × 10⁻¹⁵	1.366 × 10⁻¹⁵	1.357 × 10⁻⁴
	20	7.526 × 10⁻¹⁶	4.932 × 10⁻¹⁵	2.156 × 10⁻¹⁵	1.018 × 10⁻⁴

1.6	10	2.990 × 10⁻¹⁷	5.078 × 10⁻¹⁷	5.413 × 10⁻¹⁶	2.019 × 10⁻⁴
	15	9.369 × 10⁻¹⁷	1.415 × 10⁻¹⁶	1.166 × 10⁻¹⁵	1.360 × 10⁻⁴
	20	1.551 × 10⁻¹⁶	9.369 × 10⁻¹⁷	2.349 × 10⁻¹⁵	1.020 × 10⁻⁴

1.9	10	1.779 × 10⁻¹⁷	3.241 × 10⁻¹⁷	4.908 × 10⁻¹⁶	2.019 × 10⁻⁴
	15	1.274 × 10⁻¹⁶	1.393 × 10⁻¹⁶	1.201 × 10⁻¹⁵	1.365 × 10⁻⁴
	20	1.576 × 10⁻¹⁶	4.901 × 10⁻¹⁶	2.517 × 10⁻¹⁵	1.020 × 10⁻⁴

Table 3

CPU time in seconds of Example 1.

ν	5	10	15	20

1.2	4.483	21.596	94.719	319.047
1.6	4.159	21.204	93.422	313.045
1.9	4.39	20.312	87.171	310.252

In Fig. 1, the numerical and exact solutions are compared with values of parameters listed in its caption. For the case of θ₁ = ϑ₁ = −0.5, θ₂ = ϑ₂ = 0, ν = 1.6 and N = M = 10, the absolute error curve in x-direction of problem (45) is shown in Fig. 2 in the interval [0, 1].

$Figure 1 x-directional curves of exact and numerical solutions of Example 1 with θ1=ϑ1=−12,θ2=ϑ2=0,ν=1.6$\theta_1=\vartheta_1=-\frac{1}{2},\ \theta_2=\vartheta_2=0,\ \nu=1.6 $ and N = M = 10.$

Figure 1

x-directional curves of exact and numerical solutions of Example 1 with θ1=ϑ1=−12,θ2=ϑ2=0,ν=1.6 and N = M = 10.

$Figure 2 x-direction of absolute error of Example 1 with θ1=ϑ1=−12,θ2=ϑ2=0,ν=1.6$\theta_1=\vartheta_1=-\frac{1}{2},\ \theta_2=\vartheta_2=0,\ \nu=1.6 $ and N = M = 10.$

Figure 2

x-direction of absolute error of Example 1 with θ1=ϑ1=−12,θ2=ϑ2=0,ν=1.6 and N = M = 10.

Example 2

Consider the following periodic boundary condition problem

cDtνu(x,t)−t2Δu(x,t)=f(x,t),(x,t)∈[0,1]×[0,1],u(0,t)=u(1,t)=1,u(x,t)∣t=0,∂u(x,t)∂t∣t=0=0,(46)

with f(x,t)=(6Γ(4−ν)t(3−ν)+4π2t5)cos(2πx).

The exact solution is given by u(x, t) = t³cos(2πx).

Table 4 and Table 5 display the L²-error and L^∞-error using our method with several choices of N, M, θ₁, ϑ₁, θ₂, ϑ₂ and ν = 1.2. We see in these tables that the results are very accurate for small choice of N and M. Fig. 3 allows us to see the absolute error E(x, t) where θ1=ϑ1=−12, θ₂ = ϑ₂ = 0, N = M = 20 and ν = 2. Also, in Fig. 4, we plot the x-directional curves of the exact and numerical solutions with values of parameters listed in its captions. In the case of θ1=ϑ1=−12,θ2=ϑ2=0, N = M = 20 and ν = 2, the absolute error curve in x-direction of problem (46) is shown in Fig. 5 in the interval [0, 1].

Table 4

L²-error using our method for Example 2 at ν = 1.2 with different values of N and M.

θ₁	ϑ₁	θ₂	ϑ₂	8	12	16	20

0	0	0	0	7.936 × 10⁻⁶	7.726 × 10⁻⁹	1.084 × 10⁻¹³	2.660 × 10⁻¹⁶
0	−12	0	−12	1.251 × 10⁻⁵	2.201 × 10⁻⁹	1.361 × 10⁻¹³	3.901 × 10⁻¹⁶
0	12	0	12	2.357 × 10⁻⁵	5.997 × 10⁻⁹	5.022 × 10⁻¹³	2.134 × 10⁻¹⁵

Table 5

L^∞-error using our method for Example 2 with ν = 1.2 for different choose of N and M.

θ₁	ϑ₁	θ₂	ϑ₂	8	12	16	20

0	0	0	0	8.065 × 10⁻⁵	1.394 × 10⁻⁸	1.874 × 10⁻¹²	3.663 × 10⁻¹⁵
0	−12	0	−12	9.572 × 10⁻⁵	2.319 × 10⁻⁸	2.164 × 10⁻¹²	1.387 × 10⁻¹⁵
0	12	0	12	9.763 × 10⁻⁷	2.699 × 10⁻⁸	2.970 × 10⁻¹²	1.632 × 10⁻¹⁴

$Figure 3 The absolute error for Example 2 with θ1=ϑ1=−12,θ2=ϑ2=0$\theta_1=\vartheta_1= -\frac{1}{2},\ \theta_2=\vartheta_2=0$, N = M = 20 and ν = 2.$

Figure 3

The absolute error for Example 2 with θ1=ϑ1=−12,θ2=ϑ2=0, N = M = 20 and ν = 2.

$Figure 4 x-directional curves of exact and numerical solutions for Example 2 with θ1=ϑ1=−12,θ2=ϑ2=0$\theta_1=\vartheta_1= -\frac{1}{2},\ \theta_2=\vartheta_2=0$, N = M = 20 and ν = 2.$

Figure 4

x-directional curves of exact and numerical solutions for Example 2 with θ1=ϑ1=−12,θ2=ϑ2=0, N = M = 20 and ν = 2.

$Figure 5 x-direction of absolute error of Example 2 with θ1=ϑ1=−12,θ2=ϑ2=0$\theta_1=\vartheta_1= -\frac{1}{2},\ \theta_2=\vartheta_2=0$, N = M = 20 and ν = 2.$

Figure 5

x-direction of absolute error of Example 2 with θ1=ϑ1=−12,θ2=ϑ2=0, N = M = 20 and ν = 2.

Example 3

Consider the following time-fractional wave equation

cDtνu(x,t)−Δu(x,t)=0,(x,t)∈[0,1]×[0,1],u(0,t)=u(1,t)=0,u(x,t)∣t=0=sin(2πx),∂u(x,t)∂t∣t=0=2πsin(2πx),(47)

with the exact solution u(x, t) = sin(2πx)(sin(2πt) + cos(2πt)). Table 6 and Table 7 display L²-error and L^∞-error using our method with several choices of N, M, θ₁, ϑ₁, θ₂, and, ϑ₂ with ν = 2 and compare the achieved results with those obtained using the LDG-FE [30].

Table 6

Comparing the values of L²-error in Example 3.

ν	N = M	Our method			LDG-FE in [30]

		θ₁ = ϑ₁ = 0, θ₂ = ϑ₂ = 0,	θ₁ = ϑ₁ = 0, θ2=ϑ2=12,	θ1=ϑ1=12 θ₂ = ϑ₂ = 0

2	10	3.56888 × 10⁻⁵	6.50212 × 10⁻⁵	3.58118 × 10⁻⁵	1.51684 × 10⁻¹
	15	6.39075 × 10⁻¹⁰	1.50546 × 10⁻⁹	5.39032 × 10⁻¹⁰	9.22909 × 10⁻²
	20	7.32752 × 10⁻¹³	2.07878 × 10⁻¹³	9.7214 × 10⁻¹³	673199 × 10⁻²

Table 7

Comparing the values of L^∞-error in Example 3.

ν	N = M	Our method			LDG-FE in [30]

		θ₁ = ϑ₁ = 0, θ₂ = ϑ₂ = 0,	θ₁ = ϑ₁ = 0, θ2=ϑ2=12,	θ1=ϑ1=12 θ₂ = ϑ₂ = 0
2	10	1.22063 × 10⁻⁴	2.3827 × 10⁻⁴	1.16581 × 10⁻⁴	3.19793 × 10⁻¹
	15	3.08715 × 10⁻⁹	1.24299 × 10⁻⁸	2.27151 × 10⁻⁹	2.10735 × 10⁻¹
	20	4.6434 × 10⁻¹²	1.62675 × 10⁻¹²	1.02232 × 10⁻¹¹	1.57829 × 10⁻¹

Example 4

Finally, we will test the following two-dimensional T-FTTES

cDtνu(x,y,t)−tΔu(x,y,t)=2t2sin⁡(2πx)sin⁡(2πy)t−ν1Γ(3−ν1)+4π2t,(x,t)∈[0,1]×[0,1]×[0,1],u(0,y,t)=u(1,y,t)=u(x,0,t)=u(x,1,t)=u(x,y,0)=∂u(x,y,t)∂t∣t=0=0.(48)

The exact solution is given by u(x, t) = t²sin(2πx)sin(2πy).

Table 8 displays L²-error using our method with θ₀ = ϑ₀ = θ₁ = ϑ₁ = θ₂ = ϑ₂ = 0, and several choices of N, M, and, ν. In this table, we compare the achieved results with those obtained using the finite element method [36].

Table 8

Comparison the values of L²-error in Example 4.

ν	Our method with N=M=K=			Finite element method [36]
ν
	4	6	8	5	15	25
1.2	9.24221 × 10⁻³	1.90167 × 10⁻⁴	3.11384 × 10⁻⁶	1.17926 × 10⁻²	1.82855 × 10⁻⁴	2.21322 × 10⁻⁵
1.5	9.12823 × 10⁻³	1.91155 × 10⁻⁴	3.13409 × 10⁻⁶	1.17629 × 10⁻²	1.83819 × 10⁻⁴	2.33994 × 10⁻⁵
1.8	9.36705 × 10⁻³	1.95507 × 10⁻⁴	3.18883 × 10⁻⁶	1.176677 × 10⁻²	1.73947 × 10⁻⁴	1.65217 × 10⁻⁵

In Fig. 6, we plot the curves of the exact u(x, y, t) and numerical u_8,8,8(x, y, t) solutions of equation (48), where θ0=ϑ0=12,θ1=ϑ1=−12,θ2=ϑ2=0 at different values of t and y.

$Figure 6 x-direction curves of exact and numerical solutions for Example 4 with θ0=ϑ0=12,θ1=ϑ1=−12,θ2=ϑ2=0$\theta_0=\vartheta_0= \frac{1}{2},\theta_1=\vartheta_1= -\frac{1}{2},\ \theta_2=\vartheta_2=0$, N = M = K = 8 and ν = 2.$

Figure 6

x-direction curves of exact and numerical solutions for Example 4 with θ0=ϑ0=12,θ1=ϑ1=−12,θ2=ϑ2=0, N = M = K = 8 and ν = 2.

6 Conclusion

By means of SJ-GL-C and SJ-GR-C schemes, we have introduced a space-time spectral algorithm for solving T-FTTEs . According to the numerical results obtained above, we can concluded the high accuracy of our technique. Numerical examples were given to confirm the rightness and reliability of our method. The results display the accuracy of the novel method.

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Received: 2016-4-2

Accepted: 2016-5-24

Published Online: 2016-8-15

Published in Print: 2016-1-1

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