AMP Advances in Mathematical Physics 1687-9139 1687-9120 Hindawi 10.1155/2019/3263589 3263589 Research Article An Efficient Compact Difference Method for Temporal Fractional Subdiffusion Equations https://orcid.org/0000-0002-5790-7987 Ren Lei 1 Liu Lei 1 Mironov Andrei D. School of Mathematics and Statistics Shangqiu Normal University Shangqiu 476000 China sqnc.edu.cn 2019 2182019 2019 11 06 2019 25 07 2019 2182019 2019 Copyright © 2019 Lei Ren and Lei Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain 2-α in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the L2-1σ approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.

National Natural Science Foundation of China 11401363 Education Foundation of Henan Province 19A110030 Foundation for the Training of Young Key Teachers in Colleges and Universities in Henan Province 2018GGJS134
1. Introduction

The Black-Scholes model, proposed in 1973 by Black and Scholes  and Merton , gives a theoretical estimate of the price of European-style options. Until Now, some of Black-Scholes models involving the fractional derivatives have emerged. In , Wyss priced a European call option by a time-fractional Black-Scholes model. In , Liang et al. derive a biparameter fractional Black-Merton-Scholes equation and obtain the explicit option pricing formulas for the European call option and put option, individually. An explicit closed-form analytical solution for barrier options under a generalized time-fractional Black-Scholes model by using eigenfunction expansion method together with the Laplace transform is derived in . In , a discrete implicit numerical scheme with a spatially second-order accuracy and a temporally 2-α order accuracy is constructed; the stability and convergence of the proposed numerical scheme are analysed using Fourier analysis. In , H.Zhang et al. use some numerical technique to price a European double-knock-out barrier option, and then the characteristics of the three fractional Black-Scholes models are analysed through comparison with the classical Black-Scholes model. More recently, a numerical scheme of fourth-order in space and 2-α in time is derived in ; the solvability and convergence of the proposed numerical scheme are proved rigorously using a Fourier analysis. Some computationally efficient numerical methods have been proposed for solving fractional differential equation, for example, which include finite difference methods, finite element methods, finite volume methods, spectral methods, and meshless methods .

In this paper, we continue the work of R.H.De Staelen et al. . The class of equations is given by(1)αCS,ttα+12σ2S22CS,tS2+r-DSCS,tS=rCS,t,S,tBd,Bu×0,Twith the following boundary (barrier) and final conditions(2)CBd,t=Pt,CBu,t=Qt,t0,T,and its initial condition(3)CS,T=VS,SBd,Bu,where r is the risk free rate, D is the dividend rate, and σ>0 is the volatility of the returns. The functions P and Q are the rebates paid when the corresponding barrier is hit. The terminal playoff of the option is V(S). The fractional derivative in (1) is a Caputo derivative defined as(4)αCS,ttα=1Γ1-α0tCηS,ηt-η-αdη,0<α<1,CS,tt,α=1.As described in , we consider the transform problem of (1)(5)D0CtαUx,t=a2Ux,tx2+bUx,tx-cUx,t+fx,t,x,t0,×0,T,Ubd,t=pt,Ubu,t=qt,t0,T,Ux,0=φx,xbd,bu.The rest of the paper is organized as follows: in Section 2, an efficient implicit numerical scheme with second-order accuracy in time and fourth-order accuracy in space is constructed. The analysis of the stability and convergence are presented in Section 3. In Section 4, numerical examples are given to illustrate the accuracy of the presented scheme and to support our theoretical results. Concluding remarks are given in the last section.

2. Construction of the Compact Finite Difference Scheme

In order to simplify the computation and analysis of the following compact finite difference scheme for Black-Scholes model, we use an indirect approach by introducing a suitable transformation.

According to some simple calculations, we transform equation (5) into(6)D0CtαVx,t=a2Vx,tx2-dVx,t+gx,t,x,t0,×0,T,Vbd,t=pt,Vbu,t=qt,t0,T,Vx,0=φx,xbd,bu.where(7)pt=pt,qt=kbuqt,φx=kxφx.It is clear that U(x,t) is a solution of (5) if and only if V(x,t) is a solution of (6).

In order to construct the compact finite difference scheme for the problem (5), we consider the above equivalent form (6).

Let τ=T/N be the time step and h=bu-bd/M=L/M be the spatial step, where M,N are positive integers.

Since the grid function v={vi0iM}, we then define difference operators as follows: (8)δxvi-1/2=1hvi-vi-1,δx2vi=1h2vi+1-2vi+vi-1,Hxvi=vi+h212δx2vi,

We also define(9)a0=σ1-α,b0=0,ak=k+σ1-α-k-α21-αk1,bk=12-αk+σ2-α-k-α22-α-12k+σ1-α+k-α21-αk1. where σ=1-α/2, and(10)ck,n=a0,k=0,n=1ak+bk+1-bk,0kn-2,n2an-1-bn-1,k=n-1,n2.

Lemma 1.

It holds (see )(11)1-α2k+σ-α<ak-bk<k+σ1-α-k-α21-αk1,

In order to discretize (6) into a compact finite difference system, we introduce the following lemmas.

Lemma 2.

Assuming v(t)C3[0,T], we have(12)D0Ctαvtn-α/2=1μk=1ncn-k,nvtk-vtk-1+Oτ3-α.where μ=ταΓ(2-α).

Proof.

From Lemma 2 of , we can obtain the proof of lemma.

Lemma 3.

Assuming v(t)C2[0,T]. When n1, we obtain(13)vtn-α/2=α2vtn-1+1-α2vtn+Oτ2.

Proof.

According to some simple calculations, the proof follows from Taylor expansions of the function v(t) at the point tn-α/2 for t=tn-1 and t=tn.

Since the above lemmas, we then discretize (6) into a compact finite difference scheme. In order to analyse, we define (14)δtvn-1/2=1τvn-vn-11nN,vn,α/2=α2vn-1+1-α2vn1nN. We also define the grid functions as follows: (15)Vin=Vxi,tn,Win=Vxi,tnt,Zin=2Vxi,tnx2,gin=gxi,tn,gin-α/2=gxi,tn-α/2,p,n=ptn,q,n=qtn,φi=φxi. For the second-order spatial derivative Zin, we adopt the following fourth-order compact approximation (see )(16)HxZin=δx2vxi+Oh4,1iM-1,We consider equation (6) at the point (xi,tn-α/2); we can obtain(17)D0CtαVxi,tn-α/2=aZin-α/2-dVin-α/2+gin-α/2.

From Lemmas 2 and 3, we have(18)1μk=1ncn-k,nVik-Vik-1=aZin,α/2-dVin,α/2+gin-α/2+Rtαin,0iM,1nN,where(19)Rtαin=a-dOτ2+Oτ3-α,0iM,1nN.We apply Hx to equation (18); then we have(20)1μk=1ncn-k,nHxVik-Vik-1=aδx2Vin,α/2-dHxVin,α/2+Hxgin-α/2+Rtxαin,1iM-1,1nN,where(21)Rtxαin=HxRtαin+aRxαin,1iM-1,1nNand(22)RtxαinCRτ2+h4,1iM-1,1nN.If we omit (Rtxα)in, then we have the compact finite difference scheme:(23)1μk=1ncn-k,nHxvik-vik-1=aδx2vin,α/2-dHxvin,α/2+Hxgin-α/2,1iM-1,1nN,vbd,t=pt,vbu,t=qt,t0,T,vx,0=φx,xbd,bu.

3. Stability and Convergence of the Proposed Compact Difference Scheme Theorem 4.

The compact difference scheme (23) is uniquely solvable.

Proof.

The compact difference scheme (23) can be written in matrix form (24)AVn=bn-1, where (25)bn-1=k=0n-1ζkVk,ζkR The tridiagonal coefficient matrix A=(aij) yields (26)aii=56a0α+b1αμ+a2-αh2+52-α12d,jiaij=16a0α+b1αμ-a2-αh2+2-α12d.It is easy to see that the tridiagonal coefficient matrix A is strictly diagonally dominant. Therefore, the coefficient matrix is nonsingular and hence invertible.

Next, we consider the stability and convergence analysis of the compact difference scheme (23).

Letting Ω={uu=(u0,u1,,uM),u0=uM=0}, for grid functions u,vΩ, we define the inner product and norm as follows: (27)u,v=hi=1M-1uivi,u=u,u1/2,u=max0iMui.δxu,δxv=hi=0M-1δxui+1/2δxvi+1/2,u1=δxu,δxu1/2,u1=u2+u121/2.According to simple calculations, we obtain(28)δx2u,v=-δxu,δxv,hδx2u2u1,hu12u.

In order to analyse, we introduce the discrete inner product and norm: (29)u,v=Hxu,-δx2v=δxu,δxv-h212δx2u,δx2v,uε=u,u1/2. Based on above inner product and norm, we have the following lemmas.

Lemma 5 (see [<xref ref-type="bibr" rid="B22">29</xref>]).

Suppose uΩ, we obtain(30)Hxu2u23L216uε2,u23L8uε2,u1238+L216uε2.

Lemma 6 (see [<xref ref-type="bibr" rid="B24">27</xref>]).

Suppose uΩ, we obtain(31)uL28δx2u,uε23L216δx2u2.

Lemma 7 (see [<xref ref-type="bibr" rid="B1">9</xref>]).

Suppose uΩ, we obtain(32)k=1ncn-k,nHxuk-uk-1,-δx2un,α/212k=1ncn-k,nukε2-uk-1ε2,1nN.

In the next, we then analyse the stability and convergence of the scheme (23).

Theorem 8 (stability).

Let vn=(v0n,v1n,,vMn) be the solution of the compact difference scheme (23) with v0n=vMn=0. Assume that one of the conditions 14(4ε-1)a/3dεL2 holds for some positive constant ε>1/4.

Then it holds(33)vnε2v0ε2+4εΓ1-αTαamax1nNHxgn-α/22,1nN.

Proof.

We take the inner product of equation (23) with -δx2vn,α/2 yield (34)1μk=1ncn-k,nHxvk-vk-1,-δx2vn,α/2=-aδx2vn,α/22-dHxvn,α/2,δx2vn,α/2-Hxgn-α/2,δx2vn,α/2,1nN. Using Lemma 7,(35)12μk=1ncn-k,nvkε2-vk-1ε2-aδx2vn,α/22+dvn,α/2ε2-Hxgn-α/2,δx2un,α/2,1nN.When 14(4ε-1)a/3dεL2 for some positive constant ε>1/4, we have from the Cauchy-Schwarz inequality and Lemmas 6 that(36)dvn,α/2ε23dL216δx2vn,α/22a-a4εδx2vn,α/22(37)-Hxgn-α/2,δx2un,α/2εaHxgn-α/22+a4εδx2vn,α/22 By (35) and the Cauchy-Schwarz inequality,(38)-aδx2un,α/22+dvn,α/2ε2-Hxgn-α/2,δx2un,α/2εaHxgn-α/22.Substituting (38) into (35) leads to(39)k=1ncn-k,nukε2-uk-1ε22εμaHxgn-α/22. The above inequality can be rewritten as (40)c0,nunε2k=1n-1cn-k-1,n-cn-k,nukε2+cn-1,nu02+2εμaHxgn-α/22.Since by the definition of cn-1,n, (41)μcn-1,n=μan-1-bn-1<2Γ1-αTα, we have from (40) that(42)c0,nunε2k=1n-1cn-k-1,n-cn-k,nukε2+cn-1,nu0ε2+4εΓ1-αTαaHxgn-α/22.Letting(43)E=u0ε2+4εΓ1-αTαamax1nNHxgn-α/22and assuming ukε2E(0kn-1), we obtain(44)c0,nunε2k=1n-1cn-k-1,n-cn-k,nE+cn-1,nE=c0,nE.and we have the needed estimates.

Letting ein=Vin-vin, we get the following error equation:(45)1μk=1ncn-k,nHxeik-eik-1=aδx2ein,α/2-dHxein,α/2+Rtxαin,1iM-1,1nN,ebd,t=0,ebu,t=0,t0,T,ex,0=0,xbd,bu.Since the above error equation (45), we now obtain the following convergence results.

Theorem 9 (convergence).

Let Vin denote the value of the solution v(x,t) of (23) at the mesh point (xi,tn) and let vn=(v0n,v1n,,vMn) be the solution of the compact difference scheme (23). Then when 14(4ε-1)a/3dεL2, it holds(46)Un-unεC1τ2+h4,1nN,where(47)C1=4Γ1-αTαLCR2a1/2,

Proof.

It follows from Theorem 8 that (48)enε24εΓ1-αTαamax1nNRtxαin2,1nN, Applying (22), we get (49)enε2C12τ2+h42. The estimate (46) is proved.

Remark 10.

The constraint condition 14(4ε-1)a/3dεL2 in Theorems 8 and 9 is only for the analysis of the stability and convergence of the compact difference scheme (23). This condition is easily verifiable for practical problems.

4. Numerical Experiment

For demonstrating the efficiency of the compact difference scheme (23), we make two numerical experiments of it.

Suppose Vin=v(xi,tn) be the value of the solution v(x,t) of the problem (1)–(3) at the mesh point (xi,tn). From (22), we can see that(50)Vn-vnνC2τ2+h4,ν=1,2,where C2 is a positive constant independent. In order to check this accuracy of the compact difference scheme, we compute the following norm errors:(51)Eντ,h=max0nNVn-vnνν=1,2,.The temporal convergence order and the spatial convergence order are denoted by(52)Oνtτ,h=log2Eν2τ,hEντ,h,Oνsτ,h=log2Eντ,2hEντ,hν=1,2,.

Example 1.

We first consider a problem, which is governed by equation (1) in [0,1]×[0,1] with r=0.05,σ=0.25,a=σ2/2,b=r-a,c=r and(53)fx,t=2t2-αΓ3-α+2t1-αΓ2-αx21-x-t+12a2-6x+b2x-3x2-cx21-x.The boundary and initial conditions are given by (2) and (3) with(54)Ux,0=x21-x,U0,t=U1,t=0.It is easy to check that U(x,t)=(t+1)2x2(1-x) is the solution of this problem.

For different α, we let the spatial step h=1/100. Table 1 gives the errors Eν(τ,h)(ν=1,2,) and the temporal convergence orders Oνt(τ,h)(ν=1,2,) of the computed solution Uin for α=1/4,1/2,3/4 and different time step τ. From the table, we can see that the computed solution Uin has the second-order temporal accuracy. For comparison, the corresponding temporal convergence orders Oνt(τ,h)(ν=) given in  has only 2-α order; thus it is far less accurate than the compact difference scheme (23) given in this paper.

Next, we compute the spatial convergence order of the compact difference scheme (23). Table 2 presents the errors Eν(τ,h)(ν=1,2,) and the spatial convergence orders orderOνs(τ,h)(ν=1,2,). The table demonstrates that the compact difference scheme (23) has the fourth-order spatial accuracy.

The errors and the temporal convergence orders of the compact difference scheme (23) for Example 1  (h=1/100).

α τ E 1 ( τ , h ) O 1 t ( τ , h ) E 2 ( τ , h ) O 2 t ( τ , h ) E ( τ , h ) O t ( τ , h )
1/4 1/10 3.7361e–05 3.7314e–05 6.2608e–05
1/20 9.3611e–06 1.9968 9.3492e–06 1.9968 1.5684e–05 1.9970
1/40 2.3429e–06 1.9984 2.3399e–06 1.9984 3.9251e–06 1.9985
1/80 5.8603e–07 1.9992 5.8528e–07 1.9992 9.8176e–07 1.9993
1/160 1.4654e–07 1.9997 1.4635e–07 1.9997 2.4549e–07 1.9997
1/320 3.6634e–08 2.0001 3.6587e–08 2.0001 6.1372e–08 2.0000

1/2 1/10 6.7788e–05 6.7702e–05 1.1393e–04
1/20 1.6994e–05 1.9960 1.6972e–05 1.9960 2.8555e–05 1.9964
1/40 4.2543e–06 1.9980 4.2489e–06 1.9980 7.1480e–06 1.9981
1/80 1.0643e–06 1.9990 1.0630e–06 1.9990 1.7882e–06 1.9991
1/160 2.6617e–07 1.9995 2.6583e–07 1.9995 4.4718e–07 1.9996
1/320 6.6548e–08 1.9999 6.6463e–08 1.9999 1.1181e–07 1.9999

3/4 1/10 8.8226e–05 8.8110e–05 1.4950e–04
1/20 2.2098e–05 1.9973 2.2069e–05 1.9973 3.7435e–05 1.9976
1/40 5.5299e–06 1.9986 5.5226e–06 1.9986 9.3672e–06 1.9987
1/80 1.3832e–06 1.9993 1.3813e–06 1.9993 2.3429e–06 1.9993
1/160 3.4587e–07 1.9996 3.4542e–07 1.9996 5.8585e–07 1.9997
1/320 8.6475e–08 1.9999 8.6361e–08 1.9999 1.4647e–07 1.9999

The errors and the spatial convergence orders of the compact difference scheme (23) for Example 1  (h=1/10000).

α τ E 1 ( τ , h ) O 1 t ( τ , h ) E 2 ( τ , h ) O 2 t ( τ , h ) E ( τ , h ) O t ( τ , h )
1/4 1/2 1.1190e–04 6.4607e–05 9.1369e–05
1/4 5.4155e–06 4.3690 4.2667e–06 3.9205 5.4429e–06 4.0693
1/8 2.9041e–07 4.2209 2.6840e–07 3.9907 3.3922e–07 4.0041
1/16 1.7125e–08 4.0840 1.6758e–08 4.0015 2.1165e–08 4.0024
1/32 1.0264e–09 4.0604 1.0207e–09 4.0372 1.2889e–09 4.0375

1/2 1/2 1.0340e–04 5.9701e–05 8.4430e–05
1/4 5.0151e–06 4.3659 3.9472e–06 3.9189 4.9995e–06 4.0779
1/8 2.6907e–07 4.2202 2.4841e–07 3.9900 3.1142e–07 4.0048
1/16 1.5831e–08 4.0871 1.5487e–08 4.0036 1.9400e–08 4.0047
1/32 9.2534e–10 4.0967 9.2011e–10 4.0731 1.1538e–09 4.0716

3/4 1/2 9.3459e–05 5.3959e–05 7.6309e–05
1/4 4.5477e–06 4.3611 3.5734e–06 3.9165 4.4770e–06 4.0912
1/8 2.4420e–07 4.2190 2.2506e–07 3.9889 2.7865e–07 4.0060
1/16 1.4335e–08 4.0905 1.4014e–08 4.0053 1.7372e–08 4.0036
1/32 8.2043e–10 4.1270 8.1562e–10 4.1028 1.0150e–09 4.0971
Example 2.

In this example, we test the error and the convergence order of the compact difference scheme (23). Consider equation (1) in the domain [0,1]×[0,1] with r=0.5,a=1,b=r-a,c=r and(55)fx,t=2t2-αΓ3-α+2t1-αΓ2-αx3+x2+1-t+12a2+6x+b2x+3x2-cx3+x2+1.The boundary and initial conditions are given by (2) and (3) with(56)ϕ0t=x3+x2+1,U0,t=t+12,U1,t=3t+12.It is clear that U(x,t)=(t+1)2(x3+x2+1) is the exact analytical solution of this problem.

Apply the compact difference scheme (23) to solve the above problem. Table 3 presents the errors Eν(τ,h)(ν=1,2,) and the temporal convergence orders Oνt(τ,h)(ν=1,2,); we can see that the computed solution Uin has the second-order temporal accuracy.

From Table 4, we can obtain the errors Eν(τ,h)(ν=1,2,) and the spatial convergence orders Oνs(τ,h)(ν=1,2,). These numerical results demonstrate that the accuracy of the compact difference scheme (23) is fourth-order.

The errors and the temporal convergence orders of the compact difference scheme (23) for Example 2  (τ=1/100).

α h E 1 ( τ , h ) O 1 s ( τ , h ) E 2 ( τ , h ) O2s(τ,h) E ( τ , h ) O s ( τ , h )
1/4 1/10 9.5681e–05 9.5622e–05 1.3266e–04
1/20 2.3985e–05 1.9961 2.3971e–05 1.9961 3.3254e–05 1.9961
1/40 6.0044e–06 1.9981 6.0007e–06 1.9981 8.3244e–06 1.9981
1/80 1.5021e–06 1.9990 1.5012e–06 1.9990 2.0825e–06 1.9991
1/160 3.7564e–07 1.9996 3.7541e–07 1.9996 5.2078e–07 1.9996
1/320 9.3916e–08 1.9999 9.3859e–08 1.9999 1.3020e–07 1.9999

1/2 1/10 1.7283e–04 1.7272e–04 2.4033e–04
1/20 4.3358e–05 1.9950 4.3331e–05 1.9950 6.0286e–05 1.9951
1/40 1.0858e–05 1.9975 1.0852e–05 1.9975 1.5097e–05 1.9976
1/80 2.7169e–06 1.9987 2.7153e–06 1.9987 3.7774e–06 1.9988
1/160 6.7952e–07 1.9994 6.7910e–07 1.9994 9.4474e–07 1.9994
1/320 1.6991e–07 1.9998 1.6980e–07 1.9998 2.3622e–07 1.9998

3/4 1/10 2.2075e–04 2.2061e–04 3.0894e–04
1/20 5.5326e–05 1.9964 5.5291e–05 1.9964 7.7418e–05 1.9966
1/40 1.3849e–05 1.9981 1.3841e–05 1.9981 1.9378e–05 1.9983
1/80 3.4646e–06 1.9991 3.4624e–06 1.9991 4.8476e–06 1.9991
1/160 8.6641e–07 1.9995 8.6587e–07 1.9995 1.2123e–06 1.9996
1/320 2.1663e–07 1.9998 2.1650e–07 1.9998 3.0311e–07 1.9998

The errors and the spatial convergence orders of the compact difference scheme (23) for Example 2  (h=1/15000).

α τ E 1 ( τ , h ) O 1 t ( τ , h ) E 2 ( τ , h ) O 2 t ( τ , h ) E ( τ , h ) O t ( τ , h )
1/4 1/2 1.2909e–04 7.4532e–05 1.0540e–04
1/4 6.2469e–06 4.3691 4.9219e–06 3.9206 6.2789e–06 4.0693
1/8 3.3499e–07 4.2210 3.0961e–07 3.9907 3.9130e–07 4.0042
1/16 1.9745e–08 4.0846 1.9323e–08 4.0021 2.4403e–08 4.0031
1/32 1.1750e–09 4.0707 1.1685e–09 4.0475 1.4745e–09 4.0488

1/2 1/2 1.1929e–04 6.8871e–05 9.7399e–05
1/4 5.7849e–06 4.3660 4.5533e–06 3.9189 5.7675e–06 4.0779
1/8 3.1036e–07 4.2203 2.8654e–07 3.9901 3.5923e–07 4.0050
1/16 1.8246e–08 4.0883 1.7849e–08 4.0048 2.2357e–08 4.0061
1/32 1.0507e–09 4.1182 1.0447e–09 4.0946 1.3056e–09 4.0980

3/4 1/2 1.0782e–04 6.2247e–05 8.8031e–05
1/4 5.2457e–06 4.3613 4.1220e–06 3.9166 5.1648e–06 4.0912
1/8 2.8166e–07 4.2191 2.5959e–07 3.9891 3.2142e–07 4.0062
1/16 1.6515e–08 4.0921 1.6146e–08 4.0070 2.0016e–08 4.0053
1/32 9.2416e–10 4.1595 9.1874e–10 4.1354 1.1316e–09 4.1447
5. Concluding Remarks

In this paper, a high-order compact finite difference method for a class of time-fractional Black-Scholes equations is presented and analysed. We apply the L2-1σ approximation formula to the Caputo derivative; then we construct a fourth-order compact finite difference approximation for the spatial derivative. We have analysed the solvability, stability, and convergence of the constructed scheme and provided the optimal error estimates. The constructed scheme has the second-order temporal accuracy and the fourth-order spatial accuracy, which improves the temporal accuracy of the method given in .

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by National Natural Science Foundation of China No. 11401363, the Education Foundation of Henan Province No. 19A110030, the Foundation for the Training of Young Key Teachers in Colleges and Universities in Henan Province No. 2018GGJS134.

Black F. Scholes M. The pricing of options corporate liabilities Journal of Political Economy 1973 81 637 659 10.1086/260062 Merton R. C. Theory of rational option pricing Bell Journal of Economics and Management Science 1973 4 141 183 10.2307/3003143 Zbl1257.91043 Wyss W. The fractional Black-Scholes equation Fractional Calculus and Applied Analysis 2000 3 1 51 61 MR1743405 Zbl1058.91045 Liang J.-R. Wang J. Zhang W.-J. Qiu W.-Y. Ren F.-Y. Option pricing of a bi-fractional Black-Merton-Scholes model with the Hurst exponent H in [1/2,1] Applied Mathematics Letters 2010 23 8 859 863 10.1016/j.aml.2010.03.022 MR2651462 Chen W. Xu X. Zhu S.-P. Analytically pricing double barrier options based on a time-fractional Black-Scholes equation Computers & Mathematics with Applications 2015 69 12 1407 1419 10.1016/j.camwa.2015.03.025 MR3348967 Zhang H. Liu F. Turner I. Yang Q. Numerical solution of the time fractional Black-Scholes model governing European options Computers & Mathematics with Applications 2016 71 9 1772 1783 10.1016/j.camwa.2016.02.007 MR3490469 Zhang H. Liu F. Turner I. Chen S. The numerical simulation of the tempered fractional Black–Scholes equation for European double barrier option Applied Mathematical Modelling 2016 40 11-12 5819 5834 10.1016/j.apm.2016.01.027 De Staelen R. Hendy A. Numerically pricing double barrier options in a time-fractional Black–Scholes model Computers & Mathematics with Applications 2017 74 6 1166 1175 10.1016/j.camwa.2017.06.005 Alikhanov A. A. A new difference scheme for the time fractional diffusion equation Journal of Computational Physics 2015 280 424 438 10.1016/j.jcp.2014.09.031 MR3273144 2-s2.0-84908329447 Zbl1349.65261 Bueno-Orovio A. Kay D. Grau V. Rodriguez B. Burrage K. Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization Journal of the Royal Society Interface 2014 11, article no. 352 20140352 20140352 10.1098/rsif.2014.0352 2-s2.0-84903555093 Burrage K. Hale N. Kay D. An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations SIAM Journal on Scientific Computing 2012 34 4 A2145 A2172 10.1137/110847007 Zbl1253.65146 Chen S. Liu F. Zhuang P. Anh V. Finite difference approximations for the fractional Fokker-Planck equation Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems 2009 33 1 256 273 10.1016/j.apm.2007.11.005 MR2458510 2-s2.0-51749116733 Zbl1167.65419 Dimitrov Y. Numerical approximations for fractional differential equations Journal of Fractional Calculus and Applications 2014 5 1 45 MR3310666 Gao G.-h. Sun Z.-z. A compact finite difference scheme for the fractional sub-diffusion equations Journal of Computational Physics 2011 230 3 586 595 10.1016/j.jcp.2010.10.007 MR2745445 Podlubny I. Fractional Differential Equations 1999 New York, NY, USA Academic Press MR1658022 Li X. Xu C. A space-time spectral method for the time fractional diffusion equation SIAM Journal on Numerical Analysis 2009 47 3 2108 2131 10.1137/080718942 MR2519596 Zbl1193.35243 Hilfer R. Applications of Fractional Calculus in Physics 2000 Singapore World Scientific 10.1142/9789812817747 MR1890104 Zbl0998.26002 Langlands T. A. M. Henry B. I. The accuracy and stability of an implicit solution method for the fractional diffusion equation Journal of Computational Physics 2005 205 2 719 736 10.1016/j.jcp.2004.11.025 MR2135000 2-s2.0-17144427014 Luchko Y. Punzi A. Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations GEM - International Journal on Geomathematics 2011 1 2 257 276 10.1007/s13137-010-0012-8 Zbl1301.34104 Li C. Zeng F. Numerical Methods for Fractional Calculus 2015 Boca Raton, FL, USA Chapman and Hall/CRC MR3381791 Metzler R. Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach Physics Reports 2000 339 1 1 77 10.1016/S0370-1573(00)00070-3 MR1809268 Zbl0984.82032 Samarskii A. A. The Theory of Difference Schemes 2001 New York, NY, USA Marcel Dekker 10.1201/9780203908518 MR1818323 Zbl0971.65076 Wang Z. Vong S. Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation Journal of Computational Physics 2014 277 1 15 10.1016/j.jcp.2014.08.012 Zbl1349.65348 Zhang Y. Benson D. A. Reeves D. M. Time and space nonlocalities underlying fractional-derivative models: distinction and literature review of field applications Advances in Water Resources 2009 32 4 561 581 10.1016/j.advwatres.2009.01.008 2-s2.0-62349097511 Zhao L. Deng W. A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives Numerical Methods for Partial Differential Equations 2015 31 5 1345 1381 10.1002/num.21947 Zbl1332.65131 Guo Y. Solvability for a nonlinear fractional differential equation Bulletin of the Australian Mathematical Society 2009 80 1 125 138 10.1017/S0004972709000124 MR2520529 Zbl1195.34011 2-s2.0-77957240194 Wang Y. Ren L. Efficient compact finite difference methods for a class of time-fractional convection–reaction–diffusion equations with variable coefficients International Journal of Computer Mathematics 2018 96 2 264 297 10.1080/00207160.2018.1437262 Zhang Y. N. Sun Z. Z. Wu H. W. Error estimates of Crank-Nicolson-type difference schemes for the subdiffusion equation SIAM Journal on Numerical Analysis 2011 49 2302 2322 10.1137/100812707 Wang Y. A compact finite difference method for solving a class of time fractional convection-subdiffusion equations BIT Numerical Mathematics 2015 55 4 1187 1217 10.1007/s10543-014-0532-y Zbl1348.65120