April  2019, 24(4): 1989-2015. doi: 10.3934/dcdsb.2019026

Well-posedness and numerical algorithm for the tempered fractional differential equations

1. 

Department of Applied Mathematics, Xi'an University of Technology, Xi'an, Shaanxi 710054, China

2. 

Beijing Computational Science Research Center, Beijing 10084, China

3. 

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China

4. 

Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi 710054, China

Received  November 2015 Revised  January 2019 Published  January 2019

Trapped dynamics widely appears in nature, e.g., the motion of particles in viscous cytoplasm. The famous continuous time random walk (CTRW) model with power law waiting time distribution (having diverging first moment) describes this phenomenon. Because of the finite lifetime of biological particles, sometimes it is necessary to temper the power law measure such that the waiting time measure has convergent first moment. Then the time operator of the Fokker-Planck equation corresponding to the CTRW model with tempered waiting time measure is the so-called tempered fractional derivative. This paper focus on discussing the properties of the time tempered fractional derivative, and studying the well-posedness and the Jacobi-predictor-corrector algorithm for the tempered fractional ordinary differential equation. By adjusting the parameter of the proposed algorithm, high convergence order can be obtained and the computational cost linearly increases with time. The numerical results shows that our algorithm converges with order $ N_I $, where $ N_I $ is the number of used interpolating points.

Citation: Can Li, Weihua Deng, Lijing Zhao. Well-posedness and numerical algorithm for the tempered fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1989-2015. doi: 10.3934/dcdsb.2019026
References:
[1]

A. Z. Al-Abedeen and H. L. Arora, A global existence and uniqueness theorem for ordinary differential equation of generalized order, Canad. Math. Bull., 21 (1978), 271-276. doi: 10.4153/CMB-1978-047-1.

[2]

N. Atanasova and I. Brayanov, Computation of some unsteady flows over porous semi-infinite flat surface, in Large-Scale Scientific Computing, Lecture Notes in Computer Science, 3743, Springer, Berlin, 2006,621–628. doi: 10.1007/11666806_71.

[3]

M. BenchohraJ. HendersonS. K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 338 (2008), 1340-1350. doi: 10.1016/j.jmaa.2007.06.021.

[4]

B. Baeumera and M. M. Meerschaert, Tempered stable Lévy motion and transient super-diffusion, J. Comput. Appl. Math., 233 (2010), 2438-2448. doi: 10.1016/j.cam.2009.10.027.

[5]

R. G. Buschman, Decomposition of an integral operator by use of Mikusinski calculus, SIAM J. Math. Anal., 3 (1972), 83-85. doi: 10.1137/0503010.

[6]

H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, North-Holland Publishing Co., Amsterdam, 1986.

[7] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.
[8]

M. H. Chen and W. H. Deng, Discretized fractional substantial calculus, ESAIM: Math. Mod. Numer. Anal., 49 (2015), 373-394.

[9]

Á. Cartea and D. del-Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Phys. A, 374 (2007), 749-763. doi: 10.1063/1.2336114.

[10]

Á. Cartea and D. del-Castillo-Negrete, Fluid limit of the continuous-time random walk with general Lévy jump distribution functions, Phys. Rev. E, 76 (2007), 041105. doi: 10.1103/PhysRevE.76.041105.

[11]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

[12]

W. H. Deng, Numerical algorithm for the time fractional Fokker-Planck equation, J. Comp. Phys., 227 (2007), 1510-1522. doi: 10.1016/j.jcp.2007.09.015.

[13]

W. H. Deng, Smoothness and stability of the solutions for nonlinear fractional differential equations, Nonl. Anal., 72 (2010), 1768-1777. doi: 10.1016/j.na.2009.09.018.

[14]

K. Diethelm and N. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (1902), 229-248. doi: 10.1006/jmaa.2000.7194.

[15]

K. Diethelm and N. J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154 (2004), 621-640. doi: 10.1016/S0096-3003(03)00739-2.

[16]

K. DiethelmN. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential eqations, Nonlinear Dynam., 29 (2002), 3-22. doi: 10.1023/A:1016592219341.

[17]

A. M. A. El-Sayed, Fractional differential equations, Kyungpook Math. J., 28 (1988), 119-122.

[18]

R. Friedrich, F. Jenko, A. Baule and S. Eule, Anomalous diffusion of inertial, weakly damped particles, Phys. Rev. Lett., 96 (2006), 230601. doi: 10.1103/PhysRevLett.96.230601.

[19]

J. Gajda and M. Magdziarz, Fractional Fokker-Planck equation with tempered $\alpha$-stable waiting times: Langevin picture and computer simulation, Phys. Rev. E, 82 (2010), 011117. doi: 10.1103/PhysRevE.82.011117.

[20]

A. Hanyga, Wave propagation in media with singular memory, Math. Comput. Model., 34 (2001), 1399-1421. doi: 10.1016/S0895-7177(01)00137-6.

[21]

B. I. Henry, T. A. M. Langlands and S. L. Wearne, Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations, Phys. Rev. E, 74 (2006), 031116. doi: 10.1103/PhysRevE.74.031116.

[22]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747.

[23]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, North Holland, 2006.

[24]

A. A. Kilbas and J. J. Trujillo, Differential equation of fractional order: methods, results and problems-Ⅰ, Appl. Anal., 78 (2001), 153-192. doi: 10.1080/00036810108840931.

[25]

V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009.

[26]

C. P. Li and W. H. Deng, Remarks on fractional derivatives, Appl. Math. Comput., 187 (2007), 777-784. doi: 10.1016/j.amc.2006.08.163.

[27]

Y. J. Li and Y. J.Wang, Uniform asymptotic stability of solutions of fractional functional differential equations, Abst. Appl. Anal., 2013 (2013), 532589. doi: 10.1155/2013/532589.

[28]

C. Li and W. H. Deng, High order schemes for the tempered fractional diffusion equations, Adv. Comput. Math., 42 (2016), 543-572. doi: 10.1007/s10444-015-9434-z.

[29]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[30]

M. M. Meerschaert, Y. Zhang and B. Baeumer, Tempered anomalous diffusion in heterogeneous systems, Geophys. Res. Lett., 35 (2008), L17403. doi: 10.1029/2008GL034899.

[31]

M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, 43, Walter de Gruyter & Co., Berlin, 2012.

[32]

M. M. MeerschaertF. SabzikarM. S. Phanikumar and A. Zeleke, Tempered fractional time series model for turbulence in geophysical flows, J. Stat. Mech. Theory Exp., 14 (2014), 1742-5468. doi: 10.1088/1742-5468/2014/09/P09023.

[33] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[34]

A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Springer-Verlag, New York, 2000.

[35]

E. Pitcher and W. E. Sewell, Existence theorems for solutions of differential equations of non-integer order, Bull. Amer. Math. Soc., 44 (1938), 100-107. doi: 10.1090/S0002-9904-1938-06695-5.

[36] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[37]

M. G. W. Schmidt, F. Sagués and I. M. Sokolov, Mesoscopic description of reactions for anomalous diffusion: A case study, J. Phys.: Condens. Matter, 19 (2007), 065118. doi: 10.1088/0953-8984/19/6/065118.

[38]

S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, London, 1993.

[39]

F. SabzikarM. M. Meerschaert and J. H. Chen, Tempered fractional calculus, J. Comput. Phys., 293 (2015), 14-28. doi: 10.1016/j.jcp.2014.04.024.

[40]

J. L. Schiff, The Laplace Transform: Theory and Applications, Springer, New York, 1991. doi: 10.1007/978-0-387-22757-3.

[41]

J. Shen, T. Tang and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41, Springer-Verlag, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[42]

I. M. Sokolov, M. G. W. Schmidt and F. Sagués, Reaction-subdiffusion equations, Phys. Rev. E, 73 (2006), 031102. doi: 10.1103/PhysRevE.73.031102.

[43]

H. M. Srivastava and R. G. Buschman, Convolution Integral Equations with Special Function Kernels, John Wiley & Sons, New York, 1977.

[44]

L. Turgeman, S. Carmi and E. Barkai, Fractional Feynman-Kac equation for non-Brownian functionals, Phys. Rev. Lett., 103 (2009), 190201. doi: 10.1103/PhysRevLett.103.190201.

[45]

N. Tatar, The decay rate for a fractional differential equation, J. Math. Anal. Appl., 295 (2004), 303-314. doi: 10.1016/j.jmaa.2004.01.047.

[46]

Ž. Tomovski, Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator, Nonl. Anal., 75 (2012), 3364-3384. doi: 10.1016/j.na.2011.12.034.

[47] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[48]

L. J. Zhao and W. H. Deng, Jacobian-predictor-corrector approach for fractional differential equations, Adv. Comput. Math., 40 (2014), 137-165. doi: 10.1007/s10444-013-9302-7.

[49]

M. Zayernouri, M. Ainsworth and G. Karniadakis, Tempered fractional Sturm-Liouville eigenproblems, SIAM J. Sci. Comput., 37 (2015), A1777–A1800. doi: 10.1137/140985536.

[50]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069.

show all references

References:
[1]

A. Z. Al-Abedeen and H. L. Arora, A global existence and uniqueness theorem for ordinary differential equation of generalized order, Canad. Math. Bull., 21 (1978), 271-276. doi: 10.4153/CMB-1978-047-1.

[2]

N. Atanasova and I. Brayanov, Computation of some unsteady flows over porous semi-infinite flat surface, in Large-Scale Scientific Computing, Lecture Notes in Computer Science, 3743, Springer, Berlin, 2006,621–628. doi: 10.1007/11666806_71.

[3]

M. BenchohraJ. HendersonS. K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 338 (2008), 1340-1350. doi: 10.1016/j.jmaa.2007.06.021.

[4]

B. Baeumera and M. M. Meerschaert, Tempered stable Lévy motion and transient super-diffusion, J. Comput. Appl. Math., 233 (2010), 2438-2448. doi: 10.1016/j.cam.2009.10.027.

[5]

R. G. Buschman, Decomposition of an integral operator by use of Mikusinski calculus, SIAM J. Math. Anal., 3 (1972), 83-85. doi: 10.1137/0503010.

[6]

H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, North-Holland Publishing Co., Amsterdam, 1986.

[7] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.
[8]

M. H. Chen and W. H. Deng, Discretized fractional substantial calculus, ESAIM: Math. Mod. Numer. Anal., 49 (2015), 373-394.

[9]

Á. Cartea and D. del-Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Phys. A, 374 (2007), 749-763. doi: 10.1063/1.2336114.

[10]

Á. Cartea and D. del-Castillo-Negrete, Fluid limit of the continuous-time random walk with general Lévy jump distribution functions, Phys. Rev. E, 76 (2007), 041105. doi: 10.1103/PhysRevE.76.041105.

[11]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

[12]

W. H. Deng, Numerical algorithm for the time fractional Fokker-Planck equation, J. Comp. Phys., 227 (2007), 1510-1522. doi: 10.1016/j.jcp.2007.09.015.

[13]

W. H. Deng, Smoothness and stability of the solutions for nonlinear fractional differential equations, Nonl. Anal., 72 (2010), 1768-1777. doi: 10.1016/j.na.2009.09.018.

[14]

K. Diethelm and N. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (1902), 229-248. doi: 10.1006/jmaa.2000.7194.

[15]

K. Diethelm and N. J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154 (2004), 621-640. doi: 10.1016/S0096-3003(03)00739-2.

[16]

K. DiethelmN. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential eqations, Nonlinear Dynam., 29 (2002), 3-22. doi: 10.1023/A:1016592219341.

[17]

A. M. A. El-Sayed, Fractional differential equations, Kyungpook Math. J., 28 (1988), 119-122.

[18]

R. Friedrich, F. Jenko, A. Baule and S. Eule, Anomalous diffusion of inertial, weakly damped particles, Phys. Rev. Lett., 96 (2006), 230601. doi: 10.1103/PhysRevLett.96.230601.

[19]

J. Gajda and M. Magdziarz, Fractional Fokker-Planck equation with tempered $\alpha$-stable waiting times: Langevin picture and computer simulation, Phys. Rev. E, 82 (2010), 011117. doi: 10.1103/PhysRevE.82.011117.

[20]

A. Hanyga, Wave propagation in media with singular memory, Math. Comput. Model., 34 (2001), 1399-1421. doi: 10.1016/S0895-7177(01)00137-6.

[21]

B. I. Henry, T. A. M. Langlands and S. L. Wearne, Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations, Phys. Rev. E, 74 (2006), 031116. doi: 10.1103/PhysRevE.74.031116.

[22]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747.

[23]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, North Holland, 2006.

[24]

A. A. Kilbas and J. J. Trujillo, Differential equation of fractional order: methods, results and problems-Ⅰ, Appl. Anal., 78 (2001), 153-192. doi: 10.1080/00036810108840931.

[25]

V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009.

[26]

C. P. Li and W. H. Deng, Remarks on fractional derivatives, Appl. Math. Comput., 187 (2007), 777-784. doi: 10.1016/j.amc.2006.08.163.

[27]

Y. J. Li and Y. J.Wang, Uniform asymptotic stability of solutions of fractional functional differential equations, Abst. Appl. Anal., 2013 (2013), 532589. doi: 10.1155/2013/532589.

[28]

C. Li and W. H. Deng, High order schemes for the tempered fractional diffusion equations, Adv. Comput. Math., 42 (2016), 543-572. doi: 10.1007/s10444-015-9434-z.

[29]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[30]

M. M. Meerschaert, Y. Zhang and B. Baeumer, Tempered anomalous diffusion in heterogeneous systems, Geophys. Res. Lett., 35 (2008), L17403. doi: 10.1029/2008GL034899.

[31]

M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, 43, Walter de Gruyter & Co., Berlin, 2012.

[32]

M. M. MeerschaertF. SabzikarM. S. Phanikumar and A. Zeleke, Tempered fractional time series model for turbulence in geophysical flows, J. Stat. Mech. Theory Exp., 14 (2014), 1742-5468. doi: 10.1088/1742-5468/2014/09/P09023.

[33] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[34]

A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Springer-Verlag, New York, 2000.

[35]

E. Pitcher and W. E. Sewell, Existence theorems for solutions of differential equations of non-integer order, Bull. Amer. Math. Soc., 44 (1938), 100-107. doi: 10.1090/S0002-9904-1938-06695-5.

[36] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[37]

M. G. W. Schmidt, F. Sagués and I. M. Sokolov, Mesoscopic description of reactions for anomalous diffusion: A case study, J. Phys.: Condens. Matter, 19 (2007), 065118. doi: 10.1088/0953-8984/19/6/065118.

[38]

S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, London, 1993.

[39]

F. SabzikarM. M. Meerschaert and J. H. Chen, Tempered fractional calculus, J. Comput. Phys., 293 (2015), 14-28. doi: 10.1016/j.jcp.2014.04.024.

[40]

J. L. Schiff, The Laplace Transform: Theory and Applications, Springer, New York, 1991. doi: 10.1007/978-0-387-22757-3.

[41]

J. Shen, T. Tang and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41, Springer-Verlag, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[42]

I. M. Sokolov, M. G. W. Schmidt and F. Sagués, Reaction-subdiffusion equations, Phys. Rev. E, 73 (2006), 031102. doi: 10.1103/PhysRevE.73.031102.

[43]

H. M. Srivastava and R. G. Buschman, Convolution Integral Equations with Special Function Kernels, John Wiley & Sons, New York, 1977.

[44]

L. Turgeman, S. Carmi and E. Barkai, Fractional Feynman-Kac equation for non-Brownian functionals, Phys. Rev. Lett., 103 (2009), 190201. doi: 10.1103/PhysRevLett.103.190201.

[45]

N. Tatar, The decay rate for a fractional differential equation, J. Math. Anal. Appl., 295 (2004), 303-314. doi: 10.1016/j.jmaa.2004.01.047.

[46]

Ž. Tomovski, Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator, Nonl. Anal., 75 (2012), 3364-3384. doi: 10.1016/j.na.2011.12.034.

[47] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[48]

L. J. Zhao and W. H. Deng, Jacobian-predictor-corrector approach for fractional differential equations, Adv. Comput. Math., 40 (2014), 137-165. doi: 10.1007/s10444-013-9302-7.

[49]

M. Zayernouri, M. Ainsworth and G. Karniadakis, Tempered fractional Sturm-Liouville eigenproblems, SIAM J. Sci. Comput., 37 (2015), A1777–A1800. doi: 10.1137/140985536.

[50]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069.

Table 1.  Maximum errors and convergence orders of Example 1 solved by the scheme (56)-(57) with $ T = 1,N = 20,N_I = 7 $, and $ \alpha = 0.5 $
$ \lambda=0 $ $ \lambda=2 $ $ \lambda=6 $
$ \tau $ error order error order error order
1/10 1.5207e-004 2.3516e-005 1.4300e-006
1/20 4.6202e-007 8.3626 1.4040e-007 7.3879 3.3507e-008 5.4154
1/40 1.6877e-009 8.0967 6.3106e-010 7.7976 2.7846e-010 6.9109
1/80 8.1135e-012 7.7005 2.5491e-012 7.9517 1.4371e-012 7.5982
1/160 3.5305e-014 7.8443 1.2794e-014 7.6383 7.0913e-015 7.6629
$ \lambda=0 $ $ \lambda=2 $ $ \lambda=6 $
$ \tau $ error order error order error order
1/10 1.5207e-004 2.3516e-005 1.4300e-006
1/20 4.6202e-007 8.3626 1.4040e-007 7.3879 3.3507e-008 5.4154
1/40 1.6877e-009 8.0967 6.3106e-010 7.7976 2.7846e-010 6.9109
1/80 8.1135e-012 7.7005 2.5491e-012 7.9517 1.4371e-012 7.5982
1/160 3.5305e-014 7.8443 1.2794e-014 7.6383 7.0913e-015 7.6629
Table 2.  Maximum errors and convergence orders of Example 1 solved by the scheme (56)-(57) with $T = 1,N = 20,N_I = 6$, and $\alpha = 1.0$
$\lambda=0$ $\lambda=2$ $\lambda=6$
$\tau$ error order error order error order
1/10 8.1108e-005 1.2528e-005 1.1365e-006
1/20 7.8788e-007 6.6857 1.5673e-007 6.3207 2.3299e-008 5.6082
1/40 1.2817e-008 5.9418 2.1909e-009 6.1606 3.2657e-010 6.1567
1/80 2.2418e-010 5.8373 3.4124e-011 6.0046 4.4768e-012 6.1888
1/160 3.6193e-012 5.9528 5.3461e-013 5.9962 6.7955e-014 6.0417
$\lambda=0$ $\lambda=2$ $\lambda=6$
$\tau$ error order error order error order
1/10 8.1108e-005 1.2528e-005 1.1365e-006
1/20 7.8788e-007 6.6857 1.5673e-007 6.3207 2.3299e-008 5.6082
1/40 1.2817e-008 5.9418 2.1909e-009 6.1606 3.2657e-010 6.1567
1/80 2.2418e-010 5.8373 3.4124e-011 6.0046 4.4768e-012 6.1888
1/160 3.6193e-012 5.9528 5.3461e-013 5.9962 6.7955e-014 6.0417
Table 3.  Maximum errors and convergence orders of Example 1 solved by the scheme (56)-(57) with $ T = 1,N = 20,N_I = 6 $, and $ \alpha = 1.5 $
$ \lambda=0 $ $ \lambda=2 $ $ \lambda=6 $
$ \tau $ error order error order error order
1/10 6.6386e-005 9.6009e-006 8.5068e-007
1/20 9.2847e-007 6.1599 1.4297e-007 6.0694 1.9943e-008 5.4147
1/40 1.5767e-008 5.8799 2.1338e-009 6.0661 3.0437e-010 6.0339
1/80 2.3505e-010 6.0678 3.5138e-011 5.9242 3.8203e-012 6.3159
1/160 3.8498e-012 5.9320 5.3434e-013 6.0391 6.7433e-014 5.8241
$ \lambda=0 $ $ \lambda=2 $ $ \lambda=6 $
$ \tau $ error order error order error order
1/10 6.6386e-005 9.6009e-006 8.5068e-007
1/20 9.2847e-007 6.1599 1.4297e-007 6.0694 1.9943e-008 5.4147
1/40 1.5767e-008 5.8799 2.1338e-009 6.0661 3.0437e-010 6.0339
1/80 2.3505e-010 6.0678 3.5138e-011 5.9242 3.8203e-012 6.3159
1/160 3.8498e-012 5.9320 5.3434e-013 6.0391 6.7433e-014 5.8241
Table 4.  Maximum errors and convergence orders of Example 2 solved by the scheme (66) with $ T = 1.1, N = 26, \tilde{N} = 40, N_I = 2, T_0 = 0.1,\mu = 1 $, and $ \lambda = 5 $
$ \alpha=0.2 $ $ \alpha=0.9 $ $ \alpha=1.8 $
$ \tau $ error order error order error order
1/20 5.4805e-004 1.9043e-005 2.1461e-006
1/40 1.8749e-004 1.5475 4.3478e-006 2.1309 5.6685e-007 1.9207
1/80 5.0838e-005 1.8828 1.0851e-006 2.0025 1.5416e-007 1.8786
1/160 1.3492e-005 1.9138 3.1549e-007 1.7821 4.0386e-008 1.9324
$ \alpha=0.2 $ $ \alpha=0.9 $ $ \alpha=1.8 $
$ \tau $ error order error order error order
1/20 5.4805e-004 1.9043e-005 2.1461e-006
1/40 1.8749e-004 1.5475 4.3478e-006 2.1309 5.6685e-007 1.9207
1/80 5.0838e-005 1.8828 1.0851e-006 2.0025 1.5416e-007 1.8786
1/160 1.3492e-005 1.9138 3.1549e-007 1.7821 4.0386e-008 1.9324
Table 5.  Maximum errors and convergence orders of Example 2 solved by the scheme (66) with $ T = 1.1, N = 26, \tilde{N} = 40, N_I = 2, T_0 = 0.1,\mu = 1 $, and $ \lambda = 10 $
$ \alpha=0.2 $ $ \alpha=0.9 $ $ \alpha=1.8 $
$ \tau $ error order error order error order
1/20 2.0162e-004 7.0054e-006 4.0825e-007
1/40 8.8563e-005 1.1868 1.6897e-006 2.0517 1.0286e-007 1.9887
1/80 2.7211e-005 1.7025 4.1757e-007 2.0167 2.7730e-008 1.8912
1/160 7.4508e-006 1.8688 1.1169e-007 1.9025 7.3208e-009 1.9214
$ \alpha=0.2 $ $ \alpha=0.9 $ $ \alpha=1.8 $
$ \tau $ error order error order error order
1/20 2.0162e-004 7.0054e-006 4.0825e-007
1/40 8.8563e-005 1.1868 1.6897e-006 2.0517 1.0286e-007 1.9887
1/80 2.7211e-005 1.7025 4.1757e-007 2.0167 2.7730e-008 1.8912
1/160 7.4508e-006 1.8688 1.1169e-007 1.9025 7.3208e-009 1.9214
Table 6.  Maximum errors and convergence orders of Example 3 solved by the scheme (56)-(57) with $ T = 1,N = 20,N_I = 5 $, and $ \alpha = 0.4 $
$ \lambda=0 $ $ \lambda=3 $ $ \lambda=5 $
$ \tau $ error order error order error order
1/10 2.4208e-004 7.4482e-007 2.1996e-007
1/20 4.9371e-006 5.6157 6.8759e-008 3.4373 3.2736e-008 2.7483
1/40 1.0390e-007 5.5704 3.1715e-009 4.4383 2.1801e-009 3.9084
1/80 2.2895e-009 5.5041 1.1961e-010 4.7288 9.9159e-011 4.4585
1/160 5.3248e-011 5.4261 4.1052e-012 4.8647 3.7378e-012 4.7295
$ \lambda=0 $ $ \lambda=3 $ $ \lambda=5 $
$ \tau $ error order error order error order
1/10 2.4208e-004 7.4482e-007 2.1996e-007
1/20 4.9371e-006 5.6157 6.8759e-008 3.4373 3.2736e-008 2.7483
1/40 1.0390e-007 5.5704 3.1715e-009 4.4383 2.1801e-009 3.9084
1/80 2.2895e-009 5.5041 1.1961e-010 4.7288 9.9159e-011 4.4585
1/160 5.3248e-011 5.4261 4.1052e-012 4.8647 3.7378e-012 4.7295
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