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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[6]

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

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

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

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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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

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

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

[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.  Google Scholar

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

[38]

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

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[50]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

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

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

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

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

[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.  Google Scholar

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

[38]

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

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[50]

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

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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