Numerical solution of fractional bioheat equation by quadratic spline collocation method
Authors
Yanmei Qin
 Key Laboratory of Numerical Simulation of Sichuan Province/College of Mathematics and Information Science, Neijiang Normal University, Neijiang, 641112, P. R. China.
Kaiteng Wu
 Key Laboratory of Numerical Simulation of Sichuan Province/College of Mathematics and Information Science, Neijiang Normal University, Neijiang, 641112, P. R. China.
Abstract
Based on the quadratic spline function, a quadratic spline collocation method is presented for the time
fractional bioheat equation governing the process of heat transfer in tissues during the thermal therapy. The
corresponding linear system is given. The stability and convergence are analyzed. Some numerical examples
are given to demonstrate the efficiency of this method.
Keywords
 Quadratic spline collocation method
 fractional bioheat equation
 hyperthermia.
MSC
 65M70
 35K05
 92C50
 35R11
 35Q92
References

[1]
D. Baleanu, J. H. Asad, I. Petras, Numerical solution of the fractional EulerLagrange's equations of a thin elastica model, Nonlinear Dynam., 81 (2015), 97102

[2]
B. Bialecki, G. Fairweather, Orthogonal spline collocation methods for partial differential equations, J. Comput. Appl. Math., 128 (2001), 5582

[3]
B. Bialecki, G. Fairweather, A. Karageorghis, Q. N. Nguyen, Optimal superconvergent one step quadratic spline collocation methods , BIT, 48 (2008), 449472

[4]
N. Bouzid, M. Merad, D. Baleanu, On fractional DuffinKemmePetiau equation, FewBody Syst., 57 (2016), 265273

[5]
J. C. Chato, Reflections on the History of Heat and Mass Transfer in Bioengineering , J. Biomech. Eng., 103 (1981), 97101

[6]
C. C. Christara, Quadratic spline collocation methods for elliptic partial differential equations, BIT, 34 (1994), 3361

[7]
R. S. Damor, S. Kumar, A. K. Shukla, Numerical simulation of fractional bioheat equation in hyperthermia treatment, J. Mech. Med. Biol., 14 (2014), 15 pages

[8]
G. Fairweather, A. Karageorghis, J. Maack, Compact optimal quadratic spline collocation methods for the Helmholtz equation, J. Comput. Phys., 230 (2011), 28802895

[9]
P. K. Gupta, J. Singh, K. N. Rai , Numerical simulation for heat transfer in tissues during thermal therapy, J. Thermal Biol., 35 (2010), 295301

[10]
M. S. Hashemi, D. Baleanu, On the time fractional generalized Fisher equation: group similarities and analytical solutions, Commun. Theor. Phys., 65 (2016), 1116

[11]
J. Hristov , Approximate solutions to timefractional models by integralbalance approach, Fractional dynamics, De Gruyter Open, Berlin (2015)

[12]
J. Hristov, A unified nonlinear fractional equation of the diffusioncontrolled surfactant adsorption: Reappraisal and new solution of the WardTordai problem, J. King Saud Univ. Sci., 28 (2016), 713

[13]
M. Lakestani, M. Dehghan, Collocation and finite differencecollocation methods for the solution of nonlinear KleinGordon equation, Comput. Phys. Comm., 181 (2010), 13921401

[14]
E. K. Lenzi, D. S. Vieira, M. K. Lenzi, G. Goncalves, D. P. Leitoles, Solutions for a fractional diffusion equation with radial symmetry and integrodifferential boundary conditions, Thermal Sci., 19 (2015), 16

[15]
F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of the difference methods for the spacetime fractional advectiondiffusion equation, Appl. Math. Comput., 191 (2007), 1220

[16]
W. H. Luo, T. Z. Huang, G. C. Wu, X. M. Gu, Quadratic spline collocation method for the time fractional subdiffusion equation, Appl. Math. Comput., 276 (2016), 252265

[17]
W. H. Luo, G. C. Wu, Quadratic spline collocation method and ecient preconditioner for the Helmholtz equation with Robbins boundary condition, J. Comput. Complex. Appl., 2 (2016), 2437

[18]
S. Mobayen, D. Baleanu, Stability analysis and controller design for the performance improvement of disturbed nonlinear systems using adaptive global sliding mode control approach, Nonlinear Dynam., 83 (2016), 15571565

[19]
E. H. Ooi, W. T. Ang, A boundary element model of the human eye undergoing laser thermokeratoplasty, Comput. Biol. Med., 38 (2008), 727737

[20]
I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)

[21]
R. K. Saxena, A. M. Mathai, H. J. Haubold, On generalized fractional kinetic equations, Phys. A, 344 (2004), 657664

[22]
R. K. Saxena, A. M. Mathai, H. J. Haubold, An alternative method for solving a certain class of fractional kinetic equations, Astrophys. Space Sci. Proc., Springer, Heidelberg (2010)

[23]
J. Singh, P. K. Gupta, K. N. Rai , Solution of fractional bioheat equations by finite difference method and HPM, Math. Comput. Modelling, 54 (2011), 23162325

[24]
H. G. Sun, W. Chen, H. Wei, Y. Q. Chen, A comparative study of constantorder and variableorder fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top., 193, 185192 (2011)

[25]
M. Tunç, Ü. Çamdali, C. Parmaksizoglu, S. Çikrikçi, The bioheat transfer equation and its applications in hyperthermia treatments, Eng. Comput., 23 (2006), 451463

[26]
G. C. Wu, D. Baleanu, Z. G. Deng, S. D. Zeng, Lattice fractional diffusion equation in terms of a RieszCaputo difference, Phys. A, 438 (2015), 335339

[27]
G. C. Wu, D. Baleanu, S. D. Zeng, Z. G. Deng, Discrete fractional diffusion equation, Nonlinear Dynam., 80 (2015), 281286

[28]
L. XiaoZhou, Z. Yi, Z. Fei, G. XiuFen, Estimation of temperature elevation generated by ultrasonic irradiation in biological tissues using the thermal wave method, Chinese Phys. B, 22 (2013), 024301

[29]
Q. Yang, F. Liu, I. Turner, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Model., 34 (2010), 200218

[30]
K. Yue, X. Zhang, F. Yu, An analytic solution of onedimensional steadystate Pennes bioheat transfer equation in cylindrical coordinates, J. Thermal Sci., 13 (2004), 255258

[31]
Y. N. Zhang, Z. Z. Sun, X. Zhao, Compact alternating direction implicit scheme for the twodimensional fractional diffusionwave equation, SIAM J. Numer. Anal., 50 (2012), 15351555

[32]
X. Zhao, Z. Z. Sun, Z. P. Hao , A fourthorder compact ADI scheme for twodimensional nonlinear space fractional Schrödinger equation, SIAM J. Sci. Comput., 36 (2014), 28652886

[33]
X. Zhao, Q. Xu, Efficient numerical schemes for fractional subdiffusion equation with the spatially variable coefficient, Appl. Math. Model., 38 (2014), 38483859