# zbMATH — the first resource for mathematics

Computational methods for delay parabolic and time-fractional partial differential equations. (English) Zbl 1204.65114
Summary: This article is concerned with $$\vartheta$$-methods for delay parabolic partial differential equations. The methodology is extended to time-fractional-order parabolic partial differential equations in the sense of Caputo. The fully implicit scheme preserves delay-independent asymptotic stability and the solution continuously depends on the time-fractional order. Several numerical examples of interest are included to demonstrate the effectiveness of the method.

##### MSC:
 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 35R10 Functional partial differential equations 35R11 Fractional partial differential equations 35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
##### Software:
 [1] Cohen, Spatial structures in predator-prey communities with hereditary effects and diffusion, Math Biosci 44 pp 167– (1979) · Zbl 0399.92016 [2] Murray, Spatial structures in predator-prey comunities-a nonlinear time delay diffusional model, Math Biosci 30 pp 73– (1976) · Zbl 0335.92002 [3] Busenberg, Interaction of spatial diffusion and delays in models of genetic control by repression, J Math Biol 22 pp 313– (1985) · Zbl 0593.92010 [4] Mahaffy, Models of genetic control by repression with time delays and spatial effects, J Math Biol 20 pp 39– (1984) · Zbl 0577.92010 [5] Gyllenberg, An abstract delay-differential equation modelling size dependent cell growth and division, SIAM J Math Anal 18 pp 74– (1987) · Zbl 0634.34064 [6] Rey, Multistability and boundary layer development in a transport equation with delayed arguments, Can Appl Math Quar 1 pp 1– (1993) · Zbl 0783.92028 [7] Wang, Optimal control of parabolic systems with boundary conditions involving time delays, SIAM J Control 13 pp 274– (1975) · Zbl 0301.49009 [8] Kilbas, Theory and applications of fractional differential equations (2006) · Zbl 1138.26300 [9] A. Le Méhauté, J. A. T. Machado, J. C. Trigeassou, and J. Sabatier, Fractional differentiation and its applications, Proceedings of the 1st IFAC Workshop on Fractional Differentiation and Its Applications (FDA 04), Vol. 2004-1, ENSEIRB, Bordeaux, 2004, pp. 353-358. [10] Podlubny, Fractional Differential Equations (1999) [11] Bellen, Numerical methods for delay differential equations (2003) [12] Guglielmi, Open issues in devising software for numerical solution of implicit delay differential equations, J Comput Appl Math 185 pp 261– (2006) · Zbl 1079.65076 [13] N. Guglielmi and E. Hairer, Computing breaking points of implicit delay differential equations, Proceedings of 5th IFAC Workshop on Time-Delay Systems, Leuven, Belgium, 2004. · Zbl 1160.65041 [14] Guglielmi, Implementing Radau IIA methods for stiff delay differential equations, Computing 67 pp 1– (2001) · Zbl 0986.65069 [15] F. A. Rihan, Numerical treatment of delay differential equations in bioscience, Ph.D. Thesis, University of Manchester, 2000. [16] Samko, Fractional integrals and derivatives (1993) [17] Davidson, The effects of temporal delays in a model for a food-limited, diffusing population, J Math Anal Appl 261 pp 633– (2001) · Zbl 0992.35047 [18] Gourley, Nonlocality of reaction-diffusion equations included by delay: biological modeling and nonlinear dynamics, J Math Sci 124 pp 5119– (2004) [19] Sardar, Dynamics of constant and variable stepsize methods for a nonlinear population model with delay, Appl Numer Math 24 pp 425– (1997) · Zbl 0899.92027 [20] Wu, Theory and applications of partial functional differential equations (1996) · Zbl 0870.35116 · doi:10.1007/978-1-4612-4050-1 [21] Sanz-Serna, Some aspects of the boundary locus method, BIT 20 pp 97– (1980) · Zbl 0426.65036 [22] Bellman, On the computational solution of a class of functional differential equations, J Math Anal Appl 12 pp 495– (1965) · Zbl 0138.32103 [23] Liu, The stability of the $$\theta$$-methods in the numerical solution of delay differential equations, IMA J Numer Anal 10 pp 31– (1990) [24] Torelli, Stability of numerical methods for delay differential equations, J Comput Appl Math 25 pp 15– (1989) · Zbl 0664.65073