×

The method of fractional steps for the numerical solution of a multidimensional heat conduction equation with delay for the case of variable coefficient of heat conductivity. (English) Zbl 1454.65061

Pinelas, Sandra (ed.) et al., Differential and difference equations with applications. Selected papers based on the presentations at the fourth international conference, ICDDEA 2019, Lisbon, Portugal, July 1–5, 2019. Cham: Springer. Springer Proc. Math. Stat. 333, 105-121 (2020).
Summary: Multidimensional parabolic equations with delay effects in the time component for the case of variable coefficient of heat conductivity depending on spatial and temporal variables are considered. The method of fractional steps is constructed for the numerical solution of these equations. The order of approximation error for the constructed method, stability, and order of convergence are investigated. A theorem is obtained on the order of convergence of the method of fractional steps, which uses the methods from the general theory of difference schemes and the technique of the investigation of difference schemes for solving functional differential equations. Results of calculating test example with variable concentrated and distributed time delay are presented.
For the entire collection see [Zbl 1445.34003].

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35R07 PDEs on time scales
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Castro, M.A., Rodriguez, F., Cabrera, J., Martin, J.A.: Difference schemes for time-dependent heat conduction models with delay. Int. J. Comput. Math. 91(1), 53-61 (2014) · Zbl 1291.65259 · doi:10.1080/00207160.2013.779371
[2] Garcia, P., Castro, M.A., Martin, J.A., Sirvent, A.: Numerical solutions of diffusion mathematical models with delay. Math. Comput. Model. 50(5-6), 860-868 (2013) · Zbl 1185.65148
[3] Kropielnicka, K.: Convergence of implicit difference methods for parabolic functional differential equations. Int. J. Mat. Anal. 1(6), 257-277 (2007) · Zbl 1140.65061
[4] Lekomtsev, A.V., Pimenov, V.G.: Convergence of the alternating direction methods for the numerical solution of a heat conduction equation with delay. Proc. Steklov Inst. Math. 272(1), 101-118 (2011) · Zbl 1239.65058 · doi:10.1134/S0081543811020088
[5] Lekomtsev, A.V., Pimenov, V.G.: Convergence of the scheme with weights for the numericalsolution of a heat conduction equation with delay for the case of variable coefficient of heatconductivity. Appl. Math. Comput. 256, 83-93 (2015) · Zbl 1338.80017
[6] Pimenov, V.G.: General linear methods for numerical solving functional-differential equations. Differ. Equ. 37(1), 116-127 (2001) · Zbl 1002.65079 · doi:10.1023/A:1019232718078
[7] Pimenov, V.G., Lozhnikov, A.B.: Difference schemes for the numerical solution of the heat conduction equation with aftereffect. Proc. Steklov Inst. Math. 275(S1), 137-148 (2011) · Zbl 1301.65094
[8] Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker, New York (2001) · Zbl 0971.65076
[9] Samarskii, A.A., Gulin, A.V.: Stability of Difference Schemes. URSS, Moscow (2009). [in Russian] · Zbl 0179.20202
[10] Skeel, R.D.: Analysis of fixed-stepsize methods. SIAM J. Numer. Anal. 13(5), 664-685 (1976) · Zbl 0373.65033
[11] Tavernini, L.: Finite difference approximations for a class of semilinear volterra evolution problems. SIAM J. Numer. Anal. 14(5), 931-949 (1977) · Zbl 0374.65047
[12] Van der Houwen, P.J., Sommeijer, B.P., Baker, C.T.H.: On the stability of predictor-corrector methods for parabolic equations with delay. IMA J. Numer. Anal. 6(1), 1-23 (1986) · Zbl 0623.65094
[13] Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996) · Zbl 0870.35116
[14] Zhang, B., Zhou, Y.: Qualitative Analysis of Delay Partial Difference Equations. Hindawi Publishing Corporation, New York (2007) · Zbl 1153.35078
[15] Zubik-Kowal, B.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.