Abstract
We consider the initial-boundary value problem for nonlinear partial differential equations, the source of which are population models. Nonlinearity is contained both in the differential operator and in the inhomogeneity function. We construct a nonlinear implicit difference scheme, which requires the use of iterative solution methods on each time layer. Stability and convergence of the proposed numerical method were proved. Numerical experiments have been carried out, both on test examples and on the example of the biological model of the population.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Srivastava, V.K., Kumar, S., et al.: Two-dimensional time fractional-order biological population model and its analytical solution. Egypt J. Basic Appl. Sci. 1, 71–76 (2014)
Samarskii, A.A.: Theory of Difference Schemes. Nauka, Moscow (1989). (in Russian)
Pimenov, V.G.: General linear methods for the numerical solution of functional-differential equations. Differ. Equ. 37(1), 116–127 (2001)
Pimenov, V.G., Lozhnikov, A.B.: Difference schemes for the numerical solution of the heat conduction equation with aftereffect. Proc. Steklov Inst. Math. 275, 137–148 (2011)
Pimenov, V.G.: Difference Methods for Solving Partial Differential Equations with Heredity. Publishing House of the Ural University, Ekaterinburg (2014). (in Russian)
Solodushkin, S.I.: A difference scheme for the numerical solution of an advection equation with aftereffect. Russ. Math. 57, 65–70 (2013)
Solodushkin, S.I., Yumanova, I.F., De Staelen, R.H.: First order partial differential equations with time delay and retardation of a state variable. J. Comput. Appl. Math. 289, 322–330 (2015)
Acknowledgments
We acknowledge the support by the program 02.A03.21.0006 on 27.08.2013.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Gorbova, T.V., Pimenov, V.G., Solodushkin, S.I. (2019). Difference Schemes for the Nonlinear Equations in Partial Derivatives with Heredity. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods. Theory and Applications. FDM 2018. Lecture Notes in Computer Science(), vol 11386. Springer, Cham. https://doi.org/10.1007/978-3-030-11539-5_28
Download citation
DOI: https://doi.org/10.1007/978-3-030-11539-5_28
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-11538-8
Online ISBN: 978-3-030-11539-5
eBook Packages: Computer ScienceComputer Science (R0)