Numerical convergence of a one step approximation of an integro-differential equation. (English) Zbl 1254.65130

Bhowmik, Samir Kumar

Appl. Numer. Math. 62, No. 12, 1880-1892 (2012).

The authors discuss the numerical convergence of a one step approximation of an integro-differential equation. They present a space time discretization, stability of the scheme, analysis of the accuracy of the full discrete scheme and consider both smooth and non smooth initial functions. Later, the auhors present accuracy of the semi-discrete time dependent scheme. Numerical experiments are given.

Reviewer: Seenith Sivasundaram (Daytona Beach)

MSC:

65R20

Numerical methods for integral equations

Keywords:

stability; convergence; smoothness; full discrete problem; infinite domain; periodic domain

Software:

Matlab

References:

[1]	Atkinson, Kendall; Han, Weimin, Theoretical Numerical Analysis (2001), Springer · Zbl 0966.65001
[2]	Bates, Peter W.; Chmaj, Adam, A discrete convolution model for phase transitions, Archive for Rational Mechanics and Analysis, 150, 4, 281-305 (1999) · Zbl 0956.74037
[3]	Bates, Peter W.; Fife, Paul C.; Ren, Xiaofeng; Wang, Xuefeng, Travelling waves in a convolution model for phase transitions, Archive for Rational Mechanics and Analysis, 138, 2, 105-136 (July 1997)
[5]	Bhowmik, Samir Kumar, Numerical computation of a nonlocal double obstacle problem, International Journal of Open Problems in Computer Science and Mathematics, 2, 1, 19-36 (2009) · Zbl 1207.65155
[6]	Bhowmik, Samir Kumar, Stability and convergence analysis of a one step approximation of a linear partial integro-differential equation, Numerical Methods for Partial Differential Equations, 27, 5, 1179-1200 (September 2011)
[7]	Bhowmik, Samir Kumar, Stable numerical schemes for a partly convolutional partial integro-differential equation, Applied Mathematics and Computations, 217, 8, 4217-4226 (2010) · Zbl 1215.65193
[8]	Bhowmik, Samir Kumar, Numerical approximation of a convolution model of dot theta-neuron networks, Applied Numerical Mathematics, 61, 581-592 (2011) · Zbl 1208.65180
[9]	Chen, C.-K.; Fife, Paul C., Nonlocal models of phase transitions in solids, Advances in Mathematical Sciences and Applications, 10, 821-849 (2000) · Zbl 0977.74044
[10]	Chen, Fengxin, Uniform stability of multidimensional travelling waves for the nonlocal Allen-Cahn equation, Fifth Mississippi State Conference on Differential Equations and Computational Simulations. Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, 10, 109-113 (2003) · Zbl 1020.35013
[11]	Duffy, Daniel J., Finite Difference Methods for Financial Engineering: A Partial Differential Equation Approach (2006), Wiley Finance · Zbl 1141.91002
[12]	Duncan, Dugald B.; Grinfeld, M.; Stoleriu, I., Coarsening in an integro-differential model of phase transitions, European Journal of Applied Mathematics, 11, 511-523 (2000) · Zbl 0973.65131
[13]	Laing, Carlo R.; Troy, William C., PDE methods for nonlocal models, SIAM Journal on Applied Dynamical Systems, 2, 3, 487-516 (2003) · Zbl 1088.34011
[14]	Maleknejad, K., A comparison of Fourier extrapolation methods for numerical solution of deconvolution, New York Journal of Mathematics, 183, 533-538 (2006) · Zbl 1106.65335
[15]	Strikwerda, J. C., Finite Difference Schemes and Partial Differential Equations (1989), Wadsworth and Brooks, Cole Advanced Books and Software: Wadsworth and Brooks, Cole Advanced Books and Software Pacific Grove, California · Zbl 0681.65064
[16]	Trefethen, L. N., Spectral Methods in Matlab (2000), SIAM: SIAM Philadelphia