The method of lines for parabolic partial integro-differential equations. (English) Zbl 0763.65101

Kauthen, J.-P.

J. Integral Equations Appl. 4, No. 1, 69-81 (1992).

The author discusses a method of lines for nonlinear Volterra partial integro-differential equations of parabolic type. In the first step of discretization, a finite difference method is used in the spatial direction to obtain a system of nonlinear stiff Volterra integro- differential equations in time. Then in the second step, this system is numerically integrated in the temporal variable by the implicit Euler method. The concept of logarithmic matrix norm, earlier introduced in the study of numerical methods for nonlinear stiff ordinary differential equations, plays a key role in deriving realistic error bounds in the convergence analysis of this paper. Finally, some numerical results are cited to illustrate the error bounds and the rate of convergence.

Reviewer: A.K.Pani (Bombay)

MSC:

65R20	Numerical methods for integral equations
45K05	Integro-partial differential equations
45G10	Other nonlinear integral equations

Keywords:

method of lines; nonlinear Volterra partial integro-differential equations of parabolic type; finite difference method; implicit Euler method; logarithmic matrix norm; error bound; convergence; numerical results

References:

[1]	K. Dekker and J.G. Verwer, Stability of Runge-Kutta methods for stiff nonlinear differential equations , CWI Monographs 2 , North-Holland, Amsterdam, New York, Oxford, 1984. · Zbl 0571.65057
[2]	H. Engler, On some parabolic integro-differential equations: Existence and asymptotics of solutions , Equadiff 82, Lect. Notes Math. 1017 , 161-167, Springer, Berlin, 1983. · Zbl 0539.35074
[3]	R. Frank, J. Schneid, and C. Ueberhuber, The concept of \(B\)-convergence , SIAM J. Numer. Anal. 18 (1981), 753-780. JSTOR: · Zbl 0467.65032
[4]	E. Hairer, Ch. Lubich, and S.P. Nørsett, Order of convergence of one-step methods for Volterra integral equations of the second kind , SIAM J. Numer. Anal. 20 (1983), 569-579. JSTOR: · Zbl 0519.65088
[5]	B.G. Pachpatte, On a nonlinear diffusion system arising in reactor dynamics , J. Math. Anal. Appl. 94 (1983), 501-508. · Zbl 0524.35055
[6]	J.G. Verwer and J.M. Sanz-Serna, Convergence of method of lines approximations to partial differential equations , Computing 33 (1984), 297-313. · Zbl 0546.65064
[7]	E.G. Yanik and G. Fairweather, Finite element methods for parabolic and hyperbolic partial integro-differential equations , Nonlinear Anal. 12 (1988), 785-809. · Zbl 0657.65142