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The numerical solution of parabolic Volterra integro-differential equations on unbounded spatial domains. (English) Zbl 1078.65126
The authors show how to use the artificial boundary method for the numerical solution of parabolic Volterra integro-partial differential equations of the type \[ {\partial u\over\partial t} + \int_0^t k(x,t-\tau)u(x,\tau)\, d\tau \, = \, \Delta u + f(x,t), \] where \(x\in {\mathbb R}\), \(t\in [0,T]\), \(u| _{t=0}=u_0(x)\) with \(u\to 0\) as \(| x| \to\infty\). The functions \(f\) and \(u_0\) are assumed to be continuous with support in \([0,d]\), and the kernel \(k\) satisfies \(k(x,t)=k_0(t)\) for \(x\notin (0,d)\), where \(k_0\) is continuous or weakly singular. They first derive fully transparent artificial boundary conditions by dividing the spatial-temporal domain into appropriate subdomains. These allow to reduce the original problem to one defined on bounded spatial-temporal domains.
They then study two classes of examples for the convolution kernel which appears in the reduced problem, as well as the concrete example \(k_0(t)\equiv 1\) for the memory kernel. After explaining the connection to previous work on parabolic partial differential equations by H. Han and Z. Huang [ibid. 43, 889–900 (2002; Zbl 0999.65086); Comput. Math. Appl. 44, 655–666 (2002; Zbl 1030.35092)], they conclude with giving two numerical examples.

MSC:
65R20 Numerical methods for integral equations
45K05 Integro-partial differential equations
Software:
RODAS
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References:
[1] Brunner, H.; van der Houwen, P.J., The numerical solution of Volterra equations, CWI monographs, vol. 3, (1986), North-Holland Amsterdam · Zbl 0611.65092
[2] Chen, C.; Shih, T., Finite element methods for integrodifferential equations, Series on applied math., vol. 9, (1998), World Scientific Singapore · Zbl 0909.65142
[3] Hairer, E.; Wanner, G., Solving ordinary differential equations II: stiff and differential-algebraic problems, (1996), Springer-Verlag Berlin · Zbl 0859.65067
[4] Han, H.; Huang, Zh., A class of artificial boundary conditions for heat equation in unbounded domains, Comput. math. appl., 43, 889-900, (2002) · Zbl 0999.65086
[5] Han, H.; Huang, Zh., Exact and approximating boundary conditions for the parabolic problems on unbounded domains, Comput. math. appl., 44, 655-666, (2002) · Zbl 1030.35092
[6] Han, H.; Wu, X., Approximation of infinite boundary condition and its application to finite element methods, J. comput. math., 3, 179-192, (1985) · Zbl 0579.65111
[7] Han, H.; Wu, X., The mixed element method for Stokes equations on unbounded domains, J. systems sci. math. sci., 5, 121-132, (1985) · Zbl 0614.76029
[8] Han, H.; Wu, X., The approximation of the exact boundary condition at an artificial boundary for linear elastic equations and its application, Math. comp., 59, 21-27, (1992)
[9] Henrici, P., Applied and computational complex analysis, vol. 1, (1974), Wiley New York · Zbl 0313.30001
[10] Rosenbrock, H.H., Some general implicit processes for the numerical solution of differential equations, Comput. J., 5, 329-331, (1963) · Zbl 0112.07805
[11] Souplet, Ph., Blow-up in nonlocal reaction-diffusion equations, SIAM J. math. anal., 29, 1301-1334, (1998) · Zbl 0909.35073
[12] Yanik, E.G.; Fairweather, G., Finite element methods for parabolic partial integro-differential equations, Nonlinear anal., 12, 785-809, (1988) · Zbl 0657.65142
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