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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.

65R20 Numerical methods for integral equations
45K05 Integro-partial differential equations
Full Text: DOI
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