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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
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##### References:
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