Stable numerical schemes for a partly convolutional partial integro-differential equation. (English) Zbl 1215.65193

The author considers the initial value problem (IVP)
\[ u_t(x,t)=u_{xx}(x,t)+\int_{-\infty}^{\infty}J(x-y)(u(x,t)-u(y,t))dy, \]
with condition \(u(x,0)=u_0(x)\). This problem is reduced in two ways. Both of them correspond to the IVP for the infinite system of ordinary differential equations generated by discretization of space variable \(x\) with a uniform step. The first way of discretization of the time dependent IVP is based on the explicit Euler’s scheme applied to the ODEs, and the second way uses a mixed Euler method. Theoretical results with respect to stability and accuracy of the schemes as well as numerical illustrations are presented.


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


Full Text: DOI


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