Numerical Solution of the Cauchy Problem for a Second-Order Integro-Differential Equation
Tóm tắt
In a finite-dimensional Hilbert space, we consider the Cauchy problem for a second-order
integro-differential evolution equation with memory where the integrand is the product of a
difference kernel by a linear operator of the time derivative of the solution. The main difficulties in
finding the approximate value of the solution of such nonlocal problems at a given point in time
are due to the need to work with approximate values of the solution for all previous points in
time. A transformation of the integro-differential equation in question to a system of weakly
coupled local evolution equations is proposed. It is based on the approximation of the difference
kernel by a sum of exponentials. We state a local problem for a weakly coupled system of
equations with additional ordinary differential equations. To solve the corresponding Cauchy
problem, stability estimates of the solution with respect to the initial data and the right-hand side
are given. The main attention is paid to the construction and stability analysis of three-level
difference schemes and their computational implementation.
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