Class Of Hyperbolic Integro-differential Equations Research Articles

Optimal error estimates of mixed FEMs for second order hyperbolic integro-differential equations with minimal smoothness on initial data

In this article, mixed finite element methods are discussed for a class of hyperbolic integro-differential equations (HIDEs). Based on a modification of the nonstandard energy formulation of Baker, both semidiscrete and completely discrete implicit schemes for an extended mixed method are analyzed and optimal L∞(L2)-error estimates are derived under minimal smoothness assumptions on the initial data. Further, quasi-optimal estimates are shown to hold in L∞(L∞)-norm. Finally, the analysis is extended to the standard mixed method for HIDEs and optimal error estimates in L∞(L2)-norm are derived again under minimal smoothness on initial data.

Similarity solutions for a class of hyperbolic integrodifferential equations

For a class of hyperbolic partial integrodifferential equations of the form $u_{tt} - u_{xx} + a*u_{xx} = 0$, fundamental solutions are found that depend on the similarity variable $\xi =x(t-|x|)^{-\alpha}$, where $\alpha \in (0,1)$ and the integral kernel $a$ behaves like $t^{-\alpha}$ near $t=0$. The asymptotic behavior of these solutions in various scaling limits and their regularity is discussed. Applications to solutions of general initial-value problems of such equations are given.

Numerical solutions for a class of differential equations in linear viscoelasticity

We consider numerical solutions by finite element methods for a class of hyperbolic integro-differential equations in linear viscoelasticity. The kernel under consideration is assumed to be of positive type or monotonic. The semidiscrete and fully discrete (with positive discretization of the kernel) finite element methods are studied, andL2 error estimates are demonstrated for smooth data.