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HYPERCYCLIC AND TOPOLOGICALLY MIXING PROPERTIES OF CERTAIN CLASSES OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS
Published in Marko Kostić, Abstract Volterra Integro-Differential Equations, 2015
Remark 3.2.16. One of the most undeveloped subjects in the theory of abstract Volterra integro-differential equation is, undoubtely, the analysis of solutions in the generalized function spaces. This section provides a general information on distributional solutions of abstract second order differential equations, the corresponding results for equations of first order can be found in the third chapter of monograph [292]. On the other hand, distributional solutions of abstract Volterra equations have been considered by G. Da Prato and M. Iannelli in [135]. In connection with this, we would like to address the problem of describing the class of integrated (ga, C)-regularized resolvent families (a e (0, 2)\{1}) in terms of a corresponding class of C-distribution resolvents. Also, it could be of importance to analyze solutions of abstract Volterra equations in the spaces of vector-valued tempered ultradistributions ([327], [455], [292]). The following theorem will be useful in our further work.
An Introduction to Neutron Transport Theory
Published in Robert E. Masterson, Introduction to Nuclear Reactor Physics, 2017
The primary reason the transport equation is so difficult to solve is that it is an integro-differential equation. Integro-differential equations are more challenging than other types of equations because they have both integrals and differentials in them. In general, there is no easy way to solve an integro-differential equation. However, several methods have been developed that allow approximate solutions to be obtained under certain conditions. For example, a solution can be obtained by replacing the integrals by an infinite series expansion that can be truncated after the first few terms. Practically speaking, this approach is similar (but not identical) to expanding a trigonometric function such as sin(x), cosine(x), ex, or ln(x) in a Taylor series and retaining the first few terms. For cosine (x) and for ex, the relevant Taylor series expansions are cosinex=1–x22!+x44!−⋯+higher-order termsex=1+x+x22!+x33!+⋯+higher-order terms The series expansions for other geometric functions can be found in any elementary textbook. Ordinarily, only three or four terms are required to obtain a reasonable approximation to the function being integrated. Obviously, the more terms that are used, the more accurate the solution will be, but the additional terms also make it harder to find a general solution. And so a trade-off must be made in practice between the accuracy that is desired and the degree of complexity that is involved.
Feedback Volterra control of integro-differential equations
Published in International Journal of Control, 2022
G. Pepe, E. Paifelman, A. Carcaterra
and rearranging: Laplace anti-transform and the introduction of the new variables , , defined as produces: or assembling in a compact form: that reduces the integro-differential Equation (17) to a first-order differential equation.
On uniform asymptotic stability of nonlinear Volterra integro-differential equations
Published in International Journal of Control, 2022
Pham Huu Anh Ngoc, Le Trung Hieu
Suppose there exist and continuous functions such that and If the linear Volterra integro-differential equation is UAS, then the zero solution of (1) is UAS.
One dimensional nonlocal integro-differential model & gradient elasticity model : Approximate solutions and size effects
Published in Mechanics of Advanced Materials and Structures, 2019
B. Umesh, A. Rajagopal, J. N. Reddy
The weights ξ1 and ξ2 are termed as local and nonlocal constitutive parameters. A complete local constitutive relation is recovered for the case ξ1 = 0 and ξ2 = 1 and the another possible case is ξ1 = 0 and ξ2 = 1 corresponds to Eringen nonlocal constitutive relation. The equilibrium equation obtained for Eq. (22) is termed as integro-differential equation.