Distributed-order fractional wave equation on a finite domain. Stress relaxation in a rod

We study waves in a rod of finite length with a viscoelastic constitutive equation of fractional distributed-order type for the special choice of weight functions. Prescribing boundary conditions on displacement, we obtain case corresponding to stress relaxation. In solving system of differential and integro-differential equations we use the Laplace transformation in the time domain

On a initial value problem arising in mechanics

We study initial value problem for a system consisting of an integer order and distributed-order fractional differential equation describing forced oscillations of a body attached to a free end of a light viscoelastic rod. Explicit form of a solution for a class of linear viscoelastic solids is given in terms of a convolution integral. Restrictions on storage and loss moduli following from the Second Law of Thermodynamics play the crucial role in establishing the form of the solution. Some previous results are shown to be special cases of the present analysis

An extremum variational principle for a class of boundary value problems

AbstractA variational principle for a class of Hamiltonian boundary value problems is formulated. Conditions under which it possesses an extremum (maximum or minimum) are examined and error bounds for approximate solutions are derived. The results are illustrated by two examples

Error bounds via a new extremum variational principle, mean square residual and weighted mean square residual

AbstractError bounds for a wide class of nonlinear one-dimensional boundary value problems are derived from a new extremum variational principle. A new least-squares approximate technique, based on a weighted mean square residual, is established. Also, the value of the weighted mean square residual and value of the classical mean square residual are used for error estimate. The results are illustrated by four examples

Contribution to error estimate

AbstractThe error estimate of an approximate solution to a nonlinear ordinary differential equations of the second order is obtained. The differential equation is subject to either two-point boundary conditions or initial conditions. The independent variable interval may be finite or infinite. The theory is applied to five problems

Dynamics of a Rod Made of Generalized Kelvin–Voigt Visco-elastic Material

AbstractIn this paper we develop a new mathematical model for the lateral vibration of an axially compressed visco-elastic rod. As the basis for this model we use a fractional derivative type of stress–strain relation. We show that the dynamics of the lateral vibration is governed by two coupled linear differential equations with fractional derivatives. For a special case of the generalized Kelvin–Voigt body, this system is reduced to a single fractional derivative differential equation (Eq. (19)). For a class of problems to which (19) belongs the questions of the existence of a solution and its regularity are analyzed. Both continuous and impulsive loading are treated