Dual System Least-Squares Finite Element Method for a Hyperbolic Problem

This study investigates the dual system least-squares finite element method, namely the LL∗ method, for a hyperbolic problem. It mainly considers nonlinear hyperbolic conservation laws and proposes a combination of the LL∗ method and Newton’s iterative method. In addition, the inclusion of a stabilizing term in the discrete LL∗ minimization problem is proposed, which has not been investigated previously. The proposed approach is validated using the one-dimensional Burgers equation, and the numerical results show that this approach is effective in capturing shocks and provides approximations with reduced oscillations in the presence of shocks.

EXISTENCE AND BLOW UP FOR A NONLINEAR VISCOELASTIC HYPERBOLIC PROBLEM WITH VARIABLE EXPONENTS

Our aim in this paper is to establish the weak existence theorem andfind under suitable assumptions sufficient conditions on $m, p$ andthe initial data for which the blow up takes place for the followingboundary value problem:$$|u_t|^{\rho}u_{tt}-\Delta u-\Deltau_{tt}+\displaystyle\int_{0}^{t}g(t-s)\Delta u(s)ds+|u_{t}|^{m(x)-2}u_{t}=|u|^{p(x)-2}u.$$This paper extends some of the results obtained by the authors and it is focused on new results which are consequence of the presence ofvariable exponents.

Existence of Optimal Control for a Nonlinear Partial Differential Equation of Hyperbolic Type

In this paper, we prove the existence of an optimal control for a nonlinear hyperbolic problem, examined in [3]. An estimation is used which makes it possible to extract from a minimizable sequence of controls and from the sequence of corresponding solutions weakly convergent sub sequences. To prove the passage to the limit in a true equality for every element of the minimizable sequence, Lebesgue’s theorem on the passage to the limit under the integral sign and the theorem of immersion have been used.

The existence of a solution of the two-dimensional direct problem of propagation of the action potential along nerve fibers

In this article, we consider a generalized parabolic two-dimensional direct problem of the process of propagation of the action potential along nerve fibers. The problem is reduced to a generalized hyperbolic problem using the Laplace transform. A generalized two-dimensional direct hyperbolic problem is reduced to a regular hyperbolic problem using methods for rectifying characteristics and isolating singularities. Using the piecewise-continuous function, the existence of the solution of the last problem is proved. From the equivalence of problems it follows that there exists a generalized solution of the parabolic problem.