Generalized implicit Euler method for hyperbolic functional differential equations

Nonlinear hyperbolic functional differential equations with initial boundary conditions are considered. Theorems on the convergence of difference schemes and error estimates of approximate solutions are presented. The proof of the stability of the difference functional problem is based on a comparison technique. Nonlinear estimates of the Perron type with respect to the functional variable for given functions are used. Numerical examples are given (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized solutions of mixed problems for first-order partial functional differential equations

A theorem on the existence of solutions and their continuous dependence upon initial boundary conditions is proved. The method of bicharacteristics is used to transform the mixed problem into a system of integral functional equations of the Volterra type. The existence of solutions of this system is proved by the method of successive approximations using theorems on integral inequalities. Classical solutions of integral functional equations lead to generalized solutions of the original problem. Differential equations with deviated variables and differential integral problems can be obtained from the general model by specializing given operators. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 804–828, June, 2006.     

Numerical methods for Hamilton Jacobi functional differential equations

Initial and initial boundary value problems for first order partial functional differential equations are considered. Explicit difference schemes of the Euler type and implicit difference methods are investigated. The following theoretical aspects of the methods are presented. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. There are implicit difference schemes which are convergent and corresponding explicit difference methods are not convergent. Error estimates for both the methods are construted.     

Generic invariant measures for iterated systems of interval homeomorphisms

Archiv der Mathematik - It is well known that iterated function systems generated by orientation preserving homeomorphisms of the unit interval with positive Lyapunov exponents at its ends admit a...     

Classical solutions of hyperbolic IBVPs with state dependent delays on a cylindrical domain

We consider the initial boundary value problem for a nonlinear partial functional differential equation of the first order

_∂tz(t,x)=f(t,x,V(z;t,x),_∂xz(t,x)),${\partial }_{t}z(t,x)=f(t,x,V(z;t,x),{\partial }_{x}z(t,x))\text{,}$

where V$V$ is a nonlinear operator of Volterra type, mapping bounded subsets of the space of Lipschitz-continuously differentiable functions, into bounded subsets of the space of Lipschitz continuous functions with Lipschitz continuous spatial partial derivatives. Using the method of bicharacteristics and successive approximations, we prove the local existence, uniqueness and continuous dependence on data of classical solutions of the problem. This approach covers equations of the form

_∂tz(t,x)=f(t,x,_{zα(t,x,z(t,x))},_∂xz(t,x)),${\partial }_{t}z(t,x)=f(t,x,z_{\alpha (t,x,z_{(t,x)})},{\partial }_{x}z(t,x))\text{,}$

where (t,x)?_z(t,x)$(t,x)?z_{(t,x)}$ is the (multidimensional) Hale operator, and all the components of α$\alpha$ may depend on (t,x,_z(t,x))$(t,x,z_{(t,x)})$. More specifically, problems with deviating arguments and integro-differential equations are included.     

Implicit difference methods for Hamilton-Jacobi functional differential equations

Classical solutions of initial boundary value problems are approximated by solutions of associated implicit difference functional equations. A stability result is proved by using a comparison technique with nonlinear estimates of the Perron type for given functions. The Newton method is used to numerically solve nonlinear equations generated by implicit difference schemes. It is shown that there are implicit difference schemes which are convergent whereas the corresponding explicit difference methods are not. The results obtained can be applied to differential integral problems and differential equations with deviated variables.     

Classical solutions of hyperbolic initial boundary-value problems with state-dependent delays

We consider the initial boundary-value problem for a system of quasilinear partial functional differential equations of the first order