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Exactness of formal asymptotic solutions of a Dirichlet problem modeling the steady state of functionally-graded microperiodic nonlinear rods

Décio Roberto M.S., Pérez-Fernández Leslie D., Bravo-Castillero Julián

Journal of Applied Mathematics and Computational Mechanics

2019

Vol. 18, nr 3

45--56

In their usual form, homogenization methods produce first-order approximations of the exact solutions of problems for differential equations with rapidly oscillating coefficients which model the physical behavior of microstructured media. However, there is need of approximations containing higher-order terms when the usual first-order approximations, which are formed by superposing a macroscopic trend and a local perturbation, are not capable of reproducing the local details of the exact solutions. Here, two-scale asymptotic solutions with second-order terms are provided for a Dirichlet problem modeling the steady state of functionally-graded microperiodic nonlinear rods. The need of considering higherorder terms is illustrated through numerical examples for various power-law nonlinearities.

On some nonlinear hyperbolic p(x,t)-Laplacian equations

Ahmedatt T., Touzani A., Aberqi A., Yazough C.

Journal of Applied Analysis

2018

Vol. 24, nr 1

55--69

This paper is devoted to study the global existence of solutions of the hyperbolic Dirichlet equation Utt=Lu+f(x,t) in ΩT=Ω×(0,T), where L is a nonlinear operator and ϕ(x,t,⋅), f(x,t) and the exponents of the nonlinearities p(x,t) and μ(x,t) are given functions.

On-sided estimates for quasimonotone systems of boundary value problem

Herzog G.

Demonstratio Mathematica

2004

Vol. 37, nr 4

847--856

We prove existence and uniqueness theorems for Dirichlet boundary value problems of the form u" + f(t,u) = 0, u(0) = uo, u(1) = ui in ordered finite dimensional Banach spaces, involving one-sided estimates and quasimonotonicity.

Local in time existence of solution to the Dirichlet initial-boundary value problem in nonlinear hypoelasticity

Piskorek A., Szymaniec A.

Demonstratio Mathematica

2000

Vol. 33, nr 3

502-515

Dirichlet problems with variable boundary data for nonlinear partial differential equations

Bors D.

Demonstratio Mathematica

2000

Vol. 33, nr 2

295-312

In this paper we consider some systems of partial differential equations with variable boundary data. Some sufficient conditions under which solutions of these systems continuously depend on boundary data are given. The proofs of the main result of this work are based on some variational methods.