In this paper, we consider a nonlinear beam equation with a strong damping and the p(x)-biharmonic operator. The exponent p(·) of nonlinearity is a given function satisfying some condition to be specified. Using Faedo-Galerkin method, the local and global existence of weak solutions is established with mild assumptions on the variable exponent p(·). This work improves and extends many other results in the literature.
In this paper, we consider a nonlinear beam equation with the p-biharmonic operator, where 1 < p < ∞. Using a change of variable, we transform the problem into a system of differential equations and prove the existence, uniqueness and regularity of the weak solution by applying the Lax-Milgram theorem and classical results of functional analysis. We investigate the discrete formulation for that system and, with the aid of the Brouwer theorem, we show that the problem has a discrete solution. The uniqueness and stability of the discrete solution are obtained through classical methods. After establishing the order of convergence, we apply the mixed finite element method to obtain an algebraic system of equations. Finally, we implement the computational codes in Matlab software and perform the comparison between theory and simulations.
Resumo. Neste artigo, estudamos uma equação de viga não linear com o operador p(x)-biharmônico em um domínio unidimensional. Transformamos o problema em um sistema de duas equações diferenciais e demonstramos a existência, unicidade e regularidade da solução fraca e da solução discreta. Também investigamos a ordem de convergência e provamos algumas estimativas de erro. Em seguida, utilizamos as bases de Lagrange para obter um sistema algébrico de equações e, nalmente, implementamos os códigos computacionais no software Matlab e apresentamos dois exemplos para ilustrar a teoria.Palavras-chave. Operador p(x)-biharmônico, solução fraca, método de elementos nitos mistos, ordem de convergência, simulações numéricas.
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