Computation of power-series expansions in homogenisation of nonlinear equations

Nenad Antonić & \underline{Martin Lazar}

Abstract.

In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) effects. A number of results in that direction has been obtained for linear problems.

Tartar (1990) innitiated the study of the effective equation corresponding to nonlinear equation:

∂_{t}u_{n} + a_{n}u_{n}^{2} = f .

Significant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power-series expansion. We propose a method which overcomes that difficulty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the effective equation and the convergence of power-series expansions.

The feasibility of our method for other types of nonlinearities will be discussed as well.

Nenad Antonić, Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia.

     e–mail: nenad@anak.math.hr
     http://www.math.hr/~nenad

Martin Lazar, Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia.

     e–mail: martin@math.hr
     http://www.math.hr/~martin