328 Computational Mechanics of Composite Materials operator may be(approximately)preserved.Furthermore,if the equation is of the form of -V(e(x)Vu(x))=f(x) (7.10) or some other variable-coefficient differential equation,we should verify if the reduction procedure preserves this form,so that we may find effective coefficients of the equation on the coarse scale.This process is the basic goal of homogenisation techniques,and it extracts information from the reduced equation based on the form of the original equation.Thus,within the multiresolution approach,reduction and homogenisation are closely related but have different goals:homogenisation attempts to find effective equations and their coefficients on the coarse scale,whereas reduction merely finds a coarse-scale version of a given system of equations. The multiresolutional (MRA)homogenisation procedure is applied to the systems of ODEs,which may be written in the form Bx+q+入=K(Ax+p) (7.11) In particular,we consider equations of the form (+Bx(t)+gt)+元=∫A(s)x(s)+p(s)ds,t∈(0,1) (7.12) 0 on L(0,1),where B(t)and A(t)are n x n matrix-valued functions,p(t)and g(t)are vector forcing terms,and x(t)is the solution vector.As a differential equation this is written as +B00+g0）=A0a0+p0 (7.13) with the initial conditions (I+B(0)x(0)=-g(0)-.On V,,j<0,the projection of (7.11)is written as B+9+元=K,4x,+Pj (7.14) or Sx;=fi (7.15) where S,=B-K,A,f=KjP1-4-九，x=Px (7.16328 Computational Mechanics of Composite Materials operator may be (approximately) preserved. Furthermore, if the equation is of the form of −∇( ) e(x)∇u(x) = f (x) (7.10) or some other variable-coefficient differential equation, we should verify if the reduction procedure preserves this form, so that we may find effective coefficients of the equation on the coarse scale. This process is the basic goal of homogenisation techniques, and it extracts information from the reduced equation based on the form of the original equation. Thus, within the multiresolution approach, reduction and homogenisation are closely related but have different goals: homogenisation attempts to find effective equations and their coefficients on the coarse scale, whereas reduction merely finds a coarse-scale version of a given system of equations. The multiresolutional (MRA) homogenisation procedure is applied to the systems of ODEs, which may be written in the form Bx + q + λ = K( ) Ax + p (7.11) In particular, we consider equations of the form () ( ) + + + = ∫ + t I B t x t q t A s x s p s ds 0 ( ) ( ) ( ) λ ( ) ( ) ( ) , ) t ∈ (0,1 (7.12) on ) (0,1 2 L , where B(t) and A(t) are n x n matrix-valued functions, p(t) and q(t) are vector forcing terms, and x(t) is the solution vector. As a differential equation this is written as ( ) ( ) I B(t) x(t) q(t) A(t)x(t) p(t) dt d + + = + (7.13) with the initial conditions ( ) I + B(0) x(0) = −q(0) − λ . On Vj , j<0, the projection of (7.11) is written as ( ) j j j j j j p j B x + q + λ = K A x + (7.14) or j j j S x = f (7.15) where S j = Bj − K j Aj , f j = K j p j − q j − λ , j j j x = P x (7.16)