Scientific Research of the Institute of Mathematics and Computer Science

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1 Scietific Researc of te Istitute of Matematics ad Computer Sciece ON THE TOLERANCE AVERAGING FOR DIFFERENTIAL OPERATORS WITH PERIODIC COEFFICIENTS Jolata Borowska, Łukasz Łaciński 2, Jowita Ryclewska, Czesław Woźiak 3 Istitute of Matematics, Czestocowa Uiversity of Tecology, Polad 2 Istitute of Computer ad Iformatio Scieces, Czestocowa Uiversity of Tecology, Polad 3 Cair of Structural Mecaics, Tecical Uiversity of Lodz, Polad jolaborowska@o2.pl Abstract. Te aim of tis cotributio is to propose te tolerace averagig for differetial operators wit periodic coefficiets. Te averagig tecique preseted i tis paper is based o proper limit passages wit tolerace parameter to zero. Tis approac is a certai geeralizatio of tat preseted i []. Itroductio Te tolerace averagig of differetial operators wit periodic coefficiets is based o te cocept of slowly-varyig ad a special decompositio of ukow field. Tis cocepts ca be defied asymptotically by itroducig to equatio a small parameter λ. I te course of asymptotic omogeizatio parameter λ teds to zero but i every specific problem uder cosideratio λ as to be treated as costat. I cotrast to omogeizatio te tolerace averagig of differetial operators is a o-asymptotic tecique based o some pysical ypoteses rater ta o te formal aalytical procedures. Tat is wy we itroduce pysically reasoable o--asymptotic defiitios of te slowly-varyig fuctios. Tese defiitios take ito accout te matematical cocept of tolerace ad te pysical idea of idisceribility.. Basic otios ad cocepts Let Ω be a regular regio i R ad m be a positive iteger, m. We de- λ / 2, λ / 2... λ / 2, λ / 2 were λ > for i m ; fie te basic cell [ ] [ ] i, λ = for i > m. By λ we deote a diameter of. Moreover, we itroduce deotatio ( + for every R ad assume tat Ω ( Ω is te coected set Ω. For a arbitrary iteger m, m, we itroduce gradiet ope- i

2 8 J. Borowska, Ł. Łaciński, J. Ryclewska, C. Woźiak rators (,...,,,..., ad (,...,,,..., ( Ω v C, settig v = v, m Ω suc tat = +. Let m+ Ω, we itroduce te followig differeces ( ( ( if v z v z v z Ω ( v ( v ( v ( if z z z Ω Te tolerace averagig of differetial operators wit periodic coefficiets is based o te cocepts of slowly-varyig ad fluctuatio sape fuctios. Now tese cocepts will be defied. Defiitio Fuctio ν H (Ω will be called slowly varyig fuctio (wit respect to te cell ad tolerace parameter δ if for every Ω v H δ ( Ω Te above coditio will be writte dow i te form SV δ ( Ω; If for fuctio v H ( Ω ad for every Ω (i v H δ ( Ω (ii ( v H ( Ω te we sall write v ( Ω; gradiet. Defiitio 2 SV δ Periodic fuctio H ( FS (, if (i ( λ ( O( λ (ii =, = (iii v ( ; Remark ( v. te followig coditios old δ, i.e. v is slowly varyig togeter wit its first will be called fluctuatio sape fuctio,, for a.e. Ω ( SV δ v H ( Ω Ω δ If v SV ( Ω C ( Ω te ( Ω( z Ω ( z ( v v.

3 Remark 2 O te tolerace averagig for differetial operators wit periodic coefficiets 9 If SV ( Ω C ( Ω v te (i ( Ω( z Ω v ( z v ( (ii ( Ω( z Ω v ( z v ( (iii ( Ω( z Ω FS ( C ( Ω ( ( ( z v ( z v ( ( z ( z v ( + 2. Fudametals of averagig I tis Sectio we are to formulate averagig of a composite fuctio by usig limit passage wit te tolerace parameter (, δ ] to zero. Let ( Ω; For every v SV v SV. Ω we defie v SV ( Ω; as a family of fuctios ad ( Ω; as a family of vector fuctios suc tat Ω v z = v + O, z Ω v z = v + O, z Ω (i ( ( ( (ii ( ( ( (iii ( ( Hece if ( v ( v ( ( v ( O(, z z = z + z + z Ω te lim v ( = v ( ad lim v ( z = ( z ( Subsequetly let ϕ = ϕ z, ( z, ( z w w, z Ω, w, w C ( R be a composite fuctio suc tat ϕ (,, L α ( R + Ω v. w w ad ϕ (,, C ( Λ z for a.e. z, Λ. Te fudametal assumptio imposed o field w i te framework of te tolerace averagig approac will be give i te form of te followig -decompositio w = w = v + v A A δ A ( ( v, va SV Ω, FS, A =,..., N Uder deotatios ( N v v, v,..., v N, (,,..., =, were aforemetioed -decompositio will be give i te form ( te

4 2 J. Borowska, Ł. Łaciński, J. Ryclewska, C. Woźiak Defiitio 3 w = v (2 By te tolerace averagig of fuctio ϕ uder -decompositio (2 we sall mea ϕ ( ( % ( ( ( ( ( v, v lim ϕ z, v ( z ( z, v z z dz ( % v v v v. were (,,..., N Hece R ϕ ( ( ( ϕ ( ( ( ( ( ( ( v, % v = y, v y, v y + y % v dy It ca be see tat fuctio C ( ( N + m + +. (3 ϕ Ξ were Ξ is a bouded domai i 3. Averagig of differetial operators Te aim of tis Sectio is to derive te tolerace averagig form of differetial operator L ad equatio Lw = f were f = p, = f L 2 2 ( Ω, L ( settig ij p w ad H ( Ω w,, i, j =,...,. To tis ed we apply te -decompositio w = v, =,,..., N A (, FS, A =,..., N (4 Let Lv = f, % for,..., p p L v = f, =,..., N ad A N % for p p N = =. Te vector operator L ( L, L,..., L called te tolerace averagig of operator L. Defiitio 4 Equatio Lv=f defied by =, = will be

5 O te tolerace averagig for differetial operators wit periodic coefficiets 2 f f L v = f, =,,.., N (5 were % p r = f p ( v L = L (6 ( = L L r v (7 is said to be te tolerace averaged equatio for equatio Lw=f uder decompositio (4. Coclusios Te proposed formal modellig ca be applied to te formatio of differet matematical models for te aalysis of termomecaical processes ad peomea i microeterogeeous solids ad structures. Te problems related to some applicatios of tis approac will be studied i fortcomig papers. Refereces [] Woźiak C., Wierzbicki E., Averagig Teciques i Termomecaics of Composite Solids- Tolerace Averagig versus Homogeizatio, Częstocowa Uiversity Press, Częstocowa 2.

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