Solution of the Hyperbolic Partial Differential Equation on Graphs and Digital Spaces: a Klein Bottle a Projective Plane and a 4D Sphere

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1 Soluio of he Hyperbolic Parial Differeial Equaio o Graph ad Digial Space: a Klei Bole a Projecive Plae ad a 4D Sphere Alexader V. Evako Diae, Laboraory of Digial Techologie, Mocow, Ruia addre: evakoa@mail.ru Abrac: I may cae, aalyic oluio of parial differeial equaio may o be poible. For pracical problem, i i more reaoable o carry ou compuaioal oluio. However, he adard grid i he fiie differece approximaio i o a correc model of he coiuou domai i erm of digial opology. I order o avoid eriou problem i compuaioal oluio i i eceary o ue opologically correc digial domai. Thi paper udie he rucure of he hyperbolic parial differeial equaio o graph ad digial -dimeioal maifold, which are digial model of coiuou -maifold. Codiio for he exiece of oluio are deermied ad iveigaed. Numerical oluio of he equaio o graph ad digial -maifold are preeed. Keyword: Hyperbolic PDE, Graph, Soluio, Iiial Value Problem, Digial Space, Digial Topology. Iroducio Differeial equaio play a impora role i variou field of ciece ad echology. However i may cae, aalyic oluio of PDE (parial differeial equaio) may o be poible. For pracical problem, i i more reaoable o carry ou compuaioal or umerical oluio. I ca be doe by implemeig a domai graph ad by raferrig PDE from a coiuou area io dicree oe. A review of work devoed o parial differeial equaio o graph ca be foud i [2], [4], ad [7]. Figure. (a) 2D grid for wo idepede variable x ad y. (b) The ball U(v ) of poi v (black poi) i graph G. A a rule, radiioal umerical mehod provide good approximaio o he exac oluio of PDE. However, he adard grid i he fiie differece approximaio i o a correc model of he coiuou domai i erm of digial opology (ee [4]). I order o avoid eriou problem i compuaioal oluio i i eceary o ue opologically correc digial domai. I phyic, umerical mehod are ued i he udy of laice model o o-orieable urface uch a a Moebiu rip ad a Klei bole (ee [3]). I uch cae, o-orieable urface mu be replaced by opologically correc digial grid. Mahemaically correc grid ca be obaied i he framework of digial opology. Digial opology mehod are crucial i aalyzig - dimeioal digiized image ariig i may area of ciece icludig eurociece, medical imagig, compuer graphic, geociece ad fluid dyamic. Tradiioally, digial objec are repreeed by graph whoe edge defie eare ad coeciviy (ee, e.g., [3], [8], []). The impora feaure of a -urface i a imilariy of i properie wih properie of i coiuou couerpar i erm of algebraic opology. For example, he Euler characeriic ad he homology group of digial -phere, a Moebiu rip ad a Klei bole are he ame a oe of heir coiuou couerpar ([] ad [2]). I rece year, here ha bee a coiderable amou of work devoed o buildig wo, hree ad -dimeioal dicreizaio cheme ad digial image. I paper [6] ad [0], dicreizaio cheme are defied ad udied ha allow o build digial model of 2-dimeioal coiuou objec wih he ame opological properie a heir coiuou couerpar ad wih ay required accuracy. I hi paper, we udie he rucure of he hyperbolic parial differeial equaio o graph ad digial pace modelig coiuou objec. Secio 2 aalyze he rucure of a hyperbolic differeial equaio o a graph. Secio 3 coai a hor decripio of digial pace ad digial -urface udied i [8]- [9] uch a digial - dimeioal phere, a digial oru, a digial Klei bole, ec. Secio 4 pree a umerical oluio of a hyperbolic equaio o a digial rig, a digial Klei bole, a digial projecive plae ad a digial 4D phere. 2. Hyperbolic PDE o a Graph

2 We eed o emphaize ha he defiiio, he heorem aeme ad proof foud i hi ecio ca be uderood oly afer umerical experime. Digial pace are graph wih pecific opological rucure. Soluio of PDE o everal ype of digial pace were udied i paper [7]. Fiie differece approximaio of he PDE are baed upo replacig parial differeial equaio by fiie differece equaio uig Taylor approximaio [5]. A a example, coider a hyperbolic PDE wih wo paial idepede variable. 2 f = a 2 f + b 2 f + g () 2 x 2 y 2 where f = f(x, y, ), a = a(x, y, ), b = b(x, y, ), g = g(x, y, ). A wo-dimeioal paial orhogoal grid G wih poi v p = (pδx, Δy) i how i figure. Uig he forward differece formula for he derivaive wih repec o, ad he ceral differece formula for he ecod derivaive wih repec o x ad y, we obai he followig equivale fiie deferece equaio, f + i,j 2f i,j + f i,j Δ 2 = a f i,j 2f i,j + f i+,j i,j Δx 2 + +b f i,j i,j 2f i,j + f i,j+ Δy 2 + g i,j where x = iδx, i =,2,, y = jδy, j =,2,, = Δ, =,2,. Thi equaio ca be raformed o he form + i+ = e pk f k + Δ 2 g i,j f i,j where e i,j + e i,j j+ p=i k=j (2) + e i+,j +e i,j +e i,j+ = 0 Noice ha grid G i figure (a) i a graph. Poi (i, j) i adjace o poi (i, j ± ) ad (i ±, j). The ball of poi (i, j) i U((i, j) coaiig alo poi (i, j ± ) ad (i ±, j).. The ball of poi (i, j) i U((i, j) coaiig alo poi (i, j ± ) ad (i ±, j). Uig hee oaio, equaio (2) ca be wrie a a equaio o graph G. e pk f + p = v k U(v p ) f k + 2f p f p + g p (3) v k U(v p ) e pk = 0 The ummaio i produced over all poi belogig o ball U((i, j). Here f k i he value of he fucio f(v k, ) a poi v k of G a he mome, coefficie e pk are fucio o he pair of poi (v p, v k ) ad = (wih domai V V ). If poi v p ad v k are o adjace, he e pk = 0. Noice ha i geeral, codiio v k U(v p ) e pk = 0, i o eceary for he PDE. For example, i doe o hold for he hyperbolic PDE o a direced ework. The equaio (3) i called homogeeou. if all g p = 0. Laer o i hi paper, all g p = 0. Thu, if G(V, W) i a graph wih he e of poi V=(v, v 2,...v ), he e of edge W = ((v р v q ),...), ad U(v p ) i he ball of poi v p,,... he a hyperbolic PDE o G i he e of equaio e pk f + p = v k U(v p ) f k + 2f p f p, =, p =, (4) The fucio a =+ o a give poi v p deped o he value of he fucio a =, - o v p ad he poi adjace o v p. The ecil for equaio (4) i illuraed i figure (b). The ball U(v ) coi of black poi. Sice e pk = 0 if poi v p ad v k are o-adjace, he e (4) ha he form f + p = e pk f k + 2f p f p, p =, (5) To make hi equaio look more coveie for aalyi we ca wrie i i he form ued i [4]. I wa how ha he parabolic PDE o a graph i f + p = e pk f k + f p Thi equaio ca be wrie a f + p = c pk f k c pk where e pk = 0, where, c pk =, = e pk if p k, c pp p =, = e pp +. Therefore, Here equaio (5) ca be raformed io he followig form. Defiiio 2. Suppoe ha G(V, W) i a graph wih he e of poi V=(v, v 2,...v ), he e of edge W = ((v р v q ),...), ad U(v p ) i he ball of poi v p,,... A hyperbolic PDE o G i he e of equaio f + p = c pk f k + f p f p, where, c pk =, p =, f + p = c pk f k + f p f p, where, c pk =, p =, (6) Noice ha equaio (6) i more obviou ad coveie for udyig becaue i coai he parabolic par ( c pk f k ) ad he hyperbolic par (f p f p ). I oher word, hi equaio ca be rewrie a (f p + ) hyperbolic = (f p + ) parabolic + f p f p Equaio (6) do o deped explicily o he opology of graph G, ad ca be applied o a graph of ay dimeio or o a ework. All opological feaure are coaied i he local ad global rucure of G. Se (6) i imilar o he e of differeial equaio o a graph iveigaed by A. I. Volper i [6]. Equaio (6) ca be preeed i he marix form. f c c 2 f = [ ], C() = [ c 2 ], f c

3 f + = C()f + f f Coider ow he iiial codiio for olvig equaio (6). Iiial codiio are defied i a regular way by he e of equaio a = 0 ad =. f p 0 = f(v p, 0), f p = f(v p, ), p =,2, Defiiio 2.2 Se of equaio (6) alog wih iiial codiio i called he iiial value problem for he hyperbolic PDE o a graph G(v, v 2, v ). f + p = c pk f k + f p f p, { f 0 p = f(v p, 0), f p = f(v p, ), p =,2, Boudary codiio ca alo be e he uual way. Boudary codiio are affeced by wha happe a he ubgraph H of G. Le H be a ubgraph of G. Le he value of he fucio f(v k, ) a poi v k ϵh a he mome = be give by he e f(v k, ) = f k = k, v k ϵh, = The boudary-value problem o G ca be formulaed a follow. Defiiio 2.3 Le G(v, v 2, v ) be a graph ad he e (6) be a differeial hyperbolic equaio o G. Le H be a ubgraph of G, ad he value of he fucio f(v k, ) a poi v k ϵh a he mome be defied by boudary codiio. Equaio (6) alog wih boudary codiio i called he boudary value problem for he hyperbolic PDE o a graph G. f + p = c { pk f k + f p f p, f(v k, ) = f k = k, v k ϵh, = N c pk Coider a paricular cae ha i equaio (6), =, c pk 0, p, k =,. For he parabolic equaio, hi mea he hea or diffuio equaio o a graph a how i [4]. We will how ha if iiial codiio are uch ha 0 f p = f p = A he f p = A for every =. Theorem 2. Le f + p = c pk f k + f p f p be a hyperbolic equaio ad aume ha c pk =, k =,, ad he iiial codiio f 0 p = f(v p, 0), f p = f(v p, ), p =,2,, aify he equaio 0 f p = f p = A. The f p = A for every =. Proof S + = f p + c pk = c pk f k + f p f p = f k + A A = (7) (8) c pk f k = f k = A Thi complee he proof. We focu ow o he defiiio ad rucure of wave equaio which i a pecial cae of a hyperbolic PDE. We are equipped o iroduce he geeral wave equaio o G. Defiiio 2.4 A hyperbolic equaio i called he wave equaio if he followig codiio hold f p + = c pk =, c pk c pk f k + f p f p (9) 0, =, p =, The oluio of hi equaio hould be aalogou he oluio of he wave equaio i he coiuou cae. I mu be aociaed wih ocillaio of he fucio a poi of G ad he wave propagaio. Fir, aume ha graph G o which we deire o olve equaio (9) coi of wo adjace poi v ad v 2. Theorem 2.2 Le i he wave equaio (9) o graph G(v, v 2 ) coiig of wo adjace poi, he iiial codiio f 0 p = f(v p, 0), f p = f(v p, ), p =,2, aify he equaio 0 f p = f p = A, ad coefficie c pk = c pk do o deped o =. The he oluio f = f(v, ) ad f 2 = f(v 2, ), = N, are periodic equece wih period T > 2. Proof Coider graph G(v, v 2 ) coiig of wo adjace poi i figure 2(a). Accordig o (9), { f + = c f + c 2 f 2 + f f f + 2 = c 2 f + c 22 f 2 + f 2 f 2 By heorem 2., f + f 2 = A. Therefore, f 2 = A f. The f + = c f + c 2 (A f ) + f f = c 2 A + ( + c c 2 )f f (0) I paper [5], i wa how ha he geeralized Fiboaccilike equece *F + = A + BF F,, F 0 = a, F = b+ i periodic wih period T = 2π > 2, ω = arcco B if ω 2 B < 2. I equece (0), B = + c c 2. Sice c pk 0, c + c 2 =, c 2 + c 22 =, he B < 2. Therefore, (0) i a periodic equece wih period T = 2π > 2, ω = ω arcco +c c 2. Acig i he ame way, i i eay o prove 2 + ha f 2 i periodic wih he ame period T. Thi complee he proof. The heorem above will be ued o prove a geeral heorem abou periodiciy of he oluio of he wave equaio. The followig heorem aer ha he oluio of (9) i periodic o a graph G((v,v 2,..v ), >2. Noice ha G((v,v 2,..v ) i he uio of poi v ad ubgraph H=G-v, G((v,v 2,..v )=v H. I he proof of he ex heorem, graph H i replaced wih poi v H (ee figure 2(b), (c)). Theorem 2.3 Le i he wave equaio (9) o graph G(v, v ), he

4 iiial codiio f 0 p = f(v p, 0), f p = f(v p, ), p =,2,, aify he equaio 0 f p = f p = A, ad coefficie c pk do o deped o =. The he oluio f p = f(v p, ), p =,2,, = N, are periodic equece wih period T > 2. Proof Graph G ca be preeed a he uio of v ad ubgraph H(v 2, v ), G = v H(v 2, v ) (ee figure 2(b)). Le f H = i=2 f i be he u of f i over poi of H. Se (9) coi of equaio. Figure 2. (a) Graph G(v,v 2) coi of wo adjace poi v ad v 2. (b) G=(v,H)= =v H. Subgraph H=G-v. (c) Graph G(v,v H) coi of wo adjace poi v ad v H. f + = c f + c 2 f c f + f f { f + = c f + c 2 f c f + f f Summig up la (-) equaio, we obai where c HH { f + = c f + c H f H + f f f + H = c H f + c HH f H + f H f () H c H f H = c H c k=2 k f k, i=2 )f, f = c i f = ( c f H = c i=2 k=2 ik f k = f H c k k=2 f k. A raighforward check how ha c + c H = ad c HH + c H =. Therefore, e () i he wave equaio o he graph G(v,v H ) coaiig ju wo poi v ad v H depiced i figure 2(c). Therefore, accordig o heorem 2.2, f = f(v, ), = N, i a periodic equece wih period T = 2π ω > 2, ω = arcco +c c H 2. For he ame reao, he oluio f p = f(v p, = ), i a periodic equece a every poi v p, p = 2,. Thi complee he proof. The ocillaio of he fucio a every poi are compleely deermied if we kow he iiial value a mome =0 ad =. 3. Digial -Maifold, Digial -Sphere, a Digial Toru, a Digial Projecive Plae There i coiderable amou of lieraure devoed o he udy of differe approache o digial lie, urface ad pace ued by reearcher. I order o make hi paper elfcoaied, i i reaoable o iclude ome reul relaed o digial pace. Tradiioally, a digial image ha a graph rucure (ee [, 5, 0]). A digial pace G i a imple udireced graph G=(V,W) where V=(v,v 2,...v, ) i a fiie or couable e of poi, ad W = ((v р v q ),...) i a e of edge. The iduced ubgraph O(v) G coaiig all poi adjace o v, excludig v ielf, i called he rim or he eighborhood of poi v i G, he iduced ubgraph U(v)=v O(v) i called he ball of v. For wo graph G=(X, U) ad H=(Y, W) wih dijoi poi e X ad Y, heir joi G H i he graph ha coai G, H ad edge joiig every poi i G wih every poi i H. Coracible graph were udied i [, 2]. Coracible graph are defied recurively. Defiiio 3. A oe-poi graph i coracible. A coeced graph G wih poi i coracible if i coai a poi v o ha he rim O(v) i coracible ad G-v i coracible. A poi v of a graph G i aid o be imple if i rim O(v) i a coracible graph. Thu, a coracible graph ca be covered o a poi by equeial deleig imple poi. A digial -maifold i a pecial cae of a digial -urface defied ad iveigaed i [6]. Defiiio 3.2 A digial 0-dimeioal phere i a dicoeced graph S 0 (a,b) wih ju wo poi a ad b. A coeced graph M i called a digial -phere, >0, if for ay poi v M, he rim O(v) i a (-)-phere ad he pace M-v i a coracible graph [8]. Defiiio 3.3 A coeced graph M i called a -dimeioal maifold, >, if he rim O(v) of ay poi v i a (-)-dimeioal phere [8]. The followig reul were obaied i [6] ad [8]. Theorem 3. The joi S mi=s 0 S 0 2 S 0 + of (+) copie of he zero-dimeioal phere S 0 i a miimal -phere. Le M ad N be ad m-phere. The M N i a (+m+)-phere. Ay -phere M ca be covered o he miimal - phere S mi by coracible raformaio. Figure 3. S 0 i a digial zero-dimeioal phere. S mi, S, S 2, S 3 are digial oe-dimeioal phere. D i a digial oe-dimeioal dik coaiig e poi. A digial 0-dimeioal urface i a digial 0-dimeioal phere. Figure 3 depic digial zero ad oe-dimeioal phere. Figure 4 how digial 2-dimeioal phere. All phere are homeomorphic ad ca be covered io he miimal phere S 2 mi by coracible raformaio. Digial oru T ad a digial 2-dimeioal Klei bole K are how i figure 5. Figure 6 depic a digial projecive plae P ad digial hree ad four-dimeioal phere S 3 ad S 4 repecively. I wa how i [4] ha wo- dimeioal paial grid ued if i fiie-differece cheme are o correc from he viewpoi of digial opology. Thee grid do o reflec local opological feaure of coiuou domai which are replaced

5 by he grid. I order o obai he correc oluio of fiie differece equaio o -dimeioal domai, he grid hould have he rucure of digial -dimeioal pace. 4. Numerical Soluio of he Wave Equaio o Graph I hi ecio we demorae he umerical oluio of he iiial ad boudary value problem for he hyperbolic PDE o a graph G(v, v 2, v ) which i a digial - dimeioal maifold. f + p = c pk f k + f p f p (2) c pk =, c pk 0, k, p =,2, f p 0 = f(v p, 0), f p = f(v p, ) f(v k, ) = f k = k, v k ϵh, N (3) 4.. A Coeced Graph G wih Two Poi. Numerical Soluio of he Iiial Value Problem Figure 4. Two-dimeioal phere wih a differe umber of poi. Ay of phere ca be covered io he miimal phere S 2 mi by coracible raformaio. Coider coeced graph G wih wo poi depiced i figure 2. Defie coefficie c pk i equaio (2) i he followig way; c = 0.8, c 2 = 0.2, c 22 = 0.7, c 2 = 0.3. Iiial value are give a f 0 =2, f 2 =2. The reul of he oluio of he iiial value problem (2) a poi, 2 6 for =, 50 are diplayed i figure 7, which illurae he ime behavior of he value of he fucio f. Figure 7 pree ocillaio a poi ad 2. Figure 5. Digial 2-dimeioal oru T ad Klei bole K wih ixee poi Digial Oe-Dimeioal Dik D Numerical Soluio of he Boudary Value Problem Numerical oluio of he boudary value problem (3) o he digial -D dik D depiced i figure 3 ad modelig a rig i how i figure 8. D coai e poi. Aume ha coefficie c ik i (2) do o deped o, c i,i+ =0.3, c i,i =0.4. Ed poi of he digial rig are fixed, i.e., boudary value are f = f 0 =0. Iiial value are f 5 0 =0, f 5 =0. Two graph i figure 8 how he profile of he rig a mome =45 ad =50. Thi oluio i imilar o he oluio i he coiuou rig. Figure 6. P, S 3 ad S 4 are digial wo-dimeioal projecive plae, hreead four-dimeioal phere repecively Digial 2-Dimeioal Klei Bole K. Numerical Soluio of he Iiial Value Problem A Klei bole i a objec of iveigaio i may field. I phyic [3], a Klei bole araced aeio i udyig laice model o o-orieable urface a a realizaio ad eig of predicio of he coformal field heory ad a ew challegig uolved laice-aiical problem. A digial 2D Klei bole K depiced i figure 5 coi of ixee poi. The rim O(v k ) of every poi v k i a digial -phere coaiig ix poi. i.e., K i a homogeeou digial pace. Topological properie of K are imilar o opological properie of i coiuou couerpar. For example, he Euler characeriic ad he homology group of a coiuou ad a digial Klei bole are he ame ([] ad [2]). I (2), defie coefficie c pk i he followig way. If poi v p ad v k are adjace he c pk =0.; c kk =0.4,, 6. Iiial value are give a f 8 0 =6, f 0 =6. I figure 9, umerical oluio a poi ad 3 of he of he iiial value problem (2) are ploed a ime =, 00. Two lie how ocillaio wih period T> Digial 2D Projecive Plae P. Numerical Soluio of he Iiial Value Problem Figure 6 how a digial 2-dimeioal projecive plae P which i a digial couerpar of a coiuou projecive plae. P i a o-homogeeou digial pace coaiig eleve poi. The rim O(v k ) of every poi v k i a digial - phere. Topological properie of a digial ad a coiuou 2D projecive plae are imilar. I wa how i [] ad [2] ha he Euler characeriic ad he homology group of a coiuou ad a digial projecive plae are he ame. I i eay o check direcly ha a digial 2D projecive plae wihou a poi i homoopy equivale o a digial oedimeioal phere a i i for a coiuou projecive plae. Defie coefficie c pk i (2) i he followig way; if v k ad v p are adjace he c kp = c pk = 0., c pp =,k p c kp. Coider he umerical oluio of he iiial value problem for (2). Iiial value are give a f 0 0 =, f =. The profile of he oluio a poi ad 2 are ploed i figure 0, where =, 00. The plo demorae ocillaio wih period T>2.

6 4.5. Digial 4-Dimeioal Sphere S 4. Numerical Soluio of he Iiial Value Problem Figure 7. Numerical oluio of he iiial value problem o graph G (figure 2) wih wo poi v ad v 2. Iiial value are f 0 =2, f =2. The oluio o G are how a poi ad 2, =0,, 50,. The oluio profile are ocillaio wih period T>2. Figure 0. Numerical oluio of he iiial value problem o he projecive plae P how i figure 6. The oluio profile o P a poi ad 2, =0,, 00, f 0 0 =, f =. The oluio profile are ocillaio wih period T>2. Coider a digial 4-dimeioal -phere S 4 wih e poi depiced i figure 6. The rim O(v k ) of every poi v k i a digial 3-phere coaiig eigh poi ad depiced i figure 6. S 4 i a homogeeou digial pace coaiig he miimal umber of poi. The umber of poi ca be icreaed by uig coracible raformaio. Topological properie of a digial 4D phere are imilar o opological properie of a coiuou 4D phere. Defie he umerical oluio of he iiial value problem for (2). Le c ik =c ki =0.0 if poi v i ad v k are adjace, ad c pp =0.92 for, 0. Iiial value are give a f 0 6=0, f 0 7=0. The reul of he oluio of he iiial value problem (2) a poi, 2 6 for =0, 80 are diplayed i figure, which illurae he ime behavior of he value of he fucio f. Figure pree beaig ocillaio a poi ad 2. Obviouly, he profile of oluio are beaig ocillaio a every poi of hi phere. 5. Cocluio Figure 8. Numerical oluio of he boudary value problem o he -D dik D how i figure 3 ad modelig a rig. D coai e poi. Ed poi of he digial rig are fixed, i.e., boudary value are f = f 0 =0. Iiial value are f 5 0 =0, f 5 =0. Two graph how ocillaio of he rig a mome =45 ad =50. Thi paper iveigae he rucure of differeial hyperbolic equaio o graph, digial pace, digial - dimeioal maifold, >0, ad ework ad udie heir properie. The approach ued i he paper i mahemaically correc i erm of digial opology. A example, compuaioal oluio of he hyperbolic PDE o - dimeioal digial maifold uch a a Klei bole ad a 4- dieioal phere are preeed. Referece [] Bai, Y., Ha, X., Price, J.(2009) Digial Topology o Adapive Ocree Grid. Joural of Mahemaical Imagig ad Viio. 34 (2), [2] Borovkikh, A. ad Lazarev, K. (2004) Fourh-order differeial equaio o geomeric graph. Joural of Mahemaical Sciece, 9 (6), Figure 9. The oluio profile of he iiial value problem o he Klei bole K (figure 5) a poi ad 3, =0,, 00, f 8 0 =0, f 6 =0. Two lie how ocillaio wih period T>2. [3] Eckhard, U. ad Laecki, L. (2003) Topologie for he digial pace Z2 ad Z3. Compuer Viio ad Image Uderadig, 90, [4] Evako, A. (206) Soluio of a Parabolic Parial Differeial Equaio o Digial Space: A Klei Bole, a Projecive Plae, a 4D Sphere ad a Moebiu Bad, Ieraioal Joural of Dicree Mahemaic. Vol., No., pp doi: 0.648/j.dmah

7 [5] Evako, A. (0/207) Properie of Periodic Fiboacci-like Sequece. arxiv: , 207arXiv E. [9] Evako, A. (205) Claificaio of digial -maifold. Dicree Applied Mahemaic. 8, [0] Evako, A. (995) Topological properie of he ierecio graph of cover of -dimeioal urface. Dicree Mahemaic, 47, [] Ivahcheko, A. (993) Repreeaio of mooh urface by graph. Traformaio of graph which do o chage he Euler characeriic of graph. Dicree Mahemaic, 22, [2] Ivahcheko, A. (994) Coracible raformaio do o chage he homology group of graph. Dicree Mahemaic, 26, Figure.. Numerical oluio of he iiial value problem o he 4D phere S 4 depiced i fig. 6.. The graph are how a poi ad 2, =0,, 400, Iiial value are f 5 0 =0, f 6 =0. The oluio profile are ocillaio wih period T>2. [6] Evako, A. (204) Topology preervig dicreizaio cheme for digial image egmeaio ad digial model of he plae. Ope Acce Library Joural,, e566, hp://dx.doi.org/0.4236/oalib [7] Evako, A. (999) Iroducio o he heory of molecular pace (i Ruia laguage). Publihig Houe Paim, Mocow. [8] Evako, A., Kopperma, R. ad Mukhi, Y. (996) Dimeioal properie of graph ad digial pace. Joural of Mahemaical Imagig ad Viio, 6, [3] Lu, W. T. ad Wu, F. Y. (200) Iig model o oorieable urface: Exac oluio for he Moebiu rip ad he Klei bole. Phy. Rev., E 63, [4] Pokoryi, Y. ad Borovkikh, A. (2004) Differeial equaio o ework (Geomeric graph). Joural of Mahemaical Sciece, 9 (6), [5] Smih, G. D. (985) Numerical oluio of parial differeial equaio: fiie differece mehod (3rd ed.). Oxford Uiveriy Pre. [6] Vol per, A. (972) Differeial equaio o graph. Ma. Sb. (N. S.), 88 (30), [7] Xu Ge, Qi. ad Maoraki, N. (200) Differeial equaio o meric graph. WSEAS Pre.