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 International Journal o Discrete Mathematics 2017; 2(3): doi: /jdmath olution o the Hyperbolic Partial Dierential Equation on Graphs and Digital paces: A Klein ottle a Projective Plane and a 4D phere Alexander V Evako Dianet, Laboratory o Digital Technologies, Moscow, Russia address: evakoa@mailru To cite this article: Alexander V Evako olution o the Hyperbolic Partial Dierential Equation on Graphs and Digital paces: A Klein ottle a Projective Plane and a 4D phere International Journal o Discrete Mathematics Vol 2, No 3, 2017, pp doi: /jdmath Received: February 1, 2017; Accepted: February 28, 2017; Published: March 29, 2017 Abstract: This paper studies the structure o the hyperbolic partial dierential equation on graphs and digital n-dimensional maniolds, which are digital models o continuous n-maniolds Conditions or the existence o solutions are determined and investigated Numerical solutions o the equation on graphs and digital n-maniolds are presented Keywords: Hyperbolic PDE, Graph, olution, Initial Value Problem, Digital pace, Digital Topology 1 Introduction Dierential equations play an important role in various ields o science and technology However in many cases, analytic solutions o PDE (partial dierential equation) may not be possible For practical problems, it is more reasonable to carry out computational or numerical solutions It can be done by implementing as domains graphs and by transerring PDE rom a continuous area into discrete one A review o works devoted to partial dierential equations on graphs can be ound in [2], [14], and [17] Figure 1 (a) 2D grid or two independent variable x and y (b) The ball U(v 1) o point v 1 (black points) in graph G As a rule, traditional numerical methods provide good approximations to the exact solution o PDE However, the standard grid in the inite dierence approximation is not a correct model o the continuous domain in terms o digital topology (see [4]) In order to avoid serious problems in computational solutions it is necessary to use topologically correct digital domains In physics, numerical methods are used in the study o lattice models on non-orientable suraces such as a Moebius strip and a Klein bottle (see [13]) In such cases, non-orientable suraces must be replaced by topologically correct digital grids Mathematically correct grids can be obtained in the ramework o digital topology Digital topology methods are crucial in analyzing n- dimensional digitized images arising in many areas o science including neuroscience, medical imaging, computer graphics, geoscience and luid dynamics Traditionally, digital objects are represented by graphs whose edges deine nearness and connectivity (see, eg, [3], [8], [11]) The important eature o an n-surace is a similarity o its properties with properties o its continuous counterpart in terms o algebraic topology For example, the Euler characteristics and the homology groups o digital n-spheres, a Moebius strip and a Klein bottle are the same as ones o their continuous counterparts ([11] and [12]) In recent years, there has been a considerable amount o works devoted to building two, three and n-dimensional discretization schemes and digital images In papers [6] and [10], discretization schemes are deined and studied that allow to build digital models o 2-dimensional continuous objects with the same topological properties as their continuous counterparts and with any required accuracy In this paper, we studies the structure o the hyperbolic

2 International Journal o Discrete Mathematics 2017; 2(3): partial dierential equation on graphs and digital spaces modeling continuous objects ection 2 analyzes the structure o a hyperbolic dierential equation on a graph ection 3 contains a short description o digital spaces and digital n-suraces studied in [8]-[9] such as digital n- dimensional spheres, a digital torus, a digital Klein bottle, etc ection 4 presents a numerical solution o a hyperbolic equation on a digital string, a digital Klein bottle, a digital projective plane and a digital 4D sphere 2 Hyperbolic PDE on a Graph We need to emphasize that the deinitions, the theorem statements and proos ound in this section can be understood only ater numerical experiments Digital spaces are graphs with speciic topological structure olutions o PDE on several types o digital spaces were studied in paper [7] Finite dierence approximations o the PDE are based upon replacing partial dierential equations by inite dierence equations using Taylor approximations [15] As an example, consider a hyperbolic PDE with two spatial independent variables =a +b +g (1) where =x,y,t,a=ax,y,t,b=bx,y,t,g=gx,y,t A two-dimensional spatial orthogonal grid G with points v =pδx,sδy is shown in igure 1 Using the orward dierence ormula or the derivative with respect to t, and the central dierence ormula or the second derivatives with respect to x and y, we obtain the ollowing equivalent inite deerence equation,, 2 ", +, Δt # =a ",, 2, +, Δx # ++b," 2, +,, Δy # +g, where x=iδx,i=1,2,,y=jδy,j=1,2,,t=nδt,n= 1,2, This equation can be transormed to the orm / = e + + +Δt # g,,, 0/" -," (2) where e, +e ", +e, +e," +e, =0 Notice that grid G in igure 1(a) is a graph Point 2,3 is adjacent to points 2,3±1 and i±1,j The ball o point 2,3 is U2,3 containing also points 2,3±1 and i± 1,j The ball o point 2,3 is U2,3 containing also points 2,3±1 and i±1,j Using these notations, equation (2) can be written as an equation on graph G = v k U6v p 7e " +g (3) 8 e + v k U6v p 7 =0 The summation is produced over all points belonging to ball U2,3 Here + is the value o the unction v +,n at point v + o G at the moment n, coeicients e + are unctions on the pairs o point v,v + and t=n (with domain V V t) I points v and v + are not adjacent, then e + =0 Notice that in general, condition v k U6v p 7 e + =0, is not necessary or the PDE For example, it does not hold or the hyperbolic PDE on a directed network The equation (3) is called homogeneous i all g =0 Later on in this paper, all g =0 Thus, i G(V, W) is a graph with the set o points V=(v 1, v 2,v s ), the set o edges W = ((v р v q ),), and U6v p 7 is the ball o point v, p=1,s then a hyperbolic PDE on G is the set o equations The unction at t=n+1 on a given point v depends on the values o the unction at t=n, n-1 on v and the points adjacent to v The stencil or equation (4) is illustrated in igure 1(b) The ball U(v 1 ) consists o black points ince =0 i points v and v + are non-adjacent, then set (4) has e + the orm = e " +,p=1, s (5) To make this equation look more convenient or analysis we can write it in the orm used in [4] It was shown that the parabolic PDE on a graph is =8e This equation can be written as =8c where 8e + =0, where,8c + =1, p=1, s Here c + =e + i p k,c =e +1 Thereore, equation (5) can be transormed into the ollowing orm Deinition 21 uppose that G (V, W) is a graph with the set o points V=(v 1, v 2,v s ), the set o edges W = ((v р v q ),), and U6v p 7 is the ball o point v, p=1,s A hyperbolic PDE on G is the set o equations =8c " +,where,8c + =1, + p=1, s = v k U6v p 7e ",t=n,p=1, s (4) = c " +,where, + c + =1, p=1, s (6)

3 90 Alexander V Evako: olution o the Hyperbolic Partial Dierential Equation on Graphs and Digital paces: A Klein ottle a Projective Plane and a 4D phere Notice that equation (6) is more obvious and convenient or studying because it contains the parabolic part ( + c + + and the hyperbolic part " In other words, this equation can be rewritten as 6A 0 7 CD0EFGHI/J =6A 0 7 0KFKGHI/J +A 0 A 0 " Equations (6) do not depend explicitly on the topology o graph G, and can be applied to a graph o any dimension or to a network All topological eatures are contained in the local and global structure o G et (6) is similar to the set o dierential equations on a graph investigated by A I Volpert in [16] Equation (6) can be presented in the matrix orm c c # =L N,Ct=Pc # Q, c =Ct + " Consider now the initial conditions or solving equation (6) Initial conditions are deined in a regular way by the set o equations at t=0 and t=1 =6v,07, =6v,17,p=1,2, s Deinition 22 et o equations (6) along with initial conditions is called the initial value problem or the hyperbolic PDE on a graph Gv,v #, v = c " +, U =6v,07, =6v,17,p=1,2, s oundary conditions can also be set the usual way oundary conditions are aected by what happens at the subgraph H o G Let H be a subgraph o G Let the values o the unction v +,t at points v + ϵh at the moments t=n be given by the set v +,n= + =s +,v + ϵh,t=n The boundary-value problem on G can be ormulated as ollows Deinition 23 Let Gv,v #, v be a graph and the set (6) be a dierential hyperbolic equation on G Let H be a subgraph o G, and the values o the unction v +,t at points v + ϵh at the moments t be deined by boundary conditions Equation (6) along with boundary conditions is called the boundary value problem or the hyperbolic PDE on a graph G = c U " +, v +,n= + =s +,v + ϵh,t=n N Consider a particular case that in equation (6), c + = 1,c + 0,p,k=1, s For the parabolic equation, this means the heat or diusion equation on a graph as shown in [4] We will show that i initial conditions are such that (7) (8) = =A then =A or every t=n Theorem 21 Let = c " + be a hyperbolic equation and assume that c + =1,k=1, s, and the initial conditions =6v,07, =6v,17,p=1,2, s, satisy the equation = =A Then =A or every n=t Proo =8 =8 8 c " = c + + +A A= 8 8 c + + =8A - =] + + ^ - This completes the proo We ocus now on the deinition and structure o wave equation which is a special case o a hyperbolic PDE We are equipped to introduce the general wave equation on G Deinition 24 A hyperbolic equation is called the wave equation i the ollowing condition holds 8c + =1, + (9) = c " _ 0-0,t=n,p=1, s The solution o this equation should be analogous the solution o the wave equation in the continuous case It must be associated with oscillations o the unction at points o G and the wave propagation First, assume that graph G on which we desire to solve equation (9) consists o two adjacent points v 1 and v 2 Theorem 22 Let in the wave equation (9) on graph Gv,v # consisting o two adjacent points, the initial conditions = 6v,07, =6v,17,p=1,2, satisy the equation = =A, and coeicients _ 0- =_ 0- do not depend on t=n Then the solutions =v,n and # = v #,n,`=a b, are periodic sequences with period c>2 Proo Consider graph Gv,v # consisting o two adjacent points in igure 2(a) According to (9), e A =_ A +_ # A # +A " A A # =_ # A +_ ## A # +A " # A # y theorem 21, A +A # =] Thereore, A # =] A Then A =_ A +_ # ] A +A A " =_ # ]+1+ _ _ # A " A (10) In paper [5], it was shown that the generalized Fibonacci-

4 International Journal o Discrete Mathematics 2017; 2(3): like sequences g =]+hg g ",`Z1,g i,g jk is periodic with period c #l m d2, nio pqr # i h t2 In sequence (10), h1 # ince _ 0- Z0, # 1, _ # _ ## 1, then h t2 Thereore, (10) is a periodic sequence with period c #l d2, n m io pq J uu v "J v u Acting in the same way, it is easy to prove # that A # is periodic with the same period T This completes the proo The theorem above will be used to prove a general theorem about periodicity o the solution o the wave equation The ollowing theorem asserts that the solution o (9) is periodic on a graph G((v 1,v 2,v s ), s>2 Notice that G((v 1,v 2,v s ) is the union o point v 1 and subgraph H=G-v 1, G((v 1,v 2,v s )=v 1 wx In the proo o the next theorem, graph H is replaced with point v H (see igure 2(b), (c)) Theorem 23 Let in the wave equation (9) on graph Gv, v, the initial conditions 6v,07, 6v,17,p1,2, s, satisy the equation A, and coeicients _ 0- do not depend on t=n Then the solutions 6v,n7,p1,2, s,a`xb, are periodic sequences with period cd2 Proo Graph G can be presented as the union o v 1 and subgraph Hv #, v, yz wxv #, v (see igure 2(b)) Let A ^ { /# A / be the sun o A / over points o H et (9) consists o s equations Figure 2 (a) Graph G(v 1,v 2) consists o two adjacent points v 1 and v 2 (b) G=(v 1,H)= =v 1wx ubgraph H=G-v 1 (c) Graph G(v 1,v H) consists o two adjacent points v 1 and v H A _ A _ # A # _ A^A " ^ A A^ _^ A _^# A # _^^ A^A^ A^" umming up last (s-1) equations, we obtain where _ {{ e A _ A _ { A " { A A A { _ { A _ {{ A " { A { A (11) { _ { A { _ - _ { ^ -# A -, ^ /# A, A _ / A 1 _ A ^ ^ { ^ /# -#_ /Ḇ A - A { _ - -# A - A straightorward check shows that { 1 and _ {{ _ { 1 Thereore, set (11) is the wave equation on the graph G(v 1,v H ) containing just two points v 1 and v H depicted in igure 2(c) Thereore, according to theorem 22, v,n,`axb, is a periodic sequence with period c #l m d2, nio pqj uu v v "J u} # For the same reason, the solution 6v,tn7, is a periodic sequence at every point z 0,p2, s This completes the proo The oscillations o the unction at every point are completely determined i we know the initial values at moments t=0 and t=1 3 Digital n-maniolds, Digital n-pheres, a Digital Torus, a Digital Projective Plane There is considerable amount o literature devoted to the study o dierent approaches to digital lines, suraces and spaces used by researchers In order to make this paper selcontained, it is reasonable to include some results related to digital spaces Traditionally, a digital image has a graph structure (see [1, 5, 10]) A digital space G is a simple undirected graph G=(V,W) where V=(v 1,v 2,v n, ) is a inite or countable set o points, and W = ((v р v q ),) is a set o edges The induced subgraph O(v) G containing all points adjacent to v, excluding v itsel, is called the rim or the neighborhood o point v in G, the induced subgraph U(v)=v~O(v) is called the ball o v For two graphs G=(X, U) and H=(Y, W) with disjoint point sets X and Y, their join G H is the graph that contains G, H and edges joining every point in G with every point in H Contractible graphs were studied in [11, 12] Contractible graphs are deined recursively Deinition 31 A one-point graph is contractible A connected graph G with n points is contractible i it contains a point v so that the rim O(v) is contractible and G-v is contractible A point v o a graph G is said to be simple i its rim O(v) is a contractible graph Thus, a contractible graph can be converted to a point by sequential deleting simple points A digital n-maniold is a special case o a digital n-surace deined and investigated in [6] Deinition 32 A digital 0-dimensional sphere is a disconnected graph 0 (a,b) with just two points a and b A connected graph M is called a digital n-sphere, n>0, i or any point v M, the rim O(v) is an (n-1)-sphere and the space M-v is a contractible graph [8] Deinition 33 A connected graph M is called an n-dimensional maniold, n>1, i the rim O(v) o any point v is an (n-1)-dimensional sphere [8] The ollowing results were obtained in [6] and [8] Theorem 31 The join n min= n+1 o (n+1) copies o the zero-dimensional sphere 0 is a minimal n-sphere Let M and N be n and m-spheres Then M N is an

5 92 Alexander V Evako: olution o the Hyperbolic Partial Dierential Equation on Graphs and Digital paces: A Klein ottle a Projective Plane and a 4D phere (n+m+1)-sphere Any n-sphere M can be converted to the minimal n- sphere min by contractible transormations Consider connected graph G with two points depicted in igure 2 Deine coeicients c + in equation (12) in the ollowing way; c 08,c # 02,c ## 07,c # 03 Initial values are given as 1 =2, 2 =2 The results o the solution o the initial value problem (12) at points 1, 2 6 or t=1, 50 are displayed in igure 7, which illustrates the time behavior o the values o the unction Figure 7 present oscillations at points 1 and 2 Figure 3 0 is a digital zero-dimensional sphere 1 min, 1 1, 1 2, 1 3 are digital one-dimensional spheres D 1 is a digital one-dimensional disk containing ten points A digital 0-dimensional surace is a digital 0-dimensional sphere Figure 3 depicts digital zero and one-dimensional spheres Figure 4 shows digital 2-dimensional spheres All spheres are homeomorphic and can be converted into the minimal sphere 2 min by contractible transormations Digital torus T and a digital 2-dimensional Klein bottle K are shown in igure 5 Figure 6 depicts a digital projective plane P and digital three and our-dimensional spheres 3 and 4 respectively It was shown in [4] that two- dimensional spatial grids used i in inite-dierence schemes are not correct rom the viewpoint o digital topology These grids do not relect local topological eatures o continuous domains which are replaced by the grids In order to obtain the correct solutions o inite dierence equations on n-dimensional domains, the grids should have the structure o digital n-dimensional spaces 4 Numerical olutions o the Wave Equation on Graphs In this section we demonstrate the numerical solutions o the initial and boundary value problems or the hyperbolic PDE on a graph Gv,v #, v which is a digital n- dimensional maniold 8_ 0-0 = c + " + + (12) 1,_ 0- Z0,,p1,2, s 6v,07, 6v,17 v +,n + s +,v + VH, txn (13) 41 A Connected Graph G with Two Points Numerical olution o the Initial Value Problem Figure 4 Two-dimensional spheres with a dierent number o points Any o spheres can be converted into the minimal sphere 2 min by contractible transormations Figure 5 Digital 2-dimensional torus T and Klein bottle K with sixteen points 42 Digital One-Dimensional Disk D 1 Numerical olution o the oundary Value Problem Numerical solution o the boundary value problem (13) on the digital 1-D disk D 1 depicted in igure 3 and modeling a string is shown in igure 8 D 1 contains ten points Assume that coeicients c ik in (12) do not depend on t, c i,i+1 =03, c i,i =04 End points o the digital string are ixed, ie, boundary values are 1 t = 10 t =0 Initial values are 5 0 =10, 5 1 =10 Two graphs in igure 8 show the proile o the string at moments t=45 and t=50 This solution is similar to the solution in the continuous string Figure 6 P, 3 and 4 are digital two-dimensional projective plane, threeand our-dimensional spheres respectively 43 Digital 2-Dimensional Klein ottle K Numerical olution o the Initial Value Problem A Klein bottle is an object o investigation in many ields In physics [13], a Klein bottle attracted attention in studying lattice models on non-orientable suraces as a realization and testing o predictions o the conormal ield theory and as new challenging unsolved lattice-statistical problems A digital 2D Klein bottle K depicted in igure 5 consists o sixteen points The rim O(v k ) o every point v k is a digital 1-sphere containing six points ie, K is a homogeneous digital space Topological properties o K are similar to topological properties o its continuous counterpart For example, the Euler characteristic and the homology groups o a continuous and a digital Klein bottle are the same ([11] and [12]) In (12), deine coeicients c + in the ollowing way I points v p and v k are adjacent then c + =01; c kk =04, k=1, 16 Initial values are given as 8 =16, 10 =16 In igure 9, numerical solutions at points 1 and 3 o the o the

6 International Journal o Discrete Mathematics 2017; 2(3): initial value problem (12) are plotted at time t=1, 100 Two lines show oscillations with period T>2 44 Digital 2D Projective Plane P Numerical olution o the Initial Value Problem Figure 6 shows a digital 2-dimensional projective plane P which is a digital counterpart o a continuous projective plane P is a non-homogeneous digital space containing eleven points The rim O(vk) o every point vk is a digital 1sphere Topological properties o a digital and a continuous 2D projective plane are similar It was shown in [11] and [12] that the Euler characteristic and the homology groups o a continuous and a digital projective plane are the same It is easy to check directly that a digital 2D projective plane without a point is homotopy equivalent to a digital onedimensional sphere as it is or a continuous projective plane Deine coeicients c + in (12) in the ollowing way; i vk and vp are adjacent then c+ c + 01, c 1 +,+ƒ c+ Consider the numerical solutions o the initial value problem or (12) Initial values are given as 10 =11, 11 =11 The proiles o the solution at points 1 and 2 are plotted in igure 10, where t=1, 100 The plots demonstrate oscillations with period T>2 45 Digital 4-Dimensional phere 4 Numerical olution o the Initial Value Problem Figure 7 Numerical solution o the initial value problem on graph G (igure 2) with two points v1 and v2 Initial values are 10=2, 11=2 The solutions on G are shown at points 1 and 2, t=0, 1, 50, The solution proiles are oscillations with period T>2 Figure 8 Numerical solution o the boundary value problem on the 1-D disk D1 shown in igure 3 and modeling a string D1 contains ten points End points o the digital string are ixed, ie, boundary values are 1t= 10t=0 Initial values are 50=10, 51=10 Two graphs show oscillations o the string at moments t=45 and t=50 Figure 9 The solution proiles o the initial value problem on the Klein bottle K (igure 5) at point 1 and 3, t=0, 1, 100, 80=10, 61=10 Two lines show oscillations with period T>2 Figure 10 Numerical solution o the initial value problem on the projective plane P shown in igure 6 The solution proiles on P at points 1 and 2, t=0, 1, 100, 100=11, 111=11 The solution proiles are oscillations with period T>2 Consider a digital 4-dimensional -sphere 4 with ten points depicted in igure 6 The rim O(vk) o every point vk is a digital 3-sphere containing eight points and depicted in igure 6 4 is a homogeneous digital space containing the minimal number o points The number o points can be increased by using contractible transormation Topological properties o a digital 4D sphere are similar to topological properties o a continuous 4D sphere Deine the numerical solutions o the initial value problem or (12) Let cik=cki=001 i points vi and vk are adjacent, and cpp=092 or p=1, 10 Initial values are given as 06=10, 07=10 The results o the solution o the initial value problem (12) at points 1, 2 6 or t=0, 80 are displayed in igure 11, which illustrates the time behavior o the values o the unction Figure 11 present beating oscillations at points 1 and 2 Obviously, the proiles o solution are beating oscillations at every point o this sphere Figure 11 Numerical solution o the initial value problem on the 4D sphere 4 depicted in ig 6 The graphs are shown at points 1 and 2, t=0, 1, 400, Initial values are 50=10, 61=10 The solution proiles are oscillations with period T>2

7 94 Alexander V Evako: olution o the Hyperbolic Partial Dierential Equation on Graphs and Digital paces: A Klein ottle a Projective Plane and a 4D phere 5 Conclusion This paper investigates the structure o dierential hyperbolic equations on graphs, digital spaces, digital n- dimensional maniolds, n>0, and networks and studies their properties The approach used in the paper is mathematically correct in terms o digital topology As examples, computational solutions o the hyperbolic PDE on n- dimensional digital maniolds such as a Klein bottle and a 4- dinensional sphere are presented Reerences [1] ai, Y, Han, X, Prince, J(2009) Digital Topology on Adaptive Octree Grids Journal o Mathematical Imaging and Vision 34 (2), [2] orovskikh, A and Lazarev, K (2004) Fourth-order dierential equations on geometric graphs Journal o Mathematical cience, 119 (6), [3] Eckhardt, U and Latecki, L (2003) Topologies or the digital spaces Z2 and Z3 Computer Vision and Image Understanding, 90, [4] Evako, A (2016) olution o a Parabolic Partial Dierential Equation on Digital paces: A Klein ottle, a Projective Plane, a 4D phere and a Moebius and, International Journal o Discrete Mathematics Vol 1, No 1, pp 5-14 doi: /jdmath [5] Evako, A (01/2017) Properties o Periodic Fibonacci-like equences arxiv: , 2017arXiv E [6] Evako, A (2014) Topology preserving discretization schemes or digital image segmentation and digital models o the plane Open Access Library Journal, 1, e566, [7] Evako, A (1999) Introduction to the theory o molecular spaces (in Russian language) Publishing House Paims, Moscow [8] Evako, A, Kopperman, R and Mukhin, Y (1996) Dimensional properties o graphs and digital spaces Journal o Mathematical Imaging and Vision, 6, [9] Evako, A (2015) Classiication o digital n-maniolds Discrete Applied Mathematics 181, [10] Evako, A (1995) Topological properties o the intersection graph o covers o n-dimensional suraces Discrete Mathematics, 147, [11] Ivashchenko, A (1993) Representation o smooth suraces by graphs Transormations o graphs which do not change the Euler characteristic o graphs Discrete Mathematics, 122, [12] Ivashchenko, A (1994) Contractible transormations do not change the homology groups o graphs Discrete Mathematics, 126, [13] Lu, W T and Wu, F Y (2001) Ising model on nonorientable suraces: Exact solution or the Moebius strip and the Klein bottle Phys Rev, E 63, [14] Pokornyi, Y and orovskikh, A (2004) Dierential equations on networks (Geometric graphs) Journal o Mathematical cience, 119 (6), [15] mith, G D (1985) Numerical solution o partial dierential equations: inite dierence methods (3rd ed) Oxord University Press [16] Vol pert, A (1972) Dierential equations on graphs Mat b (N ), 88 (130), [17] Xu Gen, Qi and Mastorakis, N (2010) Dierential equations on metric graph WEA Press